# Simplicial intersection homology revisited.

David Chataur  Lamfa
Université de Picardie Jules Verne
33, rue Saint-Leu
80039 Amiens Cedex 1
France<EMAIL_ADDRESS>,  Martintxo Saralegi-Aranguren
Laboratoire de Mathématiques de Lens
EA 2462
Université d’Artois
62300 Lens Cedex
France<EMAIL_ADDRESS>and  Daniel Tanré
Département de Mathématiques
UMR 8524
Université de Lille
59655 Villeneuve d’Ascq Cedex
France<EMAIL_ADDRESS>
###### Abstract.

Intersection homology is defined for simplicial, singular and PL chains. In
the case of a filtered simplicial complex, it is well known that the three
versions are isomorphic. This isomorphism is established by using the PL case
as an intermediate between the singular and the simplicial situations. Here,
we give a proof similar to the classical proof for ordinary simplicial
complexes.

We also study the intersection blown-up cohomology that we have previously
introduced. In the case of a pseudomanifold, this cohomology owns a Poincaré
isomorphism with the intersection homology, for any coefficient ring, thanks
to a cap product with a fundamental class. We prove that the simplicial and
the singular blown-up cohomologies of a filtered simplicial complex are
isomorphic. From this result, we can now compute the blown-up intersection
cohomology of a pseudomanifold from a triangulation.

Finally, we introduce a blown-up intersection cohomology for PL-spaces and
prove that it is isomorphic to the singular ones. We also show that the cup
product in perversity $\overline{0}$ of a CS-set coincides with the cup
product of the singular cohomology of the underlying topological space.

###### Key words and phrases:

Intersection (co)homology, Simplicial, Singular

###### 2010 Mathematics Subject Classification:

55N33, 55N35

The cohomology of a triangulated manifold can be computed in many different
ways :

  * -

using the simplices of the triangulation,

  * -

using the simplicial set of singular simplices,

  * -

using sheaf theory,

  * -

using homotopy classes into Eilenberg-MacLane spaces.

This paper and our previous works on blown-up cochains [4, 7, 8] show that we
have the same picture for intersection cohomology. Furthermore, we show the
existence of an isomorphism between the singular and simplicial definitions of
intersection homology, without involving the PL structure and without
limitation to CS sets.

Let us recall some historical background of intersection homology as it was
introduced by Goresky and MacPherson. In their first paper on intersection
homology, they introduce it for a pseudomanifold $X$ together with a fixed PL-
structure and a parameter $\overline{p}$, now called Goresky-MacPherson
perversity. They define the complex of $\overline{p}$-intersection,
${I{\mathfrak{C}}}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}$, as a
subcomplex of the complex of PL-chains and the $\overline{p}$-intersection
homology as the homology of
${I{\mathfrak{C}}}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}$.

Let $\sf R$ be a commutative ring of coefficients. We could also start with a
fixed triangulation on a stratified Euclidean simplicial complex, $K$, and
define a complex of $\overline{p}$-intersection,
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$, as a subcomplex of the
simplicial chains. This is, for instance, the concrete case in [16], where
MacPherson and Vilonen give a construction of the category of perverse
constructible sheaves on a Thom-Mather stratified space with a fixed
triangulation. In an appendix to this paper, Goresky and MacPherson prove that
the two intersection homologies are isomorphic; i.e., the simplicial one on
$K$ and the previous PL one on the PL space $X$ associated to $K$. They do
that with a nice construction of a left inverse to the inclusion
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\to{I{\mathfrak{C}}}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}$,
see the book of Friedman ([9, Section 3.3]) for a detailed version of this
argumentation.

A third step is to consider the topological realization, $|K|$, of $K$ and
define a complex of $\overline{p}$-intersection,
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$, as a subcomplex of the
singular chains on $|K|$. This situation has been studied by King [15] who
proves the existence of an isomorphism between the two homologies, in the case
of CS sets. This situation is developed in a thorough way in [9, Section 5.4].
Let us emphasize that the proof is not a direct comparison between the
simplicial and the singular points of view: the PL case is using as
intermediate between them and a structure of CS set is required.

Let’s go back to the classical situation with a Euclidean simplicial complex,
$K$, of topological realization $|K|$, ${C}_{{}_{*}}{(K)}$ the complex of
simplicial chains on $K$ and ${C}_{{}_{*}}{(|K|)}$ the complex of singular
chains on $|K|$. The proof (see [14]) that the inclusion
${C}_{{}_{*}}{(K)}\to{C}_{{}_{*}}{(|K|)}$ is a quasi-isomorphism goes as
follows.

  1. (a)

If $K^{(\ell)}$ is the $\ell$-skeleton on $K$, we built a morphism of short
exact sequences,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}_{{}_{*}}{(K^{(\ell-1)})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}_{{}_{*}}{(K^{(\ell)})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}_{{}_{*}}{(K^{(\ell)},K^{(\ell-1)})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}_{{}_{*}}{(|K^{(\ell-1)}|)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}_{{}_{*}}{(|K^{(\ell)}|)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}_{{}_{*}}{(|K^{(\ell)}|,|K^{(\ell-1)}|)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

  2. (b)

Next, we prove the existence of isomorphim,

$\displaystyle{C}_{{}_{*}}{(K^{(\ell)},K^{(\ell-1)})}\cong\bigoplus_{\beta\in
K,\dim\beta=\ell}{C}_{{}_{*}}{(\beta,\partial\beta)}$

and similarly for the realization $|-|$.

  3. (c)

Finally, the result comes from an induction (for $K^{(\ell-1)}$) on the
dimension and the verification of
${H}_{{}_{*}}{(\beta)}\cong{H}_{{}_{*}}{(|\beta|)}$.

We return to the case of a filtered complex $K$, that is, a simplicial complex
endowed with a filtration

$K=K_{n}\supset K_{n-1}\supset K_{n-2}\supset\ldots\supset K_{0}\supset
K_{-1}=\emptyset,$

where each $K_{i}$ is a sub-complex. We are looking for a proof along the
previous lines which gives a _direct_ argument between the simplicial and the
singular situations, without any restriction to CS sets.

Unfortunately we do not have the expected formula

${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,K^{(\ell-1)}\right)}\cong\bigoplus_{\beta\in
K,\
\dim\beta=\ell}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta,\partial\beta\right)}$
(1)

because of the existence of allowable/non intersection simplices (see Example
1.7) . Therein lies the originality of intersection homology. A such simplex
cannot appear alone in an intersection chain since its boundary is not an
allowable chain.

Inspirited by [1] and the notion of complexity we introduce the _residual
complex_ ${\mathcal{L}}(K)$ which allow us to adapt the previous strategy:

  1. (1)

the construction of a morphism of short exact sequences,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,{\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

  2. (2)

the description of the relative complex $(K,{\mathcal{L}}(K))$ as a disjoint
union of $(\beta*L_{\beta},\partial\beta*L_{\beta})$, where each link
$L_{\beta}$ is a filtered complex of smaller complexity,

  3. (3)

and the verification of the existence of isomorphisms between the singular and
the simplicial intersection homologies of $\beta*L_{\beta}$, by computation
and induction, and of $\partial\beta*L_{\beta}={\mathcal{L}}(\beta*L_{\beta})$
by induction.

In fact, we apply this program not only to $\overline{p}$-intersection
homology but also to a _$\overline{p}$ -intersection cohomology_ obtained from
simplicial blow up, denoted ${{\mathscr{H}}}^{{}^{*}}_{{}_{\overline{p}}}$ and
called _blown-up intersection cohomology._ Since the work [12] of Goresky and
Siegel, it is known that there is no Poincaré duality on intersection homology
with coefficients in any commutative ring. However, in [6], we prove that the
cap product with the fundamental class of an oriented compact, $n$-dimensional
pseudomanifold induces an isomorphism between
${{\mathscr{H}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(X\right)}$ and the
intersection homology ${H}^{{}^{\overline{p}}}_{{}_{n-*}}{\left(X\right)}$.
(Similar versions exist for a paracompact not necessarily compact
pseudomanifold, with compact supports in cohomology or Borel-Moore homology,
[7] and [21]). This blown-up cohomology coincides with the cohomology obtained
from the linear dual of the chain complexes, when $\sf R$ is a field, or more
generally with an hypothesis of $\sf R$-local torsion free, already present in
[12]. Thus this recovers the Poincaré Duality theorem of Friedman and McClure
in [10]. Let us also mention that this blown-up cohomology owns cup and cap
products for any ring of coefficients and cohomology operations [2, 8]; their
sheafification is the Deligne sheaf [7]. The existence of the isomorphism
between simplicial and singular blown-up intersection cohomology is a new
result. In short, we prove in Theorems A and B:

Theorem. _Let $K$ be a full filtered complex $K$ and $\overline{p}$ be any
perversity, then we have isomorphisms:_

$\textstyle{{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}}$$\textstyle{\cong}$$\textstyle{{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}}$__
and
$\textstyle{{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}}$$\textstyle{\cong}$$\textstyle{{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}.}$simplicialsingularsimplicialsingular

These isomorphisms are given by direct morphisms at the chain level: inclusion
of simplicial chains into singular ones for homology and restriction of
cochains in the case of blown-up cohomology. They are compatible with cap and
cup products and higher operations defined at the chain/cochain level in our
previous works.

Our program is carried out in Sections 1-6 below whose headings are self-
explanatory. In the sequel, all the complexes are considered on a commutative
ring with unity, denoted by $\sf R$, which is not explicitly mentioned. If $X$
is a topological space, we denote by
${\mathtt{c}}X=X\times[0,1][/X\times\\{0\\}$ the cone on $X$ and by
${\mathring{{\mathtt{c}}}}X=X\times[0,1[/X\times\\{0\\}$ the open cone on $X$.
The apex of a cone is ${\mathtt{v}}$.

###### Contents

  1. 1 Simplicial setting
  2. 2 Singular setting
  3. 3 Simplicial versus Singular
  4. 4 Blown-up intersection cohomologies
  5. 5 Definitions of filtered PL spaces
  6. 6 Cup product

##  1\. Simplicial setting

> Intersection homology was introduced by Goresky-MacPherson in the PL-
> category [11]. Its simplicial version is a direct adaptation of this point
> of view (see [9, Section 3.2]).
>
> We present the simplicial version of the intersection homology associated to
> a filtered complex $K$. We study some calculations involving this homology
> used in this work. Namely
>
>   * •
>
> decomposition of the relative intersection homology of the pair
> $(K,{\mathcal{L}}(K))$ (cf. Proposition 1.3 ), and
>
>   * •
>
> computation of the intersection homology of a join (cf. Proposition 1.5).
>
>

Filtered complexes

We introduce the minimal setting for defining simplicial intersection homology
(see [17, 19, 9]).

A _simplicial complex_ $K$ is a set of simplices in $\mathbb{R}^{p}$,
$p\leq\infty$, such that

  1. 1.

if $\sigma,\tau\in K$ and $\sigma\cap\tau\neq\emptyset$ then $\sigma\cap\tau$
is a face of both $\sigma$ and $\tau$,

  2. 2.

if $\sigma\in K$ and $\tau$ is a face of $\sigma$, written
$\sigma\triangleleft\tau$, then $\tau\in K$,

  3. 3.

(local finiteness) every vertex of a simplex of $K$ belongs to a finite number
of simplices of $K$.

We say that $d\in\mathbb{N}$ is the dimension of $K$, $\dim(K)=d$, if every
simplex of $K$ has dimension lower than or equal to $d$ and $K$ has at least
one simplex of dimension $d$. If this number does not exist we say that
$\dim(K)=\infty$.

We let $|K|$ denote the topological subspace of $\mathbb{R}^{p}$ consisting of
the union of the simplices of $K$. If $\Delta$ is the standard simplex then we
identify $\Delta$ with $|\Delta|$.

A sub-collection $L\subset K$ is a _simplicial subcomplex_ of $K$ if it
verifies properties 1 and 2.

The union of the simplices of $K$ whose geometrical dimension is smaller than
or equal to a given $\ell\in\mathbb{N}$ is called the _$\ell$ -skeleton_ of
$K$ and denoted $K^{(\ell)}$.

A _filtered complex_ is a simplicial complex $K$ endowed with a filtration
made up of sub-complexes.

$K=K_{n}\supset K_{n-1}\supset K_{n-2}\supset\ldots\supset K_{0}\supset
K_{-1}=\emptyset.$

We shall say that $n$ is the _(virtual) dimension_ of the filtered complex:
$\dim_{v}K_{k}=k$.

A _stratum_ is a non-empty connected component of a
$|K_{i}|\backslash|K_{i-1}|$. Its dimension is $\dim_{v}S=i$ and its
codimension is ${\rm codim\,}_{v}S=n-i$. The family of strata of $K$ is
denoted by ${\mathcal{S}}_{K}$ or ${\mathcal{S}}_{|K|}$. The strata included
in $|K|\backslash|K_{n-1}|$ are _regular strata_ , the other strata are
_singular strata_.

The _complexity_ of a filtered complex $K\neq\emptyset$ is the pair
${\mathrm{comp}\,}(K)=(a,b)$ where:

  * -

$a=\max\\{k~\in\\{0,\ldots,n\\}\mid K_{n-k}\neq\emptyset\\}$, and

  * -

$b\in\overline{\mathbb{N}}=\mathbb{N}\cup\\{\infty\\}$ is the geometrical
dimension of the simplicial complex $K_{n-a}$.

We put ${\mathrm{comp}\,}\emptyset=(-\infty,-\infty)$. We use the associated
lexicographic order to compare the complexity of filtered complexes.

Simplicial and singular intersection homologies are not isomorphic in a
general context (see [9, Examples 3.2.6 and 3.2.10] or [16, Appendix] in the
case of PL-intersection homology). For this reason we introduce a stronger
version of filtered space.

A filtered complex $K$ is a _full filtered complex_ if for each simplex
$\sigma\in K$ and each $\ell\in\\{0,\ldots,n\\}$ we have
$\sigma\cap|K_{\ell}|=\emptyset$ or $\sigma\cap|K_{\ell}|\triangleleft\sigma$.

The standard simplex $K=\Delta$ can be endowed with the following two
different structures of filtered complex:

$\left\\{\begin{array}[]{l}\hbox{$K_{\ell}$ is an $\ell$-dimensional face of
$\Delta$ containing the face $K_{\ell-1}$}\\\
K_{\ell}=\Delta^{(\ell)}.\end{array}\right.$

While the first one is full, the second is not. We can recover the fullness
after a barycentric subdivision. But this is a general fact as the following
Lemma shows (see [16, Remark 2]).

######  Lemma 1.1.

The barycentric subdivision of a filtered complex is full.

###### Proof.

Consider a filtered complex

$K=K_{n}\supset K_{n-1}\supset K_{n-2}\supset\cdots\supset
K_{0}\supset\emptyset,$

inducing the filtered complex $K^{\prime}=K^{\prime}_{n}\supset
K^{\prime}_{n-1}\supset K^{\prime}_{n-2}\supset\cdots\supset
K^{\prime}_{0}\supset\emptyset,$ where ′ indicates barycentric subdivision.
Let $\sigma$ be a simplex of $K^{\prime}$ with
$\sigma\cap|K^{\prime}_{i}|\neq\emptyset$. We need to prove that
$\sigma\cap|K^{\prime}_{i}|$ is a face of $\sigma$. Notice that
$\sigma\cap|K^{\prime}_{i}|=\sigma\cap|K_{i}|$.

By definition of the barycentric subdivision there exists a family of
simplices of $K$, $\tau_{0}\triangleleft\cdots\triangleleft\tau_{m}$, with
$\sigma=[\widehat{\tau_{0}},\ldots,\widehat{\tau_{m}}]$. Here,
$\widehat{\tau_{i}}$ is the barycenter of $\tau_{i}$, $i\in\\{0,\ldots,m\\}$.
Let us suppose that

$\alpha\in K\Longrightarrow\sigma\cap\alpha=\emptyset\hbox{ or
}\sigma\cap\alpha=[\widehat{\tau_{0}},\ldots,\widehat{\tau_{\ell}}]\hbox{ for
some }\ell\in\\{0,\cdots,m\\}.$ (2)

On the other hand, since $K$ is a simplicial complex then
$\tau_{m}\cap|K_{i}|$ is a union of faces of $\tau_{m}$. We conclude that

$\sigma\cap|K_{i}|\stackrel{{\scriptstyle\sigma\subset\tau_{m}}}{{=}}\sigma\cap\tau_{m}\cap|K_{i}|\stackrel{{\scriptstyle\eqref{KK'}}}{{=}}[\widehat{\tau_{0}},\ldots,\widehat{\tau_{k}}]$

for some $k\in\\{0,\cdots,m\\}$. This gives
$\sigma\cap|K_{i}|\triangleleft\sigma$. It remains to prove (2).

