Let $\mathcal{C}$ be an $SO(3)$-equivariant flow category of the type
considered above (for instance, any framed instanton flow category). The Morse
complex $C^{\textup{irr}}_{*}(\mathcal{S}_{\rho}\mathcal{C})$ is obtained as
an algebraic mapping cone. Precisely, there is an isomorphism of (relatively
graded) abelian groups

$uCM_{*}(\mathcal{S}_{\rho}\mathcal{C})[-3]=uCM_{*}(\mathcal{C})[-3]\oplus\mathbb{Z}[\widetilde{i}_{\mathcal{C}}(\rho)-1]$

so that

$d^{\textup{irr}}_{\rho}=\begin{pmatrix}d^{\textup{irr}}&0\\\
-V_{2}&0\end{pmatrix},$

where $\langle V_{2}(\rho),\beta\rangle$ is the signed count of isolated
irreducible components of $\mathcal{C}(\rho,\beta)$.

###### Proof.

We have

$uCM_{*}(\mathcal{S}_{\rho}\mathcal{C})=\left(\bigoplus_{\alpha\in\mathsf{Ob}^{\textup{irr}}(\mathcal{C})}\mathbb{Z}\langle\alpha\rangle[\widetilde{i}(\alpha)]\right)\oplus\mathbb{Z}\langle
S_{\rho}\rangle[\widetilde{i}(\rho)-1];$

we obtain the additional generator from $uC_{*}(S_{\rho})[-3]$, as
$\widetilde{i}_{\mathcal{S}}(S_{\rho})=\widetilde{i}_{\mathcal{C}}(\rho)-1$.
This gives the stated isomorphism at the level of relatively graded abelian
groups.

Now it remains to compute the differential, which we may write as

$d^{\text{irr}}=\begin{pmatrix}d_{CC}&d_{C\rho}\\\ d_{\rho
C}&d_{\rho\rho}\end{pmatrix}.$

Each component is given by computing
$\\#\mathcal{S}^{\text{irr}}(\alpha,\beta)$ where $\alpha,\beta$ are
irreducible objects of $\mathcal{S}_{\rho}\mathcal{C}$; equivalently, this is
the signed count of the number of components. To enumerate these, recall the
definition of the moduli spaces $\mathcal{S}(\alpha,\beta)$ from Construction
5.3. When neither $\alpha$ nor $\beta$ is $S_{\rho}$, we have an equality
$\mathcal{S}(\alpha,\beta)=\mathcal{C}(\alpha,\beta)$, so that
$d_{CC}=d^{\text{irr}}$ is the irreducible differential coming from
$\mathcal{C}$ itself.

Because
$\mathcal{S}(\alpha,S_{\rho})=\mathcal{C}(\alpha,\rho)\times_{\rho}S_{\rho}$,
and $\mathcal{C}(\alpha,\rho)$ consists of irreducible trajectories, this
space has dimension at least 4 and hence does not contribute to the
differential; so $d_{C\rho}=0$. Even more easily,
$\mathcal{S}(S_{\rho},S_{\rho})=\varnothing$ by definition, so that
$d_{\rho\rho}=0$.

The only remaining interesting component is
$\mathcal{S}(S_{\rho},\beta)=-B(\rho,\beta)$, where $B(\rho,\beta)$ is the
real blowup of the moduli space $\mathcal{C}(\rho,\beta)$ along a certain
$SO(3)$-equivariant section. In the case we’re interested in,
$\mathcal{C}(\rho,\beta)$ is 3-dimensional — a disjoint union of free orbits
of $SO(3)$ — this section may be assumed nonzero everywhere (define it to be a
nonzero value at a point in each component, and extend the definition
everywhere by equivariance). The components of $-B(\rho,\beta)$ are thus
oriented diffeomorphic to the components of $-\mathcal{C}(\rho,\beta)$, and we
defined $V_{2}(\rho)=\sum\\#\mathcal{C}^{\text{irr}}(\rho,\beta).$ This shows
that $d_{\rho C}=V_{2}$, as desired. ∎

Because $I_{*}(Y,w,\sigma_{1})$ is isomorphic to the homology of the
irreducible flow complex attached to
$\mathcal{S}_{\rho}\mathcal{I}(Y,w,\sigma_{1})$, the preceding mapping cone
formula immediately implies the following.