Any point $P\in\sigma$ has the canonical decomposition $P=\sum_{j\in
J_{P}}t_{j}\widehat{\tau_{j}}$ where $\emptyset\neq
J_{P}\subset\\{0,\ldots,m\\}$ and $t_{j}\in]0,1]$ for each $j\in J_{P}$. We
define $\ell=\max\\{j\in J_{P}\ /\ P\in\sigma\cap\alpha\\}$, which belongs to
$\\{0,\ldots,m\\}$ since $\sigma\cap\alpha\neq\emptyset$. By construction we
have the inclusions
$\sigma\cap\alpha\subset[\widehat{\tau_{0}},\ldots,\widehat{\tau_{\ell}}]$ and
$[\widehat{\tau_{0}},\ldots,\widehat{\tau_{\ell}}]\subset\sigma$. By
definition there exists a vertex $Q$ of $\sigma\cap\alpha$ with $\ell\in
J_{Q}$. Condition $\sum_{j\in J_{Q}}t_{j}\widehat{\tau_{j}}\in\alpha$ implies
$\widehat{\tau_{\ell}}\in\alpha$, and then $\tau_{\ell}\subset\alpha$. This
gives the last inclusion
$[\widehat{\tau_{0}},\ldots,\widehat{\tau_{\ell}}]\subset\alpha$. ∎

###  1.1. Canonical decomposition of $\sigma\in K$

Let us suppose that the filtered complex $K$ is full. This implies that any
simplex $\sigma\in K$ has the _canonical decomposition_

$\sigma=\sigma_{0}*\sigma_{1}*\cdots*\sigma_{n},$

where $\sigma\cap|K_{\ell}|=\sigma_{0}*\cdots*\sigma_{\ell},$ for each
$\ell\in\\{0,\ldots,n\\}$. We use the convention $\emptyset*E=E$ for each
simplicial complex $E$. For each $\ell\in\\{0,\ldots,n\\}$ we have:

$\sigma_{\ell}\neq\emptyset\Longleftrightarrow\sigma\cap|K_{\ell}|\backslash|K_{\ell-1}|\neq\emptyset\Longleftrightarrow\exists
S\in{\mathcal{S}}_{K}\hbox{ with }\dim_{v}S=\ell\hbox{ and }\sigma\cap
S\neq\emptyset.$

By connectedness, the stratum $S$ is unique, denoted by $S=S_{\ell}$. We write

. $I_{\sigma}=\\{\ell\in\\{0,\ldots,n\\}\mid\sigma_{\ell}\neq\emptyset\\}$ and
${\mathcal{S}}^{\sigma}=\\{S\in{\mathcal{S}}_{K}\mid
S\cap\sigma\neq\emptyset\\}$. .

In fact, we have $I_{\sigma}=\\{\dim_{v}S\mid S\in{\mathcal{S}}^{\sigma}\\}$
and ${\mathcal{S}}^{\sigma}=\\{S_{\ell}\mid\ell\in I_{\sigma}\\}$.

We say that a simplex $\sigma\in K$ is a _clot_ if $\sigma=\sigma_{a}$ and
$\dim\sigma=b$, where ${\mathrm{comp}\,}(K)=(a,b)$. In other words, a clot is
a simplex $\sigma\in K$ living in $K_{n-a}$ with $\dim\sigma=\dim K_{n-a}$
(geometrical dimensions). The family of clots of $K$ will be denoted by
${\mathcal{B}}(K)$.

###  1.2. Induced filtered complexes

Let $K$ be a filtered complex. Given a simplicial sub-complex $L\subset K$ we
have the induced filtration

$L=L_{n}=L\cap K_{n}\supset L_{n-1}=L\cap K_{n-1}\supset\ldots\supset
L_{0}=L\cap K_{0}\supset L_{-1}=L\cap K_{-1}=\emptyset$

defining a filtered complex structure on $L$ (see [9]). Notice that
${\mathrm{comp}\,}(L)\leq{\mathrm{comp}\,}(K)$.

Since $L_{i}\subset K_{i}$ for any $i\in\\{1,\ldots,n\\}$ then for any stratum
$T\in{\mathcal{S}}_{L}$ there exists a unique stratum $S\in{\mathcal{S}}_{K}$
with $T\subset S$. We say that $S$ is the _source_ of $T$. Notice that
$\dim_{v}T=\dim_{v}S$.

Given $\sigma\in L$ we have a priori two canonical decompositions of $\sigma$:
as a simplex of $L$ and as a simplex of $K$. Since $\sigma\cap
L_{k}=\sigma\cap L\cap K_{k}=\sigma\cap K_{k}$, for each
$k\in\\{0,\ldots,n\\}$ then both decomposition are the same. In particular
$I^{K}_{\sigma}=I^{L}_{\sigma}$, written $I_{\sigma}$. Moreover, the family
${\mathcal{S}}_{K}^{\sigma}$ is the family made up of the sources of the
strata of ${\mathcal{S}}^{\sigma}_{L}$. We get that the source of $T_{\ell}$
is $S_{\ell}$ for each $\ell\in I_{\sigma}$.

Let us introduce the main geometrical constructions associated to a filtered
complex $K$.

###  1.3. Links and joins

The _link_ of a simplex $\beta$ of a filtered complex $K$ is the simplicial
complex

$L_{\beta}=\\{\sigma\in K/\beta*\sigma\in K\\}.$

Recall that $L_{\beta}$ inherits from $K$ a filtered complex structure.

If the simplex $\beta$ is a clot then $\left(L_{\beta}\right)_{n-k}=\emptyset$
for $k\in\\{0,\ldots,a\\}$ where ${\mathrm{comp}\,}(K)=(a,b)$. In particular,
${\mathrm{comp}\,}(L_{\beta})<{\mathrm{comp}\,}(K)$.

The _join_ $\beta*L_{\beta}$ inherits from $K$ the following structure of
filtered complex:

$(\beta*L_{\beta})_{i}=\left\\{\begin{array}[]{cl}\beta*(L_{\beta})_{i}&\hbox{if
}n-a<i\leq n\\\ \beta&\hbox{if }i=n-a\\\ \emptyset&\hbox{if }0\leq i<n-a.\\\
\end{array}\right.$ (3)

The family of strata of the join is
${\mathcal{S}}_{\beta*L_{\beta}}=\\{\beta,(\beta*S)\backslash\beta={\mathtt{c}}\beta\times
S\ \mid S\ \in{\mathcal{S}}_{L_{\beta}}\\}$.

###  1.4. Residual complex

The _residual complex_ ${\mathcal{L}}(K)$ of a filtered complex
$K\neq\emptyset$ is the simplicial sub-complex defined by

. ${\mathcal{L}}(K)=\\{\sigma\in K\mid\dim(\sigma\cap|K_{n-a}|)<b\\},$ .

where ${\mathrm{comp}\,}K=(a,b)$.

The equality ${\mathcal{L}}(K)=K$ arrives if $b=\infty$. When $b$ is finite we
have ${\mathrm{comp}\,}{\mathcal{L}}(K)<{\mathrm{comp}\,}K$.

For $a=0$ we have ${\mathcal{L}}(K)=K^{(b-1)}$. When
${\mathrm{comp}\,}K=(a,0)$ we have that ${\mathcal{L}}(K)=\emptyset$.
Moreover, the geometrical dimension of $K$ is 0 and $K_{k}=\emptyset$ for each
$k\neq n-a$.

Notice that
$K={\mathcal{L}}(K)\cup\displaystyle\bigcup_{\beta\in{\mathcal{B}}(K)}\beta*L_{\beta}.$

and

${\mathcal{L}}(K)\cap\displaystyle\bigcup_{\beta\in{\mathcal{B}}(K)}\beta*L_{\beta}=\bigcup_{\beta\in{\mathcal{B}}(K)}\partial\beta*L_{\beta}=\bigcup_{\beta\in{\mathcal{B}}(K)}{\mathcal{L}}(\beta*L_{\beta}).$
(4)

###  1.5. Perversities

A _perversity_ on a filtered complex $K$ is a map
$\overline{p}\colon{\mathcal{S}}_{K}\to\overline{\mathbb{Z}}=\mathbb{Z}\cup\\{\pm\infty\\}$
taking the value 0 on the regular strata. The couple $(K,\overline{p})$ is a
_perverse filtered complex_.

The _dual perversity_ $D\overline{p}$ is defined by $D\overline{p}(S)={\rm
codim\,}_{v}S-2-\overline{p}(S)$ for any singular stratum
$S\in{\mathcal{S}}_{K}$.

Given a simplicial sub-complex $L\subset K$ we also denote by $\overline{p}$
the perversity defined on the induced filtered space $L$ by :

$\overline{p}(T)=\overline{p}(S),$ (5)

where $T\in{\mathcal{S}}_{L}$ and $S\in{\mathcal{S}}_{K}$ is the source of
$T$.

While working in a sub-complex $L$ of $K$ we prioritize the perversities as
defined on the strata of the whole filtered complex $K$. When we have several
sub-complexes at the same time, this option produces less ambiguities.

Simplicial intersection homology

We give a succinct presentation of the simplicial intersection homology of a
filtered complex $K=K_{n}\supset K_{n-1}\supset K_{n-2}\supset\ldots\supset
K_{0}\supset K_{-1}=\emptyset$ defined in [9, Section 3.2].

An _oriented simplex_ of $K$ is a simplex of $K$ with an equivalence class of
orderings of its vertices, where two orderings are equivalent if they differ
by an even permutation. The _simplicial chain module_ ${C}_{{}_{*}}{(K)}$ is
the module generated by the oriented simplices of $K$. Furthermore, if
$\sigma$ is such an oriented simplex, then we identify the oriented simplex
that has the same underlying geometric simplex but the opposite orientation
with $-\sigma$ as an element of ${C}_{{}_{*}}{(K)}$.

A simplex $\sigma$ of $K$ is _allowable_ if, for each singular stratum
$S\in{\mathcal{S}}_{K}^{\sigma}$,

$\|\sigma\|_{S}=\dim(\sigma\cap S)\leq\dim\sigma-{\rm
codim\,}_{v}S+\overline{p}(S)=\dim\sigma-2-D\overline{p}(S).$ (6)

Notice that $\sigma\cap S$ is a union of open faces of $\sigma$, so
$\dim(\sigma\cap S)$ makes sense.

A chain $c\in{C}_{{}_{*}}{(K)}$ is an _allowable chain_ if any simplex with a
non-zero coefficient in $c$ is $\overline{p}$-allowable. It is an
_intersection chain_ if $c$ and $\partial c$ are allowable chains. The complex
of intersection chains is denoted by
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$. The associated homology is
the _intersection simplicial homology_
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$. We get the simplicial
intersection homology of [9].

When $K_{n-1}=\emptyset$ then
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$ becomes the usual
simplicial chain complex ${C}_{{}_{*}}{(K)}$.

Let us suppose that the filtered complex $K$ is full. Then, for each $\ell\in
I_{\sigma}$ the number $||\sigma||_{S_{\ell}}$ is equal to
$||\sigma||_{\ell}=\dim(\sigma_{0}*\cdots*\sigma_{\ell})$ and the allowability
condition (6) becomes

$\|\sigma\|_{\ell}\leq\dim\sigma-(n-\ell)+\overline{p}(S_{\ell})=\dim\sigma-2-D\overline{p}(S_{\ell}).$
(7)

for each $\ell\in I_{\sigma}$.

###  1.6. Relative intersection homology

Let $(K,\overline{p})$ be a perverse filtered complex. Consider $L\subset K$ a
simplicial sub-complex. We have seen that $L$ inherits from $K$ a structure of
filtered complex. The allowability condition of a simplex $\sigma\in L$ can be
understood in $L$ itself or in $K$. Both points of view are equivalent. Let us
see that. The simplex $\sigma$ is allowable in $K$ if

$||\sigma||_{\ell}\leq\dim\sigma-(n-\ell)+\overline{p}(S_{\ell}),$

for each $\ell\in I^{K}_{\sigma}$. The simplex $\sigma$ is allowable in $L$ if

$||\sigma||_{\ell}\leq\dim\sigma-(n-\ell)+\overline{p}(T_{\ell}),$

for each $\ell\in I^{L}_{\sigma}$. These conditions are equivalent since the
two decompositions of $\sigma$ are the same, $I^{K}_{\sigma}=I^{L}_{\sigma}$
and $\overline{p}(S_{\ell})=\overline{p}(T_{\ell})$ (cf. Section 1.2 and (5)
).

The natural inclusion $L\hookrightarrow K$ a gives the following exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L\right)}}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\frac{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}}{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L\right)}}={C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,L\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
(8)

defining the _relative simplicial intersection complex_
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,L\right)}$. Its homology is the
_relative simplicial intersection homology_ denoted by
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,L\right)}$.

###  1.7. Example

There is an important difference between the complex of simplicial chains and
that of intersection chains. The first one is a free module over the family of
simplices. This is not the case in the second context since there are
allowable simplices which are not intersection chains.

So, the study of the relative complex
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K^{(\ell)},K^{(\ell-1)}\right)}$ does
not go through the family of $\ell$-simplices of $K$ as in the classical case.
Let us see an Example .

We consider $K$ the suspension of $L$, where $L$ is a triangle. We have
$K^{(2)}=K$. The skeleton $K^{(1)}$ is the union of the edges.

This simplicial complex $K$ is endowed with the filtration

$\emptyset=K_{-1}\subset\\{\beta_{0},\beta_{1}\\}=K_{0}=K_{1}\subset K_{2}=K.$

A straightforward calculation-gives

${H}^{{}^{\overline{0}}}_{{}_{*}}{\left(K^{(2)},K^{(1)}\right)}={H}^{{}^{\overline{0}}}_{{}_{2}}{\left(K^{(2)},K^{(1)}\right)}={\sf
R}\oplus{\sf R}\neq 0=\bigoplus_{\sigma\in
K,\dim\sigma=2}{H}^{{}^{\overline{0}}}_{{}_{2}}{\left(\sigma,\partial\sigma\right)}$

and therefore the formula (1) is not true.

We have ${\mathcal{L}}(K)=L$ which also is the link of $\beta_{0},\beta_{1}$.
Here, we do have the formule

${H}^{{}^{\overline{0}}}_{{}_{*}}{\left(K,{\mathcal{L}}(K)\right)}={H}^{{}^{\overline{0}}}_{{}_{*}}{\left(K,L\right)}=\bigoplus_{i=0,1}{H}^{{}^{\overline{0}}}_{{}_{*}}{\left(\beta_{i}*L,L\right)}=\bigoplus_{i=0,1}{H}^{{}^{\overline{0}}}_{{}_{*}}{\left(\beta_{i}*L,{\mathcal{L}}(\beta_{i}*L)\right)}$

Following the idea developed in this Example we show how to decompose the
relative filtered complex $(K,{\mathcal{L}}(K))$ in terms of a family of
relative filtered joins $(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta}))$. We
proceed by induction on the complexity of the filtered complex. Firstly, we
need the following Lemma.

######  Lemma 1.2.

Let $(K,\overline{p})$ be a perverse full filtered complex. Consider a clot
$\beta\in{\mathcal{B}}(K)$ and a simplex $\alpha\in{L_{\beta}}$. The stratum
of ${\mathcal{S}}_{K}$ containing $\beta$ is $Q$. Then

$\beta*\alpha\hbox{ is an allowable simplex
}\Longleftrightarrow\left\\{\begin{array}[]{l}\dim\alpha\geq{\rm
codim\,}_{v}(Q)-\overline{p}(Q)-1\hbox{ and}\\\ \alpha\hbox{ is an allowable
simplex. }\end{array}\right.$

###### Proof.

We write $q=\dim_{v}Q$. We have

$\displaystyle I_{\beta*\alpha}$ $\displaystyle=$
$\displaystyle\\{\ell\in\\{0,\ldots,n\\}\mid(\beta*\alpha)\cap\left(|(\beta*L_{\beta})_{\ell}|\backslash|(\beta*L_{\beta})_{\ell-1}|\right)\neq\emptyset\\}$
$\displaystyle=$
$\displaystyle\\{q\\}\cup\\{\ell\in\\{q+1,\ldots,n\\}\mid\alpha\cap\left(|(L_{\beta})_{\ell}|\backslash|(L_{\beta})_{\ell-1}|\right)\neq\emptyset\\}=\\{q\\}\cup
I_{\alpha}.$

So,

$\displaystyle\beta*\alpha\hbox{ is an allowable
simplex}\Longleftrightarrow\left\\{\begin{array}[]{l}||\beta*\alpha||_{a}\leq\dim(\beta*\alpha)-(n-q)+\overline{p}(Q)\hbox{
and}\\\
||\beta*\alpha||_{\ell}\leq\dim(\beta*\alpha)-(n-\ell)+\overline{p}(S_{\ell})\
\forall\ell\in I_{\alpha}\end{array}\right.$
$\displaystyle\Longleftrightarrow\left\\{\begin{array}[]{l}\dim\beta\leq\dim\beta+\dim\alpha+1-(n-q)+\overline{p}(Q)\hbox{
and}\\\
\dim\beta+||\alpha||_{\ell}+1\leq\dim\beta+1+\dim\alpha-(n-\ell)+\overline{p}(S_{\ell})\
\forall\ell\in I_{\alpha}\end{array}\right.$
$\displaystyle\Longleftrightarrow\left\\{\begin{array}[]{l}0\leq\dim\alpha+1-(n-q)+\overline{p}(Q)\hbox{
and}\\\ ||\alpha||_{\ell}\leq\dim\alpha-(n-\ell)+\overline{p}(S_{\ell})\
\forall\ell\in
I_{\alpha}\end{array}\right.\Longleftrightarrow\left\\{\begin{array}[]{l}\dim\alpha\geq(n-q)-\overline{p}(Q)-1\hbox{
and}\\\ \alpha\hbox{ is an allowable simplex. }\end{array}\right.$

∎

######  Proposition 1.3.

Let $(K,\overline{p})$ be a perverse full filtered complex. Inclusion maps
induce the isomorphism:

$\bigoplus_{\beta\in{\mathcal{B}}(K)}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},\partial\beta*L_{\beta}\right)}\xrightarrow{\cong}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,{\mathcal{L}}(K)\right)}.$

###### Proof.

Notice that ${\mathcal{L}}(\beta*L_{\beta})=\partial\beta*L_{\beta}$ for each
$\beta\in{\mathcal{B}}(K)$. The inclusion map is well defined (cf. Section
1.6) and a monomorphism. It remains to prove that it is an onto map. Let us
consider a chain $c\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$. We
have the decomposition

$c=\sum_{\beta\in{\mathcal{B}}(K)}\beta*e_{\beta}+f,$

where $e_{\beta}\in{C}_{{}_{*}}{(L_{\beta})}$ is an allowable chain, for each
$\beta\in{\mathcal{B}}(K)$, and $f\in{C}_{{}_{*}}{({\mathcal{L}}(K))}$ is an
allowable chain (cf. Lemma 1.2).

The derivative

$\partial
c=\sum_{\beta\in{\mathcal{B}}(K)}\partial\beta*e_{\beta}+\sum_{\beta\in{\mathcal{B}}(K)}\pm\beta*\partial
e_{\beta}+\partial f,$

(with the convention $0*e_{\beta}=e_{\beta}$) is an allowable chain. This
implies that
$e_{\beta}\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L_{\beta}\right)}$, with
$\beta\in{\mathcal{B}}(K)$, and
$f\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathcal{L}}(K)\right)}$ (cf.
Lemma 1.2). This gives the claim. ∎

######  Remark 1.4.

This approach contains the classical one as we prove now. Let us consider a
simplicial complex $K$ with $\dim K=b<\infty$. It is a filtered simplicial
complex relatively to the trivial filtration $K=K_{n}\supset\emptyset$. This
gives ${\mathrm{comp}\,}K=(0,b)$. We have $K=K^{(b)}$, $\beta(K)=\\{\beta\in
K\mid\dim\beta=b\\}$ and ${\mathcal{L}}(K)=K^{(b-1)}$. The previous formula
becomes the classical one

${C}_{{}_{*}}{\left(K,{\mathcal{L}}(K)\right)}\cong\mathop{\bigoplus_{\beta\in
K}}_{\dim\beta=b}{C}_{{}_{*}}{\left(\beta,\partial\beta\right)}\cong{C}_{{}_{*}}{\left(K^{(b)},K^{(b-1)}\right)}$

###  1.8. Intersection homology of the join

We use the dual perversity since the result is simpler to set. Notice that the
intersection homology of a filtered complex can be 0 when there are no regular
strata.

This is the reason why the following statement is more itemized than that of
[5, Proposition 1.49] for instance. The same phenomenon appears for the
calculation of the intersection homology of a cone between the statement of
King [15, Proposition 5] and of Friedman [9, Theorem 4.2.1].

######  Proposition 1.5.

Consider a perverse full filtered complex $(K,\overline{p})$. Let
$\beta\in{\mathcal{B}}(K)$ be a clot. The stratum $Q\in{\mathcal{S}}_{K}$
containing $\beta$ is a singular stratum . Then

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}=\left\\{\begin{array}[]{cl}{H}^{{}^{\overline{p}}}_{{}_{*\leq
D\overline{p}(Q)}}{\left(L_{\beta}\right)}&\hskip 28.45274pt\hbox{if }\
D\overline{p}(Q)\geq 0\\\\[14.22636pt]
{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}={\sf R}&\hskip
28.45274pt\hbox{if }\ \left\\{\begin{array}[]{l}D\overline{p}(Q)<-1\hbox{ or
}\\\ D\overline{p}(Q)=-1\hbox{ and
}{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(L_{\beta}\right)}\neq
0\end{array}\right.\\\\[22.76228pt] 0&\hskip 28.45274pt\hbox{if }\
D\overline{p}(Q)=-1\hbox{ and
}{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(L_{\beta}\right)}=0\end{array}\right.$

The first isomorphism is given by the inclusion
$L_{\beta}\hookrightarrow\beta*L_{\beta}$. A generator for the first case $\sf
R$ (resp. second case $\sf R$) is any element of
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}$ (resp. any
generator of $L_{\beta}$).

###### Proof.

We write $q=\dim_{v}Q$. Since $Q$ is singular we have
$D\overline{p}(Q)=n-q-2-\overline{p}(Q)$. Any chain
$c\in{C}_{{}_{*}}{(\beta*L_{\beta})}$ can be written as

$c=f+\sum_{\alpha\triangleleft\beta}\alpha*e_{\alpha}+e,$ (12)

where $f$ is a chain of $\beta$ and $e_{\alpha},e$ are chains of $L_{\beta}$.
This chain is allowable if

  * a)

$D\overline{p}(Q)+2\leq 0$ if $f\neq 0$,

  * b)

$|e_{\alpha}|\geq D\overline{p}(Q)+1$ if $e_{\alpha}\neq 0$, and

  * c)

$e_{\alpha},e$ allowable chains of $L_{\beta}$,

for each $\alpha\triangleleft\beta$.

We distinguish several cases for the calculation of
${H}^{{}^{\overline{p}}}_{{}_{i}}{\left(\beta*L_{\beta}\right)}$.

$\bullet$ $i=0$. A chain
$c\in{C}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}$ can be
written $c=f+e,$ where:

  * -

the chain $f\in{C}_{{}_{0}}{(\beta)}$ is 0 if $D\overline{p}(Q)+2>0$, and

  * -

$e\in{C}^{{}^{\overline{p}}}_{{}_{0}}{\left(L_{\beta}\right)}$

(cf. a) and c)). A chain
$c\in{C}^{{}^{\overline{p}}}_{{}_{1}}{\left(\beta*L_{\beta}\right)}$ can be
written $c=f+\sum_{k=0}^{b}A_{k}*e_{k}+e,$ where $\beta=\langle
A_{0},\ldots,A_{b}\rangle$ and

  * -

the chain $f\in{C}_{{}_{1}}{\left(\beta\right)}$ is 0 if
$D\overline{p}(Q)+2>0$,

  * -

the chains $e_{0},\ldots,e_{b}\in{C}_{{}_{0}}{\left(\beta\right)}$ are 0 if
$D\overline{p}(Q)+1<0$,

  * -

$e_{0},\ldots,e_{b}\in{C}^{{}^{\overline{p}}}_{{}_{0}}{\left(K\right)}$ and
$e\in{C}^{{}^{\overline{p}}}_{{}_{1}}{\left(K\right)}$

(cf. a), b) and c)).

Consider the class
$[c]=[f+e]\in{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}$.
We distinguish three cases

  1. $\bullet\bullet$

$n-q-1-\overline{p}(Q)>0$. Here, $f=0$ and therefore
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}={H}^{{}^{\overline{p}}}_{{}_{0}}{\left(L_{\beta}\right)}$
since
${C}^{{}^{\overline{p}}}_{{}_{1}}{\left(\beta*L_{\beta}\right)}={C}^{{}^{\overline{p}}}_{{}_{1}}{\left(L_{\beta}\right)}$

  2. $\bullet\bullet$

$D\overline{p}(Q)+1=0$. Here, $f=0$ and then $[c]=[e]$. So
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}=0$ if
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(L_{\beta}\right)}=0$. We have
$A_{0}*e\in{C}^{{}^{\overline{p}}}_{{}_{1}}{\left(\beta*L_{\beta}\right)}$ and
$\partial(A_{0}*e)=e-A_{0}$. So
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}=\sf R$ if
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(L_{\beta}\right)}\neq 0$.

  3. $\bullet\bullet$

$D\overline{p}(Q)+1<0$. We have
$A_{0}*e\in{C}^{{}^{\overline{p}}}_{{}_{1}}{\left(\beta*L_{\beta}\right)}$ and
$\partial(A_{0}*e)=e-A_{0}$ then $[f+e]=[f+A_{0}]$. This gives
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(\beta*L_{\beta}\right)}={H}_{{}_{0}}{\left(\beta\right)}=R$.

$\bullet$ $i>0$ and $D\overline{p}(Q)+1\leq 0$. Let $\eta\triangleleft\beta$
be the opposite face to the vertex $A_{0}$ or $\eta=\emptyset$ when
$\dim\beta=0$. We consider a cycle
$d\in{C}^{{}^{\overline{p}}}_{{}_{i}}{\left(\beta*L_{\beta}\right)}$ and we
prove that it is a boundary. In fact,
$d=A_{0}*e_{0}+e\in{C}^{{}^{\overline{p}}}_{{}_{i}}{\left(\beta*L_{\beta}\right)}$
with $e_{0},e\in{C}_{{}_{*}}{\left(\eta*L_{\beta}\right)}$. Following a), b)
and c) we have
$e_{0},e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\eta*L_{\beta}\right)}$
since $\deg d\geq 1$. We distinguish two cases.

  1. $\bullet\bullet$

$e_{0}\neq 0$ and $\deg d=1$. We have $\partial d=-A_{0}+e_{0}+\partial e$.
Since $\partial d=0$ then $A_{0}=e_{0}+\partial
e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\eta*L_{\beta}\right)}$, which is
not possible since $A_{0}\not\in\eta*L_{\beta}$.

  2. $\bullet\bullet$

$e_{0}=0$ and $\deg d=1$ or $\deg d\geq 2$. Notice that $\partial
d=-e_{0}+A_{0}*\partial e_{0}+\partial e$. Since $\partial d=0$ then
$e_{0}=-\partial e$ and therefore $d=-\partial(A_{0}*e).$ Since
$A_{0}*e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}$
(cf. a), b) and c)) then we get the claim.

$\bullet$ $D\overline{p}(Q)+1>0$. Let
$c\in{C}^{{}^{\overline{p}}}_{{}_{i}}{\left(\beta*L_{\beta}\right)}$ be a
cycle. Following a), b) and c) we have

$c=\sum_{\alpha\triangleleft\beta}\alpha*e_{\alpha}+e,$ (13)

where $e\in{C}^{{}^{\overline{p}}}_{{}_{i}}{\left(L_{\beta}\right)}$ and
$e_{\alpha}\in{\tau
C}^{{}^{\overline{p}}}_{{}_{i-\deg\alpha-1}}{\left(L\right)}_{\beta}$. Here

${\tau
C}^{{}^{\overline{p}}}_{{}_{i}}{\left(L_{\beta}\right)}=\left\\{\begin{array}[]{cl}{C}^{{}^{\overline{p}}}_{{}_{i}}{\left(L_{\beta}\right)}&\hbox{if
}k>D\overline{p}(Q)+1\\\
{C}^{{}^{\overline{p}}}_{{}_{i}}{\left(L_{\beta}\right)}\cap\partial^{-1}(0)&\hbox{if
}k=D\overline{p}(Q)+1\\\ 0&\hbox{if }i<D\overline{p}(Q)+1.\end{array}\right.$

Let us consider the map

$f\colon{C}_{{}_{*}}{(\beta)}\otimes\tau{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L_{\beta}\right)}\to\frac{{C}^{{}^{\overline{p}}}_{{}_{*+1}}{\left(\beta*L_{\beta}\right)}}{{C}^{{}^{\overline{p}}}_{{}_{*+1}}{\left({L_{\beta}}\right)}},$

defined by $f(\alpha\otimes d)=(-1)^{\deg\alpha}[\\![\alpha*d]\\!].$ Notice
that $\deg d>0$.

$i)$ If $\deg\alpha>0$ we have:

$\displaystyle f(\partial(c\otimes d))$ $\displaystyle=$ $\displaystyle
f(\partial\alpha\otimes d)+(-1)^{\deg\alpha}f(\alpha\otimes\partial
d)=(-1)^{\deg\alpha-1}[\\![\partial\alpha*d]\\!]+[\\![\alpha*\partial d]\\!]$
$\displaystyle=$
$\displaystyle(-1)^{\deg\alpha-1}([\\![\partial\alpha*d]\\!]+(-1)^{\deg\alpha-1}[\\![\alpha*\partial
d)]\\!]=(-1)^{\deg\alpha-1}[\\![\partial(\alpha*d)]\\!]=-\overline{\partial}f(\alpha\otimes
d),$

where $\overline{\partial}$ is the induced differential on the quotient
$\frac{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}}{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L_{\beta}\right)}}$.

$ii)$ If $\deg\alpha=0$:

$f(\partial(\alpha\otimes d))=f(\alpha\otimes\partial d)=[\\![\alpha*\partial
d]\\!]=[\\![-d+\alpha*\partial
d]\\!]=-[\\![\partial(\alpha*d)]\\!]=-\overline{\partial}f(\alpha\otimes d),$

So, the map $f$ is a chain map. It is an isomorphism since
$f^{-1}([\\![c]\\!])=\sum_{\alpha\triangleleft\beta}\alpha\otimes e_{\alpha}$
(cf.(13)). The map
$I\colon\tau{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L_{\beta}\right)}\to{C}_{{}_{*}}{(\beta)}\otimes\tau{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L_{\beta}\right)}$,
defined by $I(d)=-A_{0}\otimes d$, is a quasi-isomorphism. Notice that
${H}_{{}_{*}}{({\tau
C}^{{}^{\overline{p}}}_{{}_{\cdot}}{\left({L_{\beta}}\right)})}={H}^{{}^{\overline{p}}}_{{}_{*\geq
D\overline{p}(Q)+1}}{\left(L_{\beta}\right)}.$

Using this operator we get that the long exact sequence associated to the
short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\frac{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}}{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left({L_{\beta}}\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

is

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H}^{{}^{\overline{p}}}_{{}_{k+1}}{\left(L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H}^{{}^{\overline{p}}}_{{}_{k+1}}{\left(\beta*L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H}^{{}^{\overline{p}}}_{{}_{k\geq
D\overline{p}(Q)+1}}{\left(L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{H}^{{}^{\overline{p}}}_{{}_{k}}{\left(L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots,}$

where connecting morphism $\delta$ is the inclusion. This gives
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}={H}^{{}^{\overline{p}}}_{{}_{\leq
D\overline{p}(Q)}}{\left(L_{\beta}\right)}$, where the isomorphism is induced
by the inclusion $L_{\beta}\hookrightarrow\beta*L_{\beta}$. ∎

When $Q$ is a regular stratum we have $L_{\beta}=\emptyset$ and
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}={H}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta\right)}={H}_{{}_{*}}{(\beta)}={H}_{{}_{0}}{(\beta)}=\sf
R$.

##  2\. Singular setting

> The singular version of the intersection homology goes back to King [15],
> working with CS-sets. We present the singular version of the intersection
> homology associated to the realization of a filtered complex $K$. We study
> some calculations involving this homology used in this work, namely
>
>   * •
>
> the decomposition of the relative intersection homology of the pair
> $(|K|,|{\mathcal{L}}(K)|)$ (cf. Proposition 2.3 ), and
>
>   * •
>
> the computation of the intersection homology of the realization of a join
> (cf. Proposition 2.4).
>
>

###  . Filtered spaces .

######  Definition 2.1.

A _filtered space_ is a Hausdorff topological space $X$ endowed with a
filtration by closed subspaces,

$X=X_{n}\supset X_{n-1}\supset\dots\supset X_{0}\supset X_{-1}=\emptyset.$

The integer $n$ is the dimension of $X$. The $i$-strata of $X$ are the non
empty connected components of $X_{i}\backslash X_{i-1}$. The open strata are
called _regular_ , the other ones are _singular._ The set of singular strata
of $X$ is denoted ${\mathcal{S}}_{X}$.