######  Corollary .

Suppose $\sigma_{0}<\sigma_{1}$ are adjancent signature data on $(Y,w)$,
differing at $\rho$. Write the continuation map as
$\varphi:I_{*}(Y,w,\sigma_{0})\to I_{*}(Y,w,\sigma_{1})$, and
$V_{2}:\mathbb{Z}[\widetilde{i}(\rho)]\to I_{*}(Y,w,\sigma_{0})$ for the map
sending $1\mapsto[\sum_{\alpha}\\#M(\rho,\alpha)]$. Then there is an exact
triangle

$\cdots\to\mathbb{Z}[\widetilde{i}(\rho)]\to I_{*}(Y,w,\sigma_{0})\to
I_{*}(Y,w,\sigma_{1})\to\cdots$

Recall from Definition 3.2 that $I_{*}(Y,w,\sigma)$ always carries a canonical
$\mathbb{Z}/2$ grading. When $\rho$ is reducible, the absolute
$\mathbb{Z}/2$-grading $\widetilde{i}(\rho)$ is always even; this can be seen
by an application of the index formula to a 4-manifold with reducible
connection bounding $(Y,w,\rho)$.

It follows that $I_{*}(Y,w,\sigma)$ has a well-defined Euler characteristic,
and as the signature data increases, the Euler characteristic decreases:

$\chi\big{(}I_{*}(Y,\sigma_{0})\big{)}=\chi\big{(}I_{*}(Y,\sigma_{1})\big{)}+1.$

In the final section, we will use this to define a Casson–Walker type
invariant. Through the corollary below, it also shows that our technical
passage to the suspended flow complex $S_{\rho}\widetilde{C}_{*}(Y,w,\sigma)$
was in some sense necessary; the continuation map is merely a quasi-
isomorphism, and we couldn’t have constructed an equivariant homotopy inverse.

######  Corollary .

If $\sigma_{0}\leq\sigma_{1}$ are signature data on $(Y,w)$, the continuation
map

$\varphi_{\sigma_{0}\to\sigma_{1}}:\widetilde{C}(Y,w,\sigma_{0})\to\widetilde{C}(Y,w,\sigma_{1})$

is an $SO(3)$-equivariant homotopy equivalence if and only if
$\sigma_{0}=\sigma_{1}$, even though it is always an equivariant quasi-
isomorphism.

###### Proof.

When $\sigma_{0}=\sigma_{1}$ the cylinder is unobstructed running in either
direction, and the corresponding cobordism maps give equivariant chain
homotopy inverses.

If $C$ is a dg-module over $C_{*}SO(3)$, and $f:C\to C^{\prime}$ is an
$SO(3)$-homotopy equivalence, then the corresponding map $f_{*}:uC\to
uC^{\prime}$ is a homotopy equivalence. So by Proposition 7.1, it follows that
if $\varphi_{\sigma_{0}\to\sigma_{1}}$ is an equivariant homotopy equivalence,
then $\varphi_{*}:I_{*}(Y,w,\sigma_{0})\to I_{*}(Y,w,\sigma_{1})$ is an
isomorphism.

Now by Corollary 7.1, if $\sigma_{0}\leq\sigma_{1}$, then

$\chi(I_{*}(Y,\sigma_{0}))-\chi(I_{*}(Y,\sigma_{1}))=\frac{1}{4}\sum_{\alpha\in\mathfrak{A}(Y,w)}\sigma_{1}(\alpha)-\sigma_{0}(\alpha)\geq
0$

with equality if and only if $\sigma_{0}=\sigma_{1}$. So if the continuation
map $\varphi$ is an equivariant homotopy equivalence, we have
$\sigma_{0}=\sigma_{1}$. ∎

###  7.2 An instanton Casson–Walker invariant

If $\alpha\in\mathfrak{A}(Y,w)$, there is a corresponding representation
$\text{ad}_{\alpha}:\pi_{1}(Y)\to SO(2)$. Recall from Definition 3.5 that the
perturbed rho invariant of $\alpha$ with respect to a regular perturbation
$\pi$ is the rational number

$\rho_{\pi}(\alpha)=\sigma_{\pi}(\alpha)+\rho(\text{ad}_{\alpha}),$

with $\rho$ being the relative $\eta$ invariant of Atiyah–Patodi–Singer, and
$\sigma_{\pi}$ is the signature data function of Definition 3.1.

Notice two things. First, this function depends only on the signature data
function $\sigma_{\pi}$, and does not otherwise depend on the perturbation.
Second, if $\sigma_{0}<\sigma_{1}$ are adjacent signature data differing
precisely at $\alpha$, we have
$\rho_{\sigma_{1}}(\alpha)=\rho_{\sigma_{0}}(\alpha)+4$. We can thus use this
as a correction term to counteract the change in Euler characteristic of
$I_{*}$ as we move between chambers and obtain a well-defined rational number.