The _dimension_ (resp. _codimension_) of a stratum $S\subset X_{i}\backslash
X_{i-1}$ is $\dim_{v}S=i$ (resp. ${\rm codim\,}_{v}S=i$).

Given a $n$-dimensional filtered complex $K$, the associated filtration

$|K|=|K_{n}|\supset|K_{n-1}|\supset|K_{n-2}|\supset\ldots\supset|K_{0}|\supset
K_{-1}=\emptyset,$

defines a $n$-dimensional filtered space. Notice that
${\mathcal{S}}_{|K|}={\mathcal{S}}_{K}$.

###  2.1. Induced filtered spaces

Let $X=X_{n}\supset X_{n-1}\supset X_{n-2}\supset\ldots\supset X_{0}\supset
X_{-1}=\emptyset$ be a filtered space. Given a subset $Y\subset X$ we have the
induced filtration

$Y=Y_{n}=Y\cap K_{n}\supset X_{n-1}=Y\cap X_{n-1}\supset\ldots\supset
Y_{0}=Y\cap X_{0}\supset Y_{-1}=Y\cap X_{-1}=\emptyset$

defining a filtered space structure on $Y$ (see [9]). Since $Y_{i}\subset
X_{i}$ for any $i\in\\{1,\ldots,n\\}$ then for any stratum
$T\in{\mathcal{S}}_{Y}$ there exists a unique stratum $S\in{\mathcal{S}}_{X}$
with $T\subset S$. We say that $S$ is the _source_ of $T$. Notice that
$\dim_{v}T=\dim_{v}S$.

Singular intersection homology

We give a succinct presentation of the singular intersection homology of a
filtered space $X=X_{n}\supset X_{n-1}\supset\dots\supset X_{0}\supset
X_{-1}=\emptyset$ defined in King [15] (see also [9, Section 3.4]).

For each singular stratum $S\in{\mathcal{S}}_{X}$, the _perverse degree of a
singular simplex $\sigma\colon\Delta\to X$ along the stratum $S$_ is

$\|\sigma\|_{S}=\left\\{\begin{array}[]{cl}-\infty&\text{if
}S\cap\sigma(\Delta)=\emptyset,\\\ \dim\sigma^{-1}(S)&\text{if
not.}\end{array}\right.$

The _dimension_ of a non-empty subset $A\subset\Delta$ is the smallest
$s\in\mathbb{N}$ for which the subset $A$ is included in the $s$-skeleton of
$\Delta$.

###  2.2. Perversities

A _perversity_ on a filtered space $X$ is a map
$\overline{p}\colon{\mathcal{S}}_{X}\to\overline{\mathbb{Z}}$ taking the value
0 on the regular strata. The couple $(X,\overline{p})$ is a _perverse filtered
space_.

Given a subset $Y\subset X$ we also denote by $\overline{p}$ the perversity
defined on the induced filtered space $Y$ by :

$\overline{p}(T)=\overline{p}(S),$

where $T\in{\mathcal{S}}_{Y}$ and $S\in{\mathcal{S}}_{X}$ is the source of
$T$.

We have all the ingredients necessary to the definition of the intersection
homology.

######  Definition 2.2.

Let $(X,\overline{p})$ be a perverse filtered space. A singular simplex
$\sigma\colon\Delta\to X$, is _allowable_ if,

$||\sigma||_{S}\leq\dim\Delta-{\rm codim\,}_{v}S+\overline{p}(S)$ (14)

for each singular stratum $S\in{\mathcal{S}}_{X}$. A singular chain $c$ is an
_allowable chain_ if any simplex with a non-zero coefficient in $c$ is
allowable. It is an _intersection chain_ if $c$ and $\partial c$ are allowable
chains. The associated homology is the _singular intersection homology_
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}$. We get the singular
intersection homology of [15].

When $X_{n-1}=\emptyset$ then
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}$ becomes the usual singular
chain complex ${C}_{{}_{*}}{(X)}$.

###  2.3. Relative intersection homology

Let $(X,\overline{p})$ be a perverse filtered space. Consider a subset
$Y\subset X$. We have seen that $Y$ inherits from $X$ a structure of filtered
space. The natural inclusion $Y\hookrightarrow X$ gives the following exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(Y\right)}}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\frac{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}}{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(Y\right)}}={C}^{{}^{\overline{p}}}_{{}_{*}}{\left(X,Y\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(15)

(cf. [9, Corollary 4.3.5]). The homology of the right term is the _relative
singular intersection homology_ denoted by
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(X,Y\right)}$.

######  Proposition 2.3.

Let $(K,\overline{p})$ be a perverse full filtered complex. The inclusion map
induces the isomorphism:

$\bigoplus_{\beta\in{\mathcal{B}}(K)}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta})=\partial\beta*L_{\beta}|\right)}\xrightarrow{\cong}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,|{\mathcal{L}}(K)|\right)}.$

_Proof._ We proceed in two steps.

_Step 1: Thickening_. The barycenter of $\beta\in{\mathcal{B}}(K)$ is denoted
by $b_{\beta}$. We have the equality

. $|\beta*L_{\beta}|=|b_{\beta}*\partial\beta*L_{\beta}|,$ .

where $\partial\beta=\emptyset$ if $\dim\beta=0$. So, any point of
$x~\in|\beta*L_{\beta}|$ can be written as $x=(1-t)\cdot b_{\beta}+t\cdot a$
where $t\in[0,1]$ and $a\in|\partial\beta*L_{\beta}|$. This writting is unique
when $x\neq b_{\beta}$ (i.e., $t\neq 0$).

Notice that the assignment $(1-t)\cdot b_{\beta}+t\cdot c\mapsto(c,t)$ induces
the homeomorphism

$|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}\cong|\partial\beta*L_{\beta}|\times]0,1].$
(16)

Under this homeomorphism the filtration on the subset
$|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}$ becomes the product filtration.

We define the open subset

$W=|K|\backslash\bigcup_{\beta\in{\mathcal{B}}(K)}\\{b_{\beta}\\}=|{\mathcal{L}}(K)|\cup\bigcup_{\beta\in{\mathcal{B}}(K)}\left(|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}\right)$

(see Section 1.4). Associated to this open subset, we define a map, $F\colon
W\times[0,1]\to W$ by

$F(x,s)=\left\\{\begin{array}[]{ll}x&\hbox{if }x\in|{\mathcal{L}}(K)|\\\
s\cdot x+(1-s)\cdot y&\hbox{if }x=(1-t)\cdot b_{\beta}+t\cdot
y\in|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}.\end{array}\right.$

Let us notice that $F$ preserves the filtration and that $F(-,1)$ is the
identity. With the previous notations, the map $F(-,0)$ is the identity on
$|{\mathcal{L}}(K)|$ by construction, and sends $x\notin{\mathcal{L}}(K)$ on
$y\in|\partial\beta\ast L_{\beta}|\subset|{\mathcal{L}}(K)|$. Thus, $F(-,0)$
gives a map $\nu\colon W\to|{\mathcal{L}}(K)|$. Denote by
$j\colon|{\mathcal{L}}(K))|\hookrightarrow W$ the inclusion. We have proved
$\nu\circ j={\mathrm{id}}_{{\mathcal{L}}(K)}$ and $F$ is a homotopy between
$j\circ\nu$ and $F(-,0)$. From [9, Proposition 4.1.10], we get that the
inclusion $j$ induces the isomorphism
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(W\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|{\mathcal{L}}(K)|\right)}$
and therefore the isomorphism

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,W\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,|{\mathcal{L}}(K)|\right)}.$

_Step 2: Excision_. By excision (cf. [9, Corollary 4.4.18]) we have

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,W\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\backslash|{\mathcal{L}}(K)|,W\backslash|{\mathcal{L}}(K)|\right)}.$

From Section 1.4, we get the disjoint unions

$|K|\backslash|{\mathcal{L}}(K)|=\bigsqcup_{\beta\in{\mathcal{B}}(K)}|\beta*L_{\beta}|\backslash|\partial\beta*L_{\beta}|\
\ \ \hbox{ and }\ \ \
W\backslash|{\mathcal{L}}(K)|=\bigsqcup_{\beta\in{\mathcal{B}}(K)}|\beta*L_{\beta}|\backslash(\\{b_{\beta}\\}\cup|\partial\beta*L_{\beta}|).$
(17)

which give

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\backslash|{\mathcal{L}}(K)|,W\backslash|{\mathcal{L}}(K)|\right)}\cong\bigoplus_{\beta\in{\mathcal{B}}(K)}{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\backslash|\partial\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash(\\{b_{\beta}\\}\cup|\partial\beta*L_{\beta}|\right)}.$

Using excision relatively to the closed subset $|\partial\beta*L_{\beta}|$ we
obtain

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\backslash|\partial\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash(\\{b_{\beta}\\}\cup|\partial\beta*L_{\beta}||)\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}\right)}.$

Finally, applying [9, Proposition 4.1.10] to (16) we get

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|\partial\beta*L_{\beta}|\right)}$

which gives the claim. $\clubsuit$

###  2.4. Intersection homology of the join

We use the dual perversity since the result is simpler to set. The following
calculation meets the same pattern as in Proposition 1.5.

######  Proposition 2.4.

Consider a perverse full filtered complex $(K,\overline{p})$. Let
$\beta\in{\mathcal{B}}(K)$ be a clot. The stratum $Q\in{\mathcal{S}}_{K}$
containing $\beta$ is a singular stratum . Then

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}=\left\\{\begin{array}[]{cl}{H}^{{}^{\overline{p}}}_{{}_{*\leq
D\overline{p}(Q)}}{\left(|L_{\beta}|\right)}&\hskip 28.45274pt\hbox{if }\
D\overline{p}(Q)\geq 0\\\\[14.22636pt]
{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(|\beta*L_{\beta}|\right)}=\sf R&\hskip
28.45274pt\hbox{if }\ \left\\{\begin{array}[]{l}D\overline{p}(Q)<-1\hbox{ or
}\\\ D\overline{p}(Q)=-1\hbox{ and
}{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(|L_{\beta}|\right)}\neq
0\end{array}\right.\\\\[22.76228pt] 0&\hskip 28.45274pt\hbox{if }\
D\overline{p}(Q)=-1\hbox{ and
}{H}^{{}^{\overline{p}}}_{{}_{0}}{\left(|L_{\beta}|\right)}=0\end{array}\right.$

The first isomorphism is given by the inclusion
$L_{\beta}\hookrightarrow\beta*L_{\beta}$. A generator for the first $\sf R$
(resp. second $\sf R$) is any generator of
${H}^{{}^{\overline{p}}}_{{}_{0}}{\left(|L_{\beta}|\right)}$ (resp. any point
of $\beta*L_{\beta}$).

###### Proof.

We write $q=\dim_{v}Q$. Since $Q$ is singular we have
$D\overline{p}(Q)=n-q-2-\overline{p}(Q)$. The case $\dim\beta=0$ is given by
[9, Theorem 4.2.1]. When $\dim\beta>0$ we consider $\eta=\langle A_{0}\rangle$
the 0-simplex given by the first vertex $A_{0}$ of $\beta$. It suffices to
prove that the inclusion $\eta\hookrightarrow\beta$ induces the isomorphism
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\eta*K|\right)}={H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}$.
Consider a simplicial homotopy $F\colon\beta\times[0,1]\to\beta$ between the
identity on $\beta$ and the constant map $\beta\to\eta$. The induced map is
still denoted by $F$, that is,
$F\colon|\beta*L_{\beta}|\times[0,1]\to|\beta*L_{\beta}|$. It is a homotopy
between the identity of $|\beta*L_{\beta}|$ and the map
$\xi\colon|\beta*L_{\beta}|\to|\eta*L_{\beta}|$ given by $A*x\mapsto A_{0}*x$.

From [9, Proposition 4.1.10] we get that both maps
$F(-,0),F(-,1)\colon\beta*L_{\beta}\to\beta*L_{\beta}$ induce the same
morphism in ${H}^{{}^{\overline{p}}}_{{}_{*}}$-homology. Since $F(-,0)$ is the
identity then
$(F(-,1))_{*}\colon{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}\to{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\eta*L_{\beta}|\right)}$
is an isomorphism.

Finally, since
${\mathrm{id}}\colon|\eta*L_{\beta}|\hookrightarrow|\beta*L_{\beta}|\xrightarrow{\xi}|\eta*L_{\beta}|$
and
$F(-,1)\colon|\beta*L_{\beta}|\xrightarrow{\xi}|\eta*L_{\beta}|\hookrightarrow|\beta*L_{\beta}|$
we conclude that the inclusion
$|\eta*L_{\beta}|\hookrightarrow|\beta*L_{\beta}|$ induces an isomorphism
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\eta*L_{\beta}|\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}$.
∎

When $Q$ is a regular stratum we have $L_{\beta}=\emptyset$ and
${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}={H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta|\right)}={H}_{{}_{*}}{(|\beta|)}={H}_{{}_{0}}{(|\beta|)}=\sf
R$.

##  3\. Simplicial versus Singular

> Any simplex of a filtered complex $K$, is in fact a singular simplex. So,
> the natural inclusion
> $\iota\colon{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\to{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$
> is well defined. We prove that this map induces an isomorphism between the
> singular and simplicial versions of the intersection homology.
>
> The filtered complex $K$ must be full as is pointed out in [16, Remark 2].

Consider a full filtered complex $(K,\overline{p})$. We construct the map
$\iota\colon{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\to{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$
which will enable us to prove that both intersection homologies, singular and
simplicial, are isomorphic (cf. Theorem A). In order to preserve the canonical
decomposition of a simplex of $K$ we need to strengthen the order used in [9,
Section 4.4.2]. Let us consider an order $\leq$ on the vertices of $K$
verifying the following property:

$v\leq w\hbox{ and }w\in K_{k}\Rightarrow v\in K_{k}.$ (18)

Such an order always exits by the Well-Ordering Theorem applied first to
$K_{0}$, after to $K_{1}\backslash K_{0}$ and so on.

Let us consider an oriented simplex $\sigma\in K$ whose vertices are
$\\{v_{0},\ldots,v_{i}\\}$ with $v_{0}<v_{1}<\cdots<v_{i}$. If $\sigma=\langle
v_{0},\ldots,v_{i}\rangle$ we say that $\sigma$ is an _ordered simplex_.
Notice that $-\sigma$ is an ordered simplex if $\sigma$ is not and vice-versa.
So, the chain complex ${C}_{{}_{*}}{(K)}$ is generated by the ordered
simplices.

Associated to such simplex we have the singular simplex
$\iota(\sigma)\colon\Delta\to|K|$ defined by:

$\iota(\sigma)\left(\sum_{j=0}^{m}t_{i}\cdot
a_{j}\right)=\sum_{j=0}^{m}t_{j}v_{j}.$ (19)

where $\Delta=\langle a_{0},\dots,a_{m}\rangle$ is the standard simplex.

######  Proposition 3.1.

Let $(K,\overline{p})$ be a perverse full filtered complex. The map
$\iota\colon{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\to{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$,
associated to an order verifying (18), is a chain map.

###### Proof.

We know from [9, Section 4.4.2 ] that
$\iota\colon{C}_{{}_{*}}{(K)}\to{C}_{{}_{*}}{(|K|)}$ is a chain map. So, it
suffices to prove that, given an oriented simplex $\sigma\in K$, we have
$||\iota(\sigma)||_{S}=||\sigma||_{S}$ for each singular stratum
$S\in{\mathcal{S}}_{|K|}$ verifying
$S\cap\underbrace{\iota(\sigma)(\Delta)}_{=\sigma(\Delta)}\neq\emptyset$.

We write $\sigma=\langle v_{0},\ldots,v_{i}\rangle$, with
$v_{0}<v_{1}<\cdots<v_{i}$. The canonical decomposition of $\sigma$ is defined
by

$\sigma\cap|K_{\ell}|=\sigma_{0}*\cdots*\sigma_{\ell},$ (20)

for each $\ell\in\\{0,\ldots,n\\}$ (cf. Section 1.1). Notice that
$\iota(\sigma)\colon\Delta\to|K|$ is a linear map. We write
$\Delta_{j}=\iota(\sigma)^{-1}(\sigma_{i})$ which are empty-sets or faces of
$\Delta$. We have

$\iota(\sigma)^{-1}(|K_{\ell}|)=\iota(\sigma)^{-1}(\sigma\cap|K_{\ell}|)=\iota(\sigma)^{-1}(\sigma_{0}*\cdots*\sigma_{\ell})=\Delta_{0}*\cdots*\Delta_{\ell}.$

Let $j=\dim_{v}S$. Since $S$ is the only $j$-dimensional stratum meeting
$\sigma$ (cf. (20)), then

$||\sigma||_{S}=\dim(\sigma\cap
S)=\dim(\sigma\cap|K_{j}|\backslash|K_{j-1}|)=_{\eqref{caraS}}\dim(\sigma_{0}*\cdots*\sigma_{j})=\dim(\Delta_{0}*\cdots*\Delta_{j})=||\iota(\sigma)||_{S}.$

We get the claim. ∎

Next result proves that this inclusion induces an isomorphism between the
singular and the simplicial intersection homologies. This result has been
proved in [9, Corollary 5.4.6] using the PL-intersection homology.

######  Theorem A.

Let $(K,\overline{p})$ be a perverse full filtered complex. Then, the
inclusion
$\iota\colon{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\to{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$,
associated to an order verifying (18), induces the isomorphism

${H}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}{\cong}{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}.$

###### Proof.

We proceed in two steps.

_First Step._ _$K=K^{(\ell)}$ for some $\ell\in\mathbb{N}$_. Given a sub-
complex $L\subset K$ we also have $L=L^{(\ell)}$. If
${\mathrm{comp}\,}L=(a_{L},b_{L})$ then $b_{L}\leq\dim L<\infty$ and therefore
${\mathrm{comp}\,}{\mathcal{L}}(L)<{\mathrm{comp}\,}L$ (cf. Section 1.4).

We use the induction on the complexity $(a,b)$ of $K$. When $b=0$ then $K$ is
a discrete family of 0-dimensional simplices. So,
${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}={C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$
and we get the claim.

Let us suppose $b\geq 1$.

We have the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,{\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the vertical maps are induced by the inclusion map $\iota$ (cf. (8),
(15)). From the induction hypothesis we know that the left arrow is a quasi-
isomorphism. So, it suffices to prove that the right arrow is a quasi-
isomorphism. Using Propositions 1.3 and 2.3 this assertion is equivalent to
the fact that the inclusion map $\iota$ induces the quasi-isomorphism

$\bigoplus_{\beta\in{\mathcal{B}}(K)}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta}\right)}\to\bigoplus_{\beta\in{\mathcal{B}}(K)}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta}|\right)}.$

In fact, it suffices to prove that the restriction map $\iota$ induces, for
each $\beta\in{\mathcal{B}}(K)$, the quasi-isomorphism

${C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta}\right)}\to{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta}|\right)}.$

Again (8) and (15) give the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathcal{L}}(\beta*L_{\beta})\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\i}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta})\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|{\mathcal{L}}(\beta*L_{\beta}|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\j}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta})|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the vertical maps are induced by the inclusion map $\iota$. We have that
${\mathrm{comp}\,}{\mathcal{L}}(\beta*L_{\beta})<{\mathrm{comp}\,}(\beta*L_{\beta})\leq{\mathrm{comp}\,}K$
(cf. Section 1.2). From the induction hypothesis we know that the left arrow
is a quasi-isomorphism. It remains to prove that the middle arrow is a quasi-
isomorphism. We distinguish two cases.

$\bullet$ _$\beta$ is included in a regular stratum_. We have
$L_{\beta}=\emptyset$ and the middle arrow becomes
$\iota_{*}\colon{C}_{{}_{*}}{\left(\beta\right)}\to{C}_{{}_{*}}{\left(|\beta|\right)}.$
The claim comes from the classical situation (see [14, Theorem 2.27] for
instance).

$\bullet$ _$\beta$ is included in a singular stratum_. We have
${\mathrm{comp}\,}L_{\beta}<{\mathrm{comp}\,}K$ (cf. Section 1.3) From
Propositions 1.5, 2.4 and the induction hypothesis we know that the middle
arrow is a quasi-isomorphism.

_Second Step_. We consider the map
$\iota_{*}\colon{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\longrightarrow{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$.
We proceed in two points.

$\bullet$ _$\iota_{*}$ is an epimorphism._ Consider a cycle
$c\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$. Since $c$ is a
finite sum, then there exists an integer $\ell\in\mathbb{N}$ with
$c\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K^{(\ell)|}\right)}$. Applying
the first step of the proof, there exist
$f\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K^{(\ell)}|\right)}$ and
$e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K^{(\ell)}\right)}$ with $\partial
e=0$ and $\iota(e)=c+\partial f$. Since
$f\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$ and
$e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$ then
$\iota_{*}([e])=[c]$ and we get the claim.

$\bullet$ _$\iota_{*}$ is a monomorhism._ Consider a cycle
$e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$ and a chain
$f\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$ with
$\iota(e)=\partial f$. Since $e$ and $f$ are finite sums, then there exists an
integer $\ell\in\mathbb{N}$ with
$e\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K^{(\ell)}\right)}$ and
$f\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K^{(\ell)|}\right)}$. Applying
the first step of the proof, there exists
$g\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K^{(\ell)}\right)}$ with
$e=\partial g$. Since $g\in{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}$
then we get $[e]=0$ and we get the claim. ∎

######  Remark 3.2.

The existence of an isomorphism between simplicial intersection homology and
PL intersection homology has been established by Goresky and MacPherson in the
Appendix of [16]. For that, they prove the existence of an isomorphism between
the simplicial intersection homologies of a (full) filtered Euclidean
simplicial complex $K$ and of a subdivision $K^{\prime}$ of it. The proof goes
like this:

By construction, there is a map
$j\colon{\mathrm{Vert}\,}(K)\to{\mathrm{Vert}\,}(K^{\prime})$ between the sets
of vertices. Suppose there is a well order on the set ${\mathrm{Vert}\,}(K)$.
If $\sigma$ is a simplex of $K$ and $S_{\sigma}$ the stratum containing the
interior of $\sigma$, we choose the maximal vertex ${\mathtt{v}}_{\sigma}$ of
$\sigma$ which belongs to $S_{\sigma}$. As each vertex $a$ of
$K^{\prime}\backslash K$ belongs to the interior of a simplex $\sigma_{a}\in
K$, a map $\varphi\colon{\mathrm{Vert}\,}(K^{\prime})\to{\mathrm{Vert}\,}(K)$
is defined by $a\mapsto{\mathtt{v}}_{\sigma_{a}}$. The last step consists in
the proof that $\varphi$ and the subdivision map induce isomorphisms in
intersection homology.

The isomorphism between the singular and the PL intersection homologies is set
up by King in [15] for a CS-set. Let us also mention that these two
isomorphisms are taken over with details and comments in [9, Sections 3.3 and
5.4]. In conclusion, an isomorphism between the simplicial and the singular
intersection homologies is established using the PL situation as intermediary.

The intersection cohomology ${H}^{{}^{*}}_{{}_{\overline{p}}}{\left(-\right)}$
of [10] is defined as the cohomology of the dual complex ${\rm
Hom}({C}^{{}^{\overline{p}}}_{{}_{*}}{\left(-\right)};R)$. Since the Universal
Coefficient Theorem is true in this context [9] then we conclude that the
singular and simplicial version of the intersection cohomology are isomorphic.

##  4\. Blown-up intersection cohomologies

> Intersection homology and cohomology do not verify the Poincaré duality for
> any ring of coefficient (see [10]). The blown-up intersection cohomology has
> been introduced in order to recover this property through a cap product with
> a fundamental class (see [4]). We present the singular and simplicial
> versions of the blown-up intersection cohomology associated to a filtered
> complex $K$.
>
> We provide some calculations involving these cohomologies used in this work
> in order to establish an isomorphism between their singular and simplicial
> versions. Namely
>
>   * •
>
> the proof of the existence of a long exact sequence in intersection
> cohomology for the pairs $(|K|,|{\mathcal{L}}(K)|)$ and
> $(K,{\mathcal{L}}(K))$ (cf. Proposition 4.10),
>
>   * •
>
> the decomposition of the relative intersection cohomology of the above pairs
> (cf. Proposition 4.11), and
>
>   * •
>
> the computation of the intersection cohomology of $|\beta*L_{\beta}|$ (resp.
> $\beta*L_{\beta}$) in function of the intersection cohomology of
> $|L_{\beta}|$ (resp. $L_{\beta}$) (cf. Proposition 4.12),
>
>

>
> Finally, we prove the isomorphism between the two versions of the blown-up
> intersection cohomology in Theorem B.

######  Definition 4.1.

Let $X$ be a filtered space. A _filtered singular simplex_ is a continuous
map, $\sigma\colon\Delta\to X$, such that $\sigma^{-1}(X_{i})$ is a face of
$\Delta$, for $i\in\\{0,\dots,n\\}$, or the empty set. The family of these
simplices is denoted $\mathrm{Sing}^{{\mathcal{F}}}X$. A filtered singular
simplex $\sigma$ is _regular_ if $\sigma^{-1}(X_{n}\backslash
X_{n-1})\neq\emptyset$. The subfamily of regular simplices is denoted
$\mathrm{Sing}^{{\mathcal{F}}}_{+}X$.

######  Definition 4.2.

A _filtered simplex_ is a decomposition $\Delta=\Delta_{0}*\cdots*\Delta_{n}$,
where each $\Delta_{i}$ is a face of the standard simplex $\Delta$ or the
empty set. If $\Delta_{n}\neq\emptyset$ we say that the simplex is _regular_.
The _blow up of_ a regular simplex is the prism
${\widetilde{\Delta}}={\mathtt{c}}\Delta_{0}\times\cdots\times{\mathtt{c}}\Delta_{n-1}\times\Delta_{n}.$

The domain of a filtered singular simplex, $\sigma\colon\Delta\to X$, inherits
a decomposition

$\Delta_{\sigma}=\Delta_{0}\ast\Delta_{1}\ast\dots\ast\Delta_{n},$

with $\sigma^{-1}(X_{i})=\Delta_{0}\ast\dots\ast\Delta_{i}$, and $\Delta_{i}$
possibly empty, for each $i\in\\{0,\dots,n\\}$. We call it the _$\sigma$
-decomposition of $\Delta$._ This simplex $\sigma$ is regular when
$\Delta_{n}\neq\emptyset$. Notice that each filtered singular simplex
$\sigma\colon\Delta\to X$ (resp. regular) induces a filtered simplex (resp.
regular) on $\Delta$.

The family $\mathrm{Sing}^{{\mathcal{F}}}X$ is a $\Delta$-set in the sense of
Rourke and Sanderson ([20]) for the usual face maps $\delta_{i}$.

The blown-up construction, we describe above, is the key point in the
construction of our cohomology. Let us begin with the local situation.

######  Definition 4.3.

A _face_ of ${\widetilde{\Delta}}$ is represented by
$(F,\varepsilon)=(F_{0},\varepsilon_{0})\times\dots\times(F_{n},\varepsilon_{n}),$
with $\varepsilon_{i}\in\\{0,1\\}$, $\varepsilon_{n}=0$ and $F_{i}$ a face of
$\Delta_{i}$ for $i\in\\{0,\dots,n\\}$, or the empty set with
$\varepsilon_{i}=1$.

More precisely, we have

  * •

$\varepsilon_{i}=0$ and $F_{i}\triangleleft\Delta_{i}$, that is,
$(F_{i},0)=F_{i}$ is a face of $\Delta_{i}$, or

  * •

$\varepsilon_{i}=1$ and $F_{i}\triangleleft\Delta_{i}$, that is,
$(F_{i},1)={\mathtt{c}}F_{i}$ is the cone of a face of $\Delta_{i}$, or

  * •

$\varepsilon_{i}=1$ and $F_{i}=\emptyset$, that is, $(\emptyset,1)$ is the
apex of the cone ${\mathtt{c}}\Delta_{i}$, the _virtual apex_.

Recall that $\varepsilon_{n}=0$. We also set

$|(F,\varepsilon)|_{>j}=\dim(F_{j+1},\varepsilon_{j+1})+\cdots+\dim(F_{n},\varepsilon_{n}).$

Let us denote ${N}^{{}^{*}}{(\Delta)}$ the complex of simplicial cochains
defined on the standard simplex $\Delta$, with coefficients in the ring $\sf
R$.

######  Definition 4.4.

The _blown-up complex_ of a regular simplex
$\Delta=\Delta_{0}*\cdots*\Delta_{n}$ is the tensor product

${\widetilde{N}}^{*}(\Delta)=N^{*}({\mathtt{c}}\Delta_{0})\otimes\cdots\otimes
N^{*}({\mathtt{c}}\Delta_{n-1})\otimes N^{*}(\Delta_{n}).$

Let ${\boldsymbol{1}}_{(F_{i},\varepsilon_{i})}$ be the cochain on
${\mathtt{c}}\Delta_{i}$, taking the value $1$ on the simplex
$(F_{i},\varepsilon_{i})$ and $0$ on the other simplices of
${\mathtt{c}}\Delta_{i}$, for $i\in\\{0,\ldots,n-1\\}$. Analogously for
${\boldsymbol{1}}_{(F_{n},\varepsilon_{n})}$. A basis of
${\widetilde{N}}^{*}(\Delta)$ is given by the family

${\boldsymbol{1}}_{(F,\varepsilon)}={\boldsymbol{1}}_{(F_{0},\varepsilon_{0})}\otimes\dots\otimes{\boldsymbol{1}}_{(F_{n},\varepsilon_{n})},$

where $(F,\varepsilon)$ runs over the faces of ${\widetilde{\Delta}}$. Each
element of this basis owns an extra degree, coming from the filtration and
called perverse degree.

######  Definition 4.5.

Let $\ell\in\\{1,\dots,n\\}$. The _$\ell$ -perverse degree of the cochain
${\boldsymbol{1}}_{(F,\varepsilon)}\in{\widetilde{N}}^{*}(\Delta)$_ is equal
to

$\|{\boldsymbol{1}}_{(F,\varepsilon)}\|_{\ell}=\left\\{\begin{array}[]{ccl}-\infty&\text{if}&\varepsilon_{n-\ell}=1,\\\
|(F,\varepsilon)|_{>n-\ell}&\text{if}&\varepsilon_{n-\ell}=0.\end{array}\right.$

The _$\ell$ -perverse degree _ of
$\omega=\sum_{\mu}\lambda_{\mu}\,{\boldsymbol{1}}_{(F_{\mu}\varepsilon_{\mu})}\in{{\widetilde{N}}}^{{}^{*}}{(\Delta)}$,
with $\lambda_{\mu}\neq 0$, is equal to

$\|\omega\|_{\ell}=\max_{\mu}\|{\boldsymbol{1}}_{(F_{\mu},\varepsilon_{\mu})}\|_{\ell}.$

By convention, we set $\|0\|_{\ell}=-\infty$.

######  Remark 4.6.

Let us consider a face $(F,\varepsilon)$ of ${\widetilde{\Delta}}$ with
$F_{0}=\cdots=F_{m-1}=\emptyset$ for some $m\in\\{0,\ldots,n-1\\}$. From the
definition, we notice that the perverse degrees
$||{\boldsymbol{1}}_{(F,\varepsilon)}||_{\ell}$, $\ell\in\\{1,\ldots,n\\}$, do
not depend on the face $F_{m}$.

Let $X$ be a filtered space and ${\mathtt{P}}$ be a subset of
$\mathrm{Sing}^{{\mathcal{F}}}X$, a family of singular simplices of $X$,
stable by the face operators. The subfamily of its regular elements is denoted
${\mathtt{P}}_{+}$. In Example 4.9, we detail the examples of some subsets
${\mathtt{P}}$ that we will consider in this work. Essentially, they come from
the existence of a triangulation on the space $X$.

Let us define the blown-up cochain complex associated to
${\mathtt{P}}\subseteq\mathrm{Sing}^{{\mathcal{F}}}X$. First, to any regular
simplex, $\sigma\colon\Delta=\Delta_{0}\ast\dots\ast\Delta_{n}\to X$ in
${\mathtt{P}}$, we associate the cochain complex defined by

${{\widetilde{N}}}^{{}^{*}}_{{}_{\sigma}}={\widetilde{N}}^{*}(\Delta)={N}^{{}^{*}}{({\mathtt{c}}\Delta_{0})}\otimes\dots\otimes{N}^{{}^{*}}{({\mathtt{c}}\Delta_{n-1})}\otimes{N}^{{}^{*}}{(\Delta_{n})}.$

A face operator
$\delta_{\ell}\colon\nabla=\nabla_{0}\ast\dots\ast\nabla_{n}\to\Delta=\Delta_{0}\ast\dots\ast\Delta_{n}$
is _regular_ if $\nabla_{n}\neq\emptyset$. By restriction, we get the induced
operator

$\delta_{\ell}^{*}\colon{N}^{{}^{*}}{({\mathtt{c}}\Delta_{0})}\otimes\dots\otimes{N}^{{}^{*}}{({\mathtt{c}}\Delta_{n-1})}\otimes{N}^{{}^{*}}{(\Delta_{n})}\to{N}^{{}^{*}}{({\mathtt{c}}\nabla_{0})}\otimes\dots\otimes{N}^{{}^{*}}{({\mathtt{c}}\nabla_{n-1})}\otimes{N}^{{}^{*}}{(\nabla_{n})}.$

######  Definition 4.7.

Let $X$ be a filtered space and ${\mathtt{P}}$ be a sub $\Delta$-set of
$\mathrm{Sing}^{{\mathcal{F}}}X$. The _blown-up complex,_
${{\widetilde{N}}}^{{}^{*,{\mathtt{P}}}}{(X)}$, is the cochain complex formed
by the elements $\omega$, associating to any $\sigma\in{\mathtt{P}}$ an
element $\omega_{\sigma}\in{{\widetilde{N}}}^{{}^{*}}_{{}_{\sigma}}$, so that
$\delta_{\ell}^{*}(\omega_{\sigma})=\omega_{\sigma\circ\delta_{\ell}},$ for
any regular face operator, $\delta_{\ell}$, of ${\mathtt{P}}$. The
differential of ${{\widetilde{N}}}^{{}^{*,{\mathtt{P}}}}{(X)}$ is defined by
$(\delta\omega)_{\sigma}=\delta(\omega_{\sigma})$. The _perverse degree of
$\omega\in{{\widetilde{N}}}^{{}^{*,{\mathtt{P}}}}{(X)}$ along a singular
stratum,_ $S\in{\mathcal{S}}_{X}$, is equal to

$\|\omega\|_{S}=\sup\left\\{\|\omega_{\sigma}\|_{{\rm
codim\,}_{v}S}\mid\sigma\in{\mathtt{P}}_{+}\text{ with }{\rm Im\,}\sigma\cap
S\neq\emptyset\right\\}.$

We denote by $\|\omega\|$ the map associating to any singular stratum $S$ of
$K$ the element $\|\omega\|_{S}$ and 0 to a regular stratum.

######  Definition 4.8.

Let $\overline{p}$ be a perversity on a filtered space $X$ and ${\mathtt{P}}$
be a sub $\Delta$-set of $\mathrm{Sing}^{{\mathcal{F}}}X$. A cochain
$\omega\in{{\widetilde{N}}}^{{}^{*,{\mathtt{P}}}}{(X)}$ is _$\overline{p}$
-allowable_ if $\|\omega\|\leq\overline{p}.$ A cochain $\omega$ is of
_$\overline{p}$ -intersection_ if $\omega$ and its coboundary, $\delta\omega$,
are $\overline{p}$-allowable.

We denote by ${\widetilde{N}}^{*,{\mathtt{P}}}_{\overline{p}}(X)$ the complex
of $\overline{p}$-intersection cochains and
$\mathscr{H}_{\overline{p}}^{*,{\mathtt{P}}}(X)$ its homology, called the
_blown-up intersection cohomology_ of $X$ with coefficients in $\sf R$, for
the perversity $\overline{p}$.

######  Example 4.9.

Let $X$ be a filtered space and $\overline{p}$ be a perversity on $X$. In the
sequel, we consider the following cases of subsets
${\mathtt{P}}\subset\mathrm{Sing}^{{\mathcal{F}}}X$.

  * a)

If ${\mathtt{P}}=\mathrm{Sing}^{{\mathcal{F}}}X$, we recover the complex of
blown-up $\overline{p}$-intersection cochains of $X$ and the blown-up
intersection cohomology defined in [4]. We denote them
${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(X\right)}$ and
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(X\right)}$. If there is an
ambiguity with the simplicial situation introduced in the next item, we will
call them _the singular blown-up intersection complex and the singular blown-
up intersection cohomology._

  * b)

Let $X=|K|$ be the geometric realization of a full filtered complex $K$. The
family of singular simplices induced by the simplices of $K$ is denoted by
${\mathrm{Simp}\,}K$. Since $K$ is full, this is a sub $\Delta$-set of
$\mathrm{Sing}^{{\mathcal{F}}}|K|$ and we can apply the previous process with
${\mathtt{P}}={\mathrm{Simp}\,}K$. We denote
${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}$ and
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}$ the associated
complex and cohomology and call them _the simplicial blown-up intersection
complex and the simplicial blown-up intersection cohomology._

In Theorem B, we prove the existence of an isomorphism between
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}$ and
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}$, i.e., between
the singular and the simplicial blown-up intersection cohomologies.

######  Proposition 4.10.

Let $(K,\overline{p})$ be a perverse full filtered complex. Then, the
restrictions

  *   * a)

$\gamma\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|{\mathcal{L}}(K)|\right)}$,
and

  *   * b)

$\gamma\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathcal{L}}(K)\right)}$,

  *

are onto maps.

###### Proof.

a) It suffices to prove that the extension,
$\eta\in{{\widetilde{N}}}^{{}^{*}}{\left(|K|\right)}$, by 0 of a cochain
$\omega\in{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|{\mathcal{L}}(K)|\right)}$
belongs to ${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}$.
We clearly have $||\eta||\leq||\omega||$. Now, we want to bound
$\|\delta_{|K|}\eta\|$. For that, we write

$\delta_{|K|}\eta=\delta_{|{\mathcal{L}}(K)|}\eta+(\delta_{|K|}-\delta_{|{\mathcal{L}}(K)|})(\eta)$

and notice first that
$\|\delta_{|{\mathcal{L}}(K)|}\eta\|\leq\|\delta_{|{\mathcal{L}}(K)|}\omega||$.
As the perverse degree of a linear combination is the maximum of the perverse
degrees of its terms (see Definition 4.7), we are reduced to study
$\|(\delta_{|K|}-\delta_{|{\mathcal{L}}(K)}|)(\eta)\|$. We claim that

$\|(\delta_{|K|}-\delta_{|{\mathcal{L}}(K)|})(\eta)\|\leq||\omega||,$ (21)

which gives the result. Without loss of the generality, it suffices to prove
that

$||\delta_{|{\mathcal{L}}(K)|}{\boldsymbol{1}}_{(F,\varepsilon)}-\delta_{|K|}{\boldsymbol{1}}_{(F,\varepsilon)}||\leq||{\boldsymbol{1}}_{(F,\varepsilon)}||,$

where $\sigma\colon\Delta\to|{\mathcal{L}}(K)|$ is a regular singular simplex
and $(F,\varepsilon)$ is a face of the blown-up ${\widetilde{\Delta}}$. Notice
that $\sigma(F)\subset|{\mathcal{L}}(K)|$.

Let ${\mathrm{comp}\,}K=(a,b)$. Since $K_{n-a-1}=\emptyset$, the
$\sigma$-decomposition of $\Delta$ verifies
$\Delta_{0}=\cdots=\Delta_{n-a-1}=\emptyset$. We also have
$\sigma(\Delta_{n-a})\subset K^{(b-1)}_{n-a}$. Hence,
$F_{0}=\cdots=F_{n-a-1}=\emptyset$ and $\sigma(F_{n-a})\subset
K^{(b-1)}_{n-a}$. An element of
$\delta_{|{\mathcal{L}}(K)|}{\boldsymbol{1}}_{(F,\varepsilon)}-\delta_{|K|}{\boldsymbol{1}}_{(F,\varepsilon)}$
is of the form $\pm{\boldsymbol{1}}_{(H,\tau)}$ and we can suppose, without
loss of generality, that $(H,\tau)$ is a face of ${\widetilde{\Delta}}$. This
face verifies

  * +

$(F,\varepsilon)\triangleleft(H,\tau)$,

  * +

$\dim(H,\tau)=\dim(F,\varepsilon)+1$, and

  * +

$\sigma(H)\not\subset|{\mathcal{L}}(K)|$, that is,
$\sigma(H_{n-a}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\not\subset
K^{(b-1)}_{n-a}$.

This implies $H_{n-a}\neq F_{n-a}$, $H_{i}=F_{i}$, for $i\neq n-a$, and
$\tau=\varepsilon$. Since
$||{\boldsymbol{1}}_{(H,\tau)}||=||1_{(F,\varepsilon)}||$ (cf. Remark 4.6)
then we get the claim.

b) Same proof. ∎

We get the following exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|,|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(22)

The homology
${H}^{{}^{*}}{({{\widetilde{N}}}^{{}^{\cdot}}_{{}_{\overline{p}}}{\left(|K|,|{\mathcal{L}}(K)|\right)})}$
is isomorphic to the relative blown-up $\overline{p}$-intersection cohomology
introduced in [4]. We denote it
${\mathscr{H}}^{*}_{\overline{p}}(|K|,{\mathcal{L}}(|K|)$. The homology of the
complex
${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}$
is
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}$.

######  Proposition 4.11.

Let $(K,\overline{p})$ be a perverse full filtered complex. The restriction
map induces the isomorphisms

  * i)

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|,|{\mathcal{L}}(K)|\right)}\xrightarrow{\cong}\displaystyle\prod_{\beta\in{\mathcal{B}}(K)}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\alpha}|,|{\mathcal{L}}(\beta*L_{\beta})|\right)}$,
and

  *   * ii)

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}\xrightarrow{\cong}\displaystyle\prod_{\beta\in{\mathcal{B}}(K)}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta})\right)}$.

###### Proof.

i) We proceed as in the proof of Proposition 2.3 with the same notations.
Applying [4, Proposition 11.3], we get that the inclusion induces the
isomorphism
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(W\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|{\mathcal{L}}(K)|\right)}$
which gives an isomorphism

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|,W\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|,|{\mathcal{L}}(K)|\right)}.$

By excision (cf. [4, Proposition 12.9]) we have

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|,W\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\backslash|{\mathcal{L}}(K)|,W\backslash|{\mathcal{L}}(K)|\right)}.$

Using the equalities (17) we get

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\backslash|{\mathcal{L}}(K)|,W\backslash|{\mathcal{L}}(K)|\right)}\cong\prod_{\beta\in{\mathcal{B}}(K)}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|\backslash|\partial*L_{\beta}|,|*L_{\beta}|\backslash(\\{b_{\beta}\\}\cup|\partial*L_{\beta}|\right)}.$

Using excision relatively to the closed subset $|\partial\beta*L_{\beta}|$ we
obtain

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta})\backslash|\partial\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash(\\{b_{\beta}\\}\cup|\partial\beta*L_{\beta}||)\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}\right)}.$

Finally, applying [4, Theorem D] to (16) we get

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|,|\beta*L_{\beta}|\backslash\\{b_{\beta}\\}\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|,|\partial\beta*L_{\beta}|\right)}$

and therefore the isomorphism a).

ii) The complex
${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}$
is made up of the cochains of $K$ vanishing on ${\mathcal{L}}(K)$. From (4) we
get
${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}\cong\prod_{\beta\in{\mathcal{B}}(K)}{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta})\right)}$
and therefore

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K,{\mathcal{L}}(K)\right)}\cong\prod_{\beta\in{\mathcal{B}}(K)}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta})\right)},$

by restriction. ∎

######  Proposition 4.12.

Consider a perverse full filtered complex $(K,\overline{p})$. Let
$\beta\in{\mathcal{B}}(K)$ be a clot. The stratum $Q\in{\mathcal{S}}_{K}$
containing $\beta$ is a singular stratum . Then

  * i)

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|\right)}\cong\left\\{\begin{array}[]{ll}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|L_{\beta}|\right)}&\hbox{if
}*\leq\overline{p}(Q)\\\ 0&\hbox{if not}\end{array}\right.$

  *   * ii)

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta}\right)}\cong\left\\{\begin{array}[]{ll}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}&\hbox{if
}*\leq\overline{p}(Q)\\\ 0&\hbox{if not}\end{array}\right.$

where the isomorphism is induced by the natural inclusion
$L_{\beta}\hookrightarrow\beta*L_{\beta}$.

###### Proof.

If $L_{\beta}=\emptyset$ then
${N}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta}\right)}={N}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta\right)}=0$
since there are no regular simplices ($Q$ is singular). So, we can suppose
$L_{\beta}\neq\emptyset$. Let ${\mathrm{comp}\,}K=(a,b)$.

Let ${\mathrm{comp}\,}K=(a,b)$. Without loss of generality we can suppose that
$a=n$ (cf. [4, 3.5 Shifted filtrations]).

i) We proceed as in the proof of Proposition 2.4 with the same notations. The
case $\dim\beta=0$ is given by [4, Theorem E] since $L_{\beta}\neq\emptyset$.
When $\dim\beta>0$ and using [4, Proposition 11.3] we get that
$(F(-,1))^{*}\colon{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|\right)}\to{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|\right)}$
is the identity. Finally, since
${\mathrm{id}}\colon|\eta*L_{\beta}|\hookrightarrow|\beta*L_{\beta}|\xrightarrow{\xi}|\eta*L_{\beta}|$
and
$F(-,1)\colon|\beta*L_{\beta}|\xrightarrow{\xi}|\eta*L_{\beta}|\hookrightarrow|\beta*L_{\beta}|$
we conclude that the inclusion
$|\eta*L_{\beta}|\hookrightarrow|\beta*L_{\beta}|$ induces the isomorphism
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\eta*L_{\beta}|\right)}$.

ii) By definition, we have

${{\widetilde{N}}}^{{}^{*}}{(\beta*L_{\beta})}={N}^{{}^{*}}{({\mathtt{c}}\beta)}\otimes{{\widetilde{N}}}^{{}^{*}}{(L_{\beta})}.$

Let us study how to compute the perverse degree on the right side.
Essentially, there are two kinds of cochains:

  * -

${\boldsymbol{1}}_{(F,0)}\otimes\tau$ where $F\triangleleft\beta$ and
$\tau\in{{\widetilde{N}}}^{{}^{*}}{(L_{\beta})}$, and

  * -

${\boldsymbol{1}}_{(F,1)}\otimes\tau$ where $F\triangleleft\beta$ or
$F=\emptyset$ and $\tau\in{{\widetilde{N}}}^{{}^{*}}{(L_{\beta})}$.

Any cochain of ${{\widetilde{N}}}^{{}^{*}}{(\beta*L_{\beta})}$ is a linear
combination of them. The perverse degree is computed as follows. We have

  * -

$||{\boldsymbol{1}}_{(F,0)}\otimes\tau||_{\ell}=\left\\{\begin{array}[]{ll}\deg\tau&\hbox{if
}\ell=n\\\ ||\tau||_{\ell}&\hbox{if }\ell<n\end{array}\right.$

  *   * -

$||{\boldsymbol{1}}_{(F,1)}\otimes\tau||_{\ell}=\left\\{\begin{array}[]{ll}-\infty&\hbox{if
}\ell=n\\\ ||\tau||_{\ell}&\hbox{if }\ell<n\end{array}\right.$

for each $\ell\in\\{1,\ldots,n\\}$.

A cochain
$\omega\in{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta}\right)}$
can be written as the sum

$\sum_{F\triangleleft\beta}{\boldsymbol{1}}_{(F,0)}\otimes\omega_{F}+\sum_{F\triangleleft\beta}{\boldsymbol{1}}_{(F,1)}\otimes\tau_{F}+{\boldsymbol{1}}_{(\emptyset,1)}\otimes\tau_{\emptyset}.$

Following the previous equalities, the condition $||\omega||\leq\overline{p}$
gives

  * -

$\max(||\omega_{F}||,||\tau_{F}||,||\tau_{\emptyset}||)\leq\overline{p}$ , and

  * -

$\deg\omega_{F}\leq\overline{p}(Q)$.

for each $F\triangleleft\beta$. Now, the condition
$||\delta\omega||\leq\overline{p}$ gives

  * -

$\max(||\delta\omega_{F}||,||\delta\tau_{F}||,||\delta\tau_{\emptyset}||)\leq\overline{p}$,
and

  * -

$\deg(\delta\omega_{F})\leq\overline{p}(Q)$.

In particular,
$\omega_{F}\in\tau_{\overline{p}(Q)}{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}={{\widetilde{N}}}^{{}^{<\overline{p}(Q)}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}\oplus\left({{\widetilde{N}}}^{{}^{\overline{p}(Q)}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}\cap\delta^{-1}(0)\right)$
and
$\tau_{F},\tau_{\emptyset}\in{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}$
for each $F\triangleleft\beta$.

The $\sf R$-module ${N}^{{}^{*}}{({\mathtt{c}}\beta,\beta)}$ is made up of the
cochains on ${\mathtt{c}}\beta$ vanishing on $\beta$. It is generated by the
family $\\{{\boldsymbol{1}}_{(F,1)}\mid F=\emptyset\hbox{ or
}F\triangleleft\beta\\}$. Let us consider the two short exact sequences,

$0\to{N}^{{}^{*}}{({\mathtt{c}}\beta,\beta)}\hookrightarrow{N}^{{}^{*}}{({\mathtt{c}}\beta)}\xrightarrow{\nu}{N}^{{}^{*}}{(\beta)}\to
0,$

where
$\nu\left(\sum_{F\triangleleft\beta}u_{F}\,{\boldsymbol{1}}_{(F,0)}+\sum_{F\triangleleft\beta}v_{F}\,{\boldsymbol{1}}_{(F,1)}+v_{\emptyset}\,{\boldsymbol{1}}_{(\emptyset,1)}\right)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\sum_{F\triangleleft\beta}u_{F}\,{\boldsymbol{1}}_{F},$
and

$0\to{N}^{{}^{*}}{({\mathtt{c}}\beta,\beta)}\otimes{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}\hookrightarrow{N}^{{}^{*}}{(\beta*L_{\beta})}\xrightarrow{\upsilon}{N}^{{}^{*}}{(\beta)}\otimes\tau_{\overline{p}(Q)}{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(L_{\beta}\right)}\to
0,$

where

$\upsilon\left(\sum_{F\triangleleft\beta}{\boldsymbol{1}}_{(F,0)}\otimes\omega_{F}+\sum_{F\triangleleft\beta}{\boldsymbol{1}}_{(F,1)}\otimes\tau_{F}+{\boldsymbol{1}}_{(\emptyset,1)}\otimes\tau_{\emptyset}\right)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\sum_{F\triangleleft\beta}{\boldsymbol{1}}_{(F,0)}\otimes\omega_{F}.$

Since the map $\nu$ is acyclic then the complex
${N}^{{}^{*}}{({\mathtt{c}}\beta,\beta)}$ is acyclic. So, the map $\upsilon$
is a quasi-isomorphism. The result comes from
${H}^{{}^{*}}{(\beta)}={H}^{{}^{0}}{(\beta)}=\sf R$. ∎

When $Q$ is a regular stratum we have $L_{\beta}=\emptyset$ and
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta*L_{\beta}|\right)}={\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|\beta|\right)}={H}^{{}^{*}}{(|\beta|)}={H}^{{}^{0}}{(|\beta|)}=R$
and similarly
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta*L_{\beta}\right)}={\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(\beta\right)}={H}^{{}^{*}}{(\beta)}={H}^{{}^{0}}{(\beta)}=R$

The second main result of this work establishes an isomorphism between the
singular and the simplicial blown-up intersection cohomologies. These
cohomolgies are related by the following map.

######  Definition 4.13.

Let $(K,\overline{p})$ be a perverse full filtered complex . The cochain map
$\rho\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}$
comes form the natural inclusion
${\mathrm{Simp}\,}K\subset\mathrm{Sing}^{{\mathcal{F}}}|K|$ (cf. (19)).

######  Theorem B.

Let $(K,\overline{p})$ be a perverse full filtered complex. Then, the map
$\rho$ induces the isomorphism

${\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}{\cong}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}.$
(23)

###### Proof.

We proceed in two steps.

_First Step : ${\mathrm{comp}\,}K=(a,b)$ with $b<\infty$_. Given a sub-complex
$L\subset K$ we also have $b_{L}\leq\dim L<\infty$, with
${\mathrm{comp}\,}L=(a_{L},b_{L})$, and therefore
${\mathrm{comp}\,}{\mathcal{L}}(L)<{\mathrm{comp}\,}L$ (cf. Section 1.4).

We use the induction on the complexity $(a,b)$ of $K$. When $b=0$ then $K$ is
a discrete family of 0-dimensional simplices. So,
${N}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}={N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}$
and we get the claim.

We consider the inductive step. From (22) we get the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|,|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|K|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{C}^{{}^{\overline{p}}}_{{}_{*}}{\left(|{\mathcal{L}}(K)|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(K,{\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathcal{L}}(K)\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the vertical maps are induced by the map $\rho$. From the induction
hypothesis we know that the right arrow is a quasi-isomorphism. So, it
suffices to prove that the left arrow is a quasi-isomorphism. Using
Proposition 4.11 this assertion is equivalent to the fact that the operator
map $\rho$ induces the quasi-isomorphism

$\bigoplus_{\beta\in{\mathcal{B}}(K)}{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta}|\right)}\to\bigoplus_{\beta\in{\mathcal{B}}(K)}{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta}\right)}.$

In fact, it suffices to prove that $\rho$ induces the quasi-isomorphism

${N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta}|\right)}\to{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta}\right)},$

for each $\beta\in{\mathcal{B}}(K)$. From (22) we get the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|,|{\mathcal{L}}(\beta*L_{\beta})|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|\beta*L_{\beta}|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(|{\mathcal{L}}(\beta*L_{\beta})|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta},{\mathcal{L}}(\beta*L_{\beta})\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left(\beta*L_{\beta}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{N}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathcal{L}}(\beta*L_{\beta})\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the vertical maps are induced by $\rho$. We have that
${\mathrm{comp}\,}{\mathcal{L}}(\beta*L_{\beta})<{\mathrm{comp}\,}(\beta*L_{\beta})\leq{\mathrm{comp}\,}K$
(cf. Section 1.2). From the induction hypothesis we know that the right arrow
is a quasi-isomorphism. It remains to prove that the middle arrow is a quasi-
isomorphism. We distinguish two cases.

$\bullet$ _$\beta$ is included in a regular stratum_. We have
$L_{\beta}=\emptyset$ and the middle arrow becomes
$\rho\colon{N}^{{}^{*}}{\left(|\beta|\right)}={\rm
Hom}\left({C}_{{}_{*}}{(|\beta|)};R\right)\to{N}^{{}^{*}}{\left(\beta\right)}={\rm
Hom}\left({C}_{{}_{*}}{(\beta)};R\right)$ which is an isomorphism (see [14,
Theorem 2.27, Corollary 3.4] for instance).

$\bullet$ _$\beta$ is included in a singular stratum_. We have
${\mathrm{comp}\,}L_{\beta}<{\mathrm{comp}\,}K$ (cf. Section 1.3) From
Proposition 4.12 and the induction hypothesis we know that the middle arrow is
a quasi-isomorphism.

_Second Step : ${\mathrm{comp}\,}K=(a,\infty)$_. Notice that we have
$\sigma_{0}=\cdots=\sigma_{n-a-1}=\emptyset$ for each $\sigma\in K$.

We proceed by induction on $a\in\\{0,\ldots,n\\}$. When $a=0$ then $K$ does
not have a singular part, that is, $K_{n-1}=\emptyset$. So, we need to prove
that the $\rho$ induces the isomorphism
${H}^{{}^{*}}{\left(|K|\right)}{\cong}{H}^{{}^{*}}{\left(K\right)}$. This is
the classical situation we can find in [14, Theorem 2.27, Corollary 3.4] for
instance.

We consider the inductive step $a>0$.

Given $k\in\mathbb{N}$ we define $K^{k}=\\{\sigma\in K\mid\dim\sigma_{n-a}\leq
k\\}$ and $K^{-1}=\\{\sigma\in K\mid\sigma_{n-a}=\emptyset\\}$. They are
simplicial sub-complexes of $K$. We have the infinity sequence

$K^{-1}\subset K^{0}\subset\cdots\subset K^{k}\subset K^{k+1}\subset\cdots,$

with $K=\bigcup_{k\geq-1}K^{k}$.

We consider on each of theses complexes the induced structure defined in
Section 1.2. We prove that the filtered complex $K^{k}$, with
$k\in\mathbb{N}\cup\\{-1\\}$, verifies (23). Given $k\in\mathbb{N}$ with
$K^{k}\neq K^{k-1}$ we have ${\mathrm{comp}\,}(K^{k})=(a,k)$. Following the
First Step we conclude that $K^{k}$ verifies (23). Let us study the case
$k=-1$. By construction, we have
${\mathrm{comp}\,}(K^{-1})=(a_{-},b_{-})<(a,0)$. When $b_{-}<\infty$ then the
First Step gives the claim. On the other hand, if $b_{-}=\infty$ then the
inductive step assures us that $K^{-1}$ verifies (23).

Given $k\in\mathbb{N}$ with $K^{k}\neq K^{k-1}$ then
${\mathcal{L}}(K^{k})=K^{k-1}$ by definition. We get from Proposition 4.10
that the morphisms

$\rho^{*}\colon{N}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{k}\right)}\to{N}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{k-1}\right)}\hskip
28.45274pt\hbox{ and }\hskip
28.45274pt\rho^{*}\colon{N}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K^{k}|\right)}\to{N}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K^{k-1}|\right)},$
(24)

induced by the inclusion $K^{k-1}\hookrightarrow K^{k}$ are onto maps.

Associated to the directed set $K=\bigcup_{k\geq-1}K^{k}=\varinjlim K^{k}$ we
have the towers
$({N}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{k}\right)})_{k\geq-1}$ and
$({N}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K^{k}|\right)})_{k\geq-1}$. The
previous property (24) says that that they are Mittag-Leffler towers. So, we
have the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\varprojlim}^{1}{\mathscr{H}}^{{}^{*-1}}_{{}_{\overline{p}}}{\left(|K^{k}|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(|K^{k}|\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\varprojlim}^{1}{\mathscr{H}}^{{}^{*-1}}_{{}_{\overline{p}}}{\left(K^{k}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{k}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the vertical maps are induced by $\rho$. We have seen that the left and
right maps are isomorphisms by the first step. The Five’s Lemma ends the
proof. ∎

##  5\. Definitions of filtered PL spaces

######  Definition 5.1.

Let $K$ and $L$ be Euclidean simplicial complexes. A simplicial map, $f\colon
K\to L$ is a continuous function $f\colon|K|\to|L|$ that sends vertices of $K$
on vertices of $L$ and is linear on each simplex of $K$; i.e., $f\colon\langle
a_{0},\dots,a_{k}\rangle\to L$ maps $x=\sum_{i=0}^{k}t_{i}a_{i}$ to
$f(x)=\sum_{i=0}^{k}t_{i}f(a_{i})$.

######  Definition 5.2.

A _triangulation_ of $X$ is a pair $(K,h)$ of a Euclidean simplicial complex,
$K$, and a homeomorphism, $h\colon|K|\to X$. A _subdivision_ of $(K,h)$ is a
pair $(L,h)$ where $L$ is a subdivision of $K$, denoted $L\vartriangleleft K$.
Two triangulations, $(K,h)$ and $(L,f)$, of $X$ are _equivalent_ if
$f^{-1}h\colon|K|\to X\to|L|$ is induced by a simplicial isomorphism $K\to L$.

######  Definition 5.3.

A _PL space_ is a second countable, Hausdorff topological space, $X$, endowed
with a family, ${\mathcal{T}}$, of triangulations of $X$ such that the
following properties are satisfied.

  1. (a)

If $(K,h)$ in ${\mathcal{T}}$ and $L\vartriangleleft K$, then
$(L,h)\in{\mathcal{T}}$.

  2. (b)

If $(K,h)\in{\mathcal{T}}$ and $(L,f)\in{\mathcal{T}}$, then there is a common
subdivision of $K$ and $L$ in ${\mathcal{T}}$.

######  Definition 5.4.

Let $(X,{\mathcal{T}})$ and $(Y,{\mathcal{S}})$ be two PL spaces. A _PL map_ ,
$\psi\colon(X,{\mathcal{T}})\to(Y,{\mathcal{S}})$, is a continuous map,
$\psi\colon X\to Y$, such that for any $(K,h)\in{\mathcal{T}}$ and any
$(L,f)\in{\mathcal{S}}$, there is a subdivision $K^{\prime}$ of $K$ for which
$j^{-1}\psi h$ takes each simplex of $K^{\prime}$ linearly _into_ a simplex of
$L$.

A _PL subspace_ of $(X,{\mathcal{T}})$ is a PL space
$(X^{\prime},{\mathcal{T}}^{\prime})$, such that $X^{\prime}$ is a subspace of
$X$ and the inclusion map, $X^{\prime}\hookrightarrow X$, a PL map.

######  Definition 5.5.

A _filtered PL space_ is a PL space $(X,{\mathcal{T}})$, filtered by a
sequence of closed PL subspaces,

$X=X_{n}\supsetneq X_{n-1}\supset\dots\supset X_{0}\supset X_{-1}=\emptyset.$

From [9, Subsection 2.5.2], we may suppose that there is a triangulation
$(K,h)$ of $X$ with respect to which each of the $X_{i}$ is the image under
$h$ of a subcomplex pf $K$.

###  5.1. Blow up and subdivision

The Euclidean simplicial complexes that we use in this section, are not
supposed to be oriented, but we use oriented simplices. As a simplex is
determined by its vertices we denote it by $[v_{0},\dots,v_{\ell}]$. This
notation exhibits an order on the set of vertices. Thus, we identify two
writings which correspond to an even permutation of the vertices. For an odd
permutation, $\nu$, we set
$[v_{0},\dots,v_{\ell}]=-[v_{\nu(0)},\dots,v_{\nu(\ell)}]$, which makes sense
at the level of chains.

Let us connect the blow up cochain complexes of a Euclidean simplicial complex
and of one of its subdivisions.

######  Proposition 5.6.

Let $K^{\prime}$ be a subdivision of a full filtered Euclidean simplicial
complex $K$ and $\overline{p}$ be a perversity on $|K|$. There exist cochain
maps

$j_{K^{\prime}K}\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{\prime}\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\quad\text{and}\quad\varphi_{KK^{\prime}}\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{\prime}\right)}$

such that $j_{K^{\prime}K}\circ\varphi_{KK^{\prime}}={\mathrm{id}}$.

###### Proof.

Let $\sigma\colon\Delta_{\sigma}\to K$ be an oriented, regular, filtered
simplex of $K$. The filtration on $K$ induces a decomposition in join product,
$\Delta_{\sigma}=\Delta_{\sigma,0}\ast\dots\ast\Delta_{\sigma,n}$, where each
$\Delta_{\sigma,i}$ is included in a stratum $S_{i}$. We identify
$\Delta_{\sigma}$ with its image.

The simplex $\sigma$ is subdivided in some oriented simplices
$(\sigma\\{\ell\\})_{1\leq\ell\leq k}$ of $K^{\prime}$, of equal dimension.
Let $\ell\in\\{1,\dots,k\\}$. We can suppose that the orientations of $\sigma$
and $\sigma\\{\ell\\}$ are compatible. Each simplex of the subdivision is
obtained by adding new vertices to a subset of ${\mathrm{Vert}\,}(K)$. Any of
these new vertices belongs to a stratum and we can write
$\Delta_{\sigma\\{\ell\\}}$ as a join product
$\Delta_{\sigma\\{\ell\\}}=\Delta_{\sigma\\{\ell\\},0}\ast\dots\ast\Delta_{\sigma\\{\ell\\},n}$.

$\bullet$ _Construction of $j_{K^{\prime}K}$._ For any $\varepsilon$ as in
Definition 4.3 and any $\ell\in\\{1,\dots,k\\}$, we set

$j\left({\boldsymbol{1}}_{(\Delta_{\sigma\\{\ell\\}},\varepsilon)}\right)={\boldsymbol{1}}_{(\Delta_{\sigma},\varepsilon)}.$

We get a cochain map
$j\colon{{\widetilde{N}}}^{{}^{*}}{(K^{\prime})}\to{{\widetilde{N}}}^{{}^{*}}{(K)}$.
Concerning the perverse degrees, let us notice that
$\dim(\Delta_{\sigma\\{\ell\\}}\cap S)\leq\dim(\Delta_{\sigma}\cap S)$. Thus,
we have
$\|{\boldsymbol{1}}_{(\Delta_{\sigma},\varepsilon)}\|\leq\|{\boldsymbol{1}}_{(\Delta_{\sigma\\{\ell\\}},\varepsilon)}\|$
for any $\ell\in\\{1,\dots,k\\}$. Therefore the association $j$ gives a
cochain map

$j_{K^{\prime}K}\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{\prime}\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}.$

$\bullet$ _Construction of $\varphi_{KK^{\prime}}$._ We use a simplicial map,
$\nu\colon K^{\prime}\to K$, built by Goresky and MacPherson in [16,
Appendix]. Let us recall its construction, using [9, Subsection 3.3.4],
assuming that $K$ is full and that the set ${\mathrm{Vert}\,}(K)$ is well
ordered.

The map $\nu$ is defined from a map
$\overline{\nu}\colon{\mathrm{Vert}\,}(K^{\prime})\to{\mathrm{Vert}\,}(K)$
between the set of vertices of $K$ and $K^{\prime}$, that is extended linearly
on simplices. If the vertex $v^{\prime}\in K^{\prime}$ is already in $K$, we
set $\overline{\nu}(v^{\prime})=v^{\prime}$. Otherwise, $v^{\prime}$ is in the
interior of a simplex $\sigma$ of $K$. Denote by $S$ the stratum of $|K|$
containing the interior of $\sigma$.

One knows (see [16] or [9, Lemma 3.3.25]) that the interior of a simplex is
contained in the stratum $S$ if, and only if, all the vertices of $\sigma$ are
contained in the closure $\overline{S}$ and at least one vertex of $\sigma$ is
in $S$. We define $\overline{\nu}(v^{\prime})$ as the vertex of $\sigma$ in
$S$ that is greatest in the selected order. As the vertices $v^{\prime}$ and
$\overline{\nu}(v^{\prime})$ are in the same stratum (see [9, Proof of Lemma
3.3.21]), the map $\overline{\nu}$ is compatible with the strata
decomposition. Also, from the same proof, it is explicit that _only one_ of
the simplices $\Delta_{\sigma\\{\ell\\}}$ of the subdivision has an image by
$\nu$ which is of the same dimension than $\Delta_{\sigma}$. Let us call it
$\Delta_{\sigma\\{m\\}}$ and set

$\varphi_{KK^{\prime}}\left({\boldsymbol{1}}_{(\Delta_{\sigma},\varepsilon)}\right)={\boldsymbol{1}}_{(\Delta_{\sigma\\{m\\}},\varepsilon)}.$

As $\nu\colon K^{\prime}\to K$ is a simplicial map, compatible with the
strata, we have
$\|{\boldsymbol{1}}_{(\Delta_{\sigma},\varepsilon)}\|=\|{\boldsymbol{1}}_{(\Delta_{\sigma\\{m\\}},\varepsilon)}\|$
and a cochain map

$\varphi_{KK^{\prime}}\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K^{\prime}\right)}.$

The equality $j_{K^{\prime}K}\circ\varphi_{KK^{\prime}}={\mathrm{id}}$ follows
directly from the definitions of the two maps. ∎

###  5.2. PL blown-up cohomology and simplicial cohomology

Let $(X,{\mathcal{T}})$ be a PL filtered space and $K$ be any triangulation of
${\mathcal{T}}$. We denote by ${\mathrm{sd}}\;K$ the barycentric subdivision
of $K$ and by $({\mathrm{sd}}^{i}\;K)_{i\in\mathbb{N}}$ the family of iterated
barycentric subdivisions, with the convention
${\mathrm{sd}}^{0}={\mathrm{id}}$. The subdivision maps give a direct system
${\mathrm{Sub}}_{i}\colon{\mathrm{sd}}^{i+1}\;K\to{\mathrm{sd}}^{i}\;K$, for
$i\in\mathbb{N}$. Let $\overline{p}$ be a perversity on $X$. In [9, Lemma
5.4.1], Friedman proves that the PL homology is obtained as the homology of
the inductive limit of this direct system,
${H}^{{}^{\overline{p},{\mathrm{PL}}}}_{{}_{*}}{\left(X\right)}\cong{H}_{{}_{*}}{(\varinjlim_{i}{C}^{{}^{\overline{p}}}_{{}_{*}}{\left({\mathrm{sd}}^{i}\;K\right)})}$.

We suppose $K$ full and denote by
$j_{i}\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}\;K\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i-1}\;K\right)}$
and
$\varphi_{i}\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}\;K\right)}$
the maps of Proposition 5.6, for $i\in\mathbb{N}$. The maps $j_{i}$ define a
projective system which allows the following definition.

######  Definition 5.7.

Let $(X,{\mathcal{T}})$ be a PL filtered space and $\overline{p}$ be a
perversity on $X$. The complex of $\overline{p}$-intersection PL cochains is
the inverse (projective) limit,

${{\widetilde{N}}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}=\varprojlim_{i}{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}\;K\right)}.$

We denote
${\mathscr{H}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}$ the
corresponding cohomology, and call it the _blow up $\overline{p}$-intersection
PL cohomology._

From Lemma 1.1, we can suppose that $K$ is full. Let’s now prove that the blow
up $\overline{p}$-intersection PL cohomology is isomorphic to the blow up
singular and the blow up simplicial ones.

######  Theorem C.

Let $(X,{\mathcal{T}})$ be a PL filtered space and $\overline{p}$ be a
perversity on $X$. Then for a pure $K\in{\mathcal{T}}$, there are
isomorphisms,

${\mathscr{H}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(X\right)}\text{
and
}{H}^{{}^{\overline{p}}}_{{}_{*,{\mathrm{PL}}}}{\left(X\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(K\right)}\cong{H}^{{}^{\overline{p}}}_{{}_{*}}{\left(X\right)}.$

###### Proof.

Let $j_{\leq i}$ be the map defined by $j_{\leq 1}=j_{1}$ and $j_{\leq
i}=j_{\leq i-1}\circ j_{i}$. We also denote
$p_{i}\colon{{\widetilde{N}}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}(K)\right)}$
the projection given by the projective limit and
$\varphi_{i}\colon{{\widetilde{N}}}^{{}^{*}}{(K)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}(K)\right)}$.

|
---|---
$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}}$$\textstyle{{{\widetilde{N}}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{i-1}}$$\scriptstyle{p_{i}}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(sd^{i-1}\;K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\leq
i-1}}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{id}}}$$\scriptstyle{\varphi_{i-1}}$$\scriptstyle{\varphi_{i}}$$\scriptstyle{\Psi}$$\textstyle{{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}\;K\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{i}}$
(25)

From the equalities $j_{i}\circ\varphi_{i}=\varphi_{i-1}$ and the universal
property of inverse limits, we get a cochain map
$\Psi\colon{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\to{{\widetilde{N}}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}$
such that $p_{0}\circ\Psi={\mathrm{id}}$ and $p_{i}\circ\Psi=\varphi_{i}$ for
any $i\in\mathbb{N}$. From Proposition 5.6 applied to ${\mathrm{sd}}^{i}\;K$
and ${\mathrm{sd}}^{i-1}\;K$, we deduce that each $j_{i}$ and each
${H}_{{}_{*}}{(j_{i})}$ is surjective. Thus [13, Proposition 13.2.3] and
Theorem A give the existence of isomorphisms,

${\mathscr{H}}^{{}^{*,{\mathrm{PL}}}}_{{}_{\overline{p}}}{\left(X\right)}={H}_{{}_{*}}{(\varprojlim_{i}{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}\;K\right)})}\cong\varprojlim_{i}{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left({\mathrm{sd}}^{i}\;K\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(K\right)}\cong{\mathscr{H}}^{{}^{*}}_{{}_{\overline{p}}}{\left(X\right)}.$

The existence of the isomorphisms in $\overline{p}$-intersection homology
follows directly from Theorem A and the commutativity of homology with
inductive limits. ∎

##  6\. Cup product

> Given a connected CS set $X$, we exhibit a morphism, which is an isomorphism
> when $X$ is normal, between the usual cohomology
> ${H}^{{}^{*}}{\left(X\right)}$ and the blown-up intersection cohomology
> ${\mathscr{H}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$ preserving the
> $\smile$ product. More than that, if we limit ourselves to the family of
> filtered simplices then we have an isomorphism at the complex level.

###  6.1. The $\smile$ product on a topological space

Let us consider a topological space $X$. The $\smile$-product is defined on
the complex ${C}^{{}^{*}}{(X)}={\rm Hom}({C}_{{}_{*}}{(X)};R)$ of singular
cochains. We prefer the following presentation of this product. First of all,
the complex ${C}^{{}^{*}}{(X)}$ can be thought as formed by the elements
$\omega$, associating to any singular simplex $\sigma\colon\Delta\to X$ an
element $\omega_{\sigma}\in{N}^{{}^{*}}{(\Delta)}$, so that
$\delta_{\ell}^{*}(\omega_{\sigma})=\omega_{\sigma\circ\delta_{\ell}},$ for
any face operator, $\delta_{\ell}$. Under this standpoint, the $\smile$
product is defined as follows:

$(\omega\smile\eta)_{\sigma}=\omega_{\sigma}\smile\eta_{\sigma},$

where $\omega,\eta\in{C}^{{}^{*}}{(X)}$ and $\sigma\colon\Delta\to X$ is a
singular simplex. The product $\omega_{\sigma}\smile\eta_{\sigma}$ is the
usual $\smile$ product on ${N}^{{}^{*}}{(\Delta)}$.

###  6.2. The $\smile$ product on a filtered space

We do not need to work with the complete family of singular simplices, it is
enough to consider the family $\mathrm{Sing}^{{\mathcal{F}}}$ of filtered
simplices of the filtered space $X$. We define

${N}^{{}^{*}}{(X)}=\left\\{(\omega_{\sigma})\mid\begin{array}[]{l}\sigma\colon\Delta\to
X\hbox{ regular simplex }\omega_{\sigma}\in{N}^{{}^{*}}{(\Delta)},\hbox{ so
that}\\\
\delta_{\ell}^{*}(\omega_{\sigma})=\omega_{\sigma\circ\delta_{\ell}},\hbox{
for any face operator, }\delta_{\ell}\end{array}\right\\}$

The $\smile$ product is defined as follows:

$(\omega\smile\eta)_{\sigma}=\omega_{\sigma}\smile\eta_{\sigma},$

where $\omega,\eta\in{N}^{{}^{*}}{(X)}$ and $\sigma\colon\Delta\to X$ is a
filtered simplex. The restriction map ${C}^{{}^{*}}{(X)}\to{N}^{{}^{*}}{(X)}$
preserves the $\smile$ product. Proceeding as in [4, Proposition 13.5], one
shows that this map is a quasi-isomorphism when $X$ is a paracompact separable
CS-set.

###  6.3. The $\smile$ product for the $\overline{0}$ perversity

The singular blown-up intersection cohomology
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$ is endowed with a
cup product $\smile$ defined as follows (cf. [4]):

$(\omega\smile\eta)_{\sigma}=\omega_{\sigma}\smile\eta_{\sigma},$

where
$\omega,\eta\in{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$
and $\sigma\colon\Delta_{\sigma}\to X$ is a regular simplex. The product
$\omega_{\sigma}\smile\eta_{\sigma}$ is the usual $\smile$ product on the
tensor product
${{\widetilde{N}}}^{{}^{*}}{(\Delta_{\sigma})}={N}^{{}^{*}}{({\mathtt{c}}\Delta_{0})}\otimes\cdots\otimes{N}^{{}^{*}}{({\mathtt{c}}\Delta_{n-1})}\otimes{N}^{{}^{*}}{(\Delta_{n})}$,
where $\Delta_{\sigma}=\Delta_{0}\ast\cdots\ast\Delta_{n}$ is the
$\sigma$-decomposition of $\Delta$.

###  6.4. The blown-up

Given a regular simplex $\Delta=\Delta_{0}*\cdots*\Delta_{n}$ we define the
_blown-up_

$\pi\colon{\widetilde{\Delta}}={\mathtt{c}}\Delta_{0}\times\cdots\times{\mathtt{c}}\Delta_{n-1}\times\Delta_{n}\longrightarrow\Delta=\Delta_{0}\ast\cdots\ast\Delta_{n},$

by

$\pi([P_{0},t_{0}],\ldots,[P_{n-1},t_{n-1}],P_{n})=(1-t_{0})P_{0}+t_{0}(1-t_{1})P_{1}+\cdots+t_{0}\cdots
t_{n-1}(1-t_{n})P_{n}.$

Recall that any point of a cone ${\mathtt{c}}\Delta_{i}$,
$i\in\\{0,\ldots,n-1\\}$, is written $(1-t)P+t{\mathtt{v}}_{i}=[P,t]$, where
$P\in\Delta_{i}$, $t\in[0,1]$ and ${\mathtt{v}}_{i}$ is the apex of the cone
${\mathtt{c}}\Delta_{i}$. This map preserves the faces and induce the operator

$\pi^{*}\colon{N}^{{}^{*}}{(\Delta)}\to{{\widetilde{N}}}^{{}^{*}}{(\Delta)},$

defined by

$\pi^{*}\left({\boldsymbol{1}}_{F_{0}\ast\cdots\ast
F_{n}}\right)=s\cdot\sum_{Q\in\widetilde{\eta}}{\boldsymbol{1}}_{(F_{0},1)}\otimes\cdots\otimes{\boldsymbol{1}}_{(F_{k-1},1)}\otimes{\boldsymbol{1}}_{(F_{k},0)}\otimes{\boldsymbol{1}}_{Q},$

if $F_{k}\neq\emptyset$ and $F_{k+1}=\ldots=F_{n}=\emptyset$. Here,
$s=(-1)^{\sum_{j=0}^{k-1}|(F_{j},1)|\cdot|F_{j+1}\ast\cdots\ast F_{k}|}$ and
$\eta=\Delta_{k+1}\ast\cdots\ast\Delta_{n}$.

We have seen that this operator is an isomorphism (cf. [3, Proposition 3.1]).
It comes from the transformation
$\Delta_{0}*\cdots*\Delta_{n}\rightsquigarrow\Delta$ which is a
$\mu$-amalgamation in the sense of [4, Section 8.1]. Then, we know that the
operator $\pi^{*}$ preserves the $\smile$ product (cf. [4, Proposition 7.5]).

The global blown-up

$\pi^{*}\colon{N}^{{}^{*}}{(X)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$

is given by $\pi^{*}(\omega)_{\sigma}=\pi^{*}(\omega_{\sigma})$, where
$\sigma\colon\Delta\to X$ is a regular simplex. This operator also preserves
the $\smile$ product since it is a local property.

The surprising point is that the global blown-up is also an isomorphism.

######  Proposition 6.1.

Let us consider a normal CS set $X$. The blown-up

$\pi^{*}\colon{N}^{{}^{*}}{(X)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$

is an isomorphism preserving the $\smile$ product.

If $X$ is also paracompact and separable we have already proof in [4,
Proposition 13.5] that $\pi^{*}$ induces the isomorphism of algebras
${\mathscr{H}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}\cong{H}^{{}^{*}}{(X)}$.

_Proof of Proposition 6.1_. Since the local blown-up is a $\smile$-preserving
isomorphisme then the map
$\pi^{*}\colon{N}^{{}^{*}}{(X)}\to{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$
is a monomorphism preserving the $\smile$ product. It remains to prove that
this map is an onto map.

Let $\omega$ be a cochain of
${{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{0}}}{\left(X\right)}$. In fact,
$\omega$ is a family of compatible cochains
$\omega_{\sigma}\in{{\widetilde{N}}}^{{}^{*}}_{{}_{\overline{0}}}{\left(\Delta_{\sigma}\right)},$
where $\sigma\colon\Delta_{\sigma}\to X$ are regular simplices. For each
$\sigma\in\mathrm{Sing}^{{\mathcal{F}}}_{+}$ we define the cochain
$\eta_{\sigma}\in{N}^{{}^{*}}{(\Delta_{\sigma})}$ verifying
$\pi^{*}(\eta_{\sigma})=\omega_{\sigma}$. These cochains are compatible on
regular simplicial simplices. These two facts come from the fact that the
local blown-up is an isomorphism.

It remains to define $\eta_{\tau}\in{N}^{{}^{*}}{(\Delta_{\tau})}$, where the
singular simplex $\tau\colon\Delta_{\tau}\to X$ is not regular.

The $\tau$-decomposition is $\Delta_{\tau}=\nabla_{0}*\cdots*\nabla_{k}$, with
$\nabla_{k}\neq\emptyset$ and $k<n$. Let us consider a minimal regular simplex
containing $\tau$ of the form

$\sigma_{a}\colon\Delta_{\tau,a}=\nabla_{0}*\cdots*\nabla_{k}*\underbrace{\emptyset*\cdots*\emptyset}_{n-k-1\
terms}*\langle a\rangle\to X$

where $\sigma_{a}(a)\in X\backslash\Sigma$ is a regular point. We define
$\eta_{\tau}$ as the restriction of
$\eta_{\sigma_{a}}\in{N}^{{}^{*}}{(\Delta_{\tau,a})}$ to $\Delta_{\tau}$, that
is, $\eta_{\tau}=(\eta_{\sigma_{a}})_{\tau}$. The proof is finished if this
definition does not depend on the choice of $a$.

We choose two points $a,b\in X\backslash\Sigma$ verifying the previous
properties. The normality of $X$ implies the connectedness of
$X\backslash\Sigma$ (see [18]). The local conical structure of $X$ gives the
existence of a regular simplex $\sigma\colon\eta_{0}*\cdots*\eta_{k}*[a,b]\to
X$ with $\sigma_{a}\triangleleft\sigma$ and $\sigma_{b}\triangleleft\sigma$.
We have finished since
$(\eta_{\sigma_{a}})_{\tau}=((\eta_{\sigma})_{\sigma_{a}})_{\tau}=(\eta_{\sigma})_{\tau}=((\eta_{\sigma})_{\sigma_{b}})_{\tau}=(\eta_{\sigma_{b}})_{\tau}.$
∎

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