######  Definition .

Let $(Y,w)$ be a weakly admissible pair. We say that the instanton
Casson–Walker invariant is the number

$\lambda_{I}(Y,w)=\frac{1}{|\textup{Tors
}H_{1}Y|}\left(\chi\big{(}I_{*}(Y,w,\pi)\big{)}+\frac{1}{4}\sum_{\alpha\in\mathfrak{A}(Y)}\rho_{\pi}(\alpha)\right),$

for any choice of regular perturbation $\pi$. ∎

This is independent of $\pi$ because the change in the displayed quantities
cancel out as we move between adjacent signature data. We expect this to be an
instanton-theoretic definition of the Casson–Walker invariant.

######  Conjecture .

For any rational homology sphere $Y$, the instanton and Casson–Walker
invariants coincide; we have $\lambda_{I}(Y,w)=\lambda_{CW}(Y)$ for all $w$.

In particular, this would mean that $\lambda_{I}(Y,w)$ is independent of $w$.

######  Remark .

The Casson invariant and its relatives have a long history. Soon after Casson
defined his invariant of integer homology spheres, Taubes [Taubes] gave a
definition in terms of signed counts of flat connections modulo gauge; the
signs amount to saying that the Casson invariant computes the Euler
characteristic of Floer’s instanton homology. Around the same time, Walker
published his generalization to rational homology spheres [Walker]; a signed
count of points no longer gives an invariant, and instead one must add
canonical rational-valued correction terms (as we do above) to get a well-
defined invariant.

The Seiberg–Witten analogue of the Casson–Walker invariant is defined in
[Chen:SW]. There, even in the case of integer homology spheres (indeed, even
in the case of $S^{3}$) one needs to introduce correction terms to get a well-
defined invariant; these correction terms only fail to be integers for
rational homology spheres. The resulting quantity is indeed equal to the
Casson–Walker invariant, as proved in [Lim:SW] for integer homology spheres
and [MWCW] for rational homology spheres.

Finally, a version of the Casson–Walker invariant in terms of a precursor to
Heegaard Floer homology was provided in [OS:casson]; here again one uses a
topological correction term, and their Casson–Walker invariant is given as a
sum over spinc structures. ∎

As yet we have neither a surgery exact triangle or a connected sum theorem, so
we cannot say much presently outside of some specific examples; but we can at
least verify it in those few examples. To set convention, the lens space
$L(p,q)$ is $-p/q$ surgery on the unknot.

######  Theorem .

Conjecture 7.2 is true when $Y$ is an integer homology sphere or a lens space
$L(p,q)$.

###### Proof.

When $Y$ is an integer homology sphere, the sum over abelian flat connections
is tautologically zero and the choice of signature data is vacuous, so we
simply have $\lambda_{I}(Y)=\chi I_{*}(Y)$. The main result of [Taubes] is
that we have an equality $\chi I_{*}(Y)=2\lambda_{C}(Y)$ with half of the
Casson invariant of $Y$. Walker’s normalization conventions have
$\lambda_{CW}(Y)=2\lambda_{C}(Y)$, giving us the desired result.

For lens spaces, the trivial perturbation is a regular perturbation, so we may
take $\sigma=0$. Then the invariant $\lambda_{I}(L(p,q),w)$ is given as

$\frac{1}{4p}\sum_{\alpha\in\mathfrak{A}(L(p,q),w)}\rho(\alpha).$

Write $H^{2}\big{(}L(p,q)\big{)}\cong\mathbb{Z}/p$. In the case
$w=\varnothing$, the abelian reducibles can be identified with those integers
$\ell$ with $0<\ell<p/2$, and have adjoint representation given by
$\text{ad}_{\ell}(i)=e^{2\pi i(2\ell)/p}$. In the case $p$ is even and $[w]$
is nonzero in $\mathbb{Z}/2$-homology, the abelian reducibles can be
identified with the odd integers $0<2\ell+1<p$, and
$\text{ad}_{\ell}(i)=e^{2\pi i(2\ell+1)/p}$.

Let us handle the first case. Using the formula of [HK, Proposition 2.18], we
have

$\rho(\text{ad}_{\ell})=\frac{4}{p}\sum_{k=1}^{p-1}\cot\left(\frac{\pi
k}{p}\right)\cot\left(\frac{\pi kq}{p}\right)\sin^{2}\left(\frac{2\pi
k\ell}{p}\right).$

Summing over $0<\ell<p/2$ and multiplying by $1/4p$, we see that
$\lambda_{I}(L(p,q),\varnothing)$ is

$\frac{1}{p^{2}}\sum_{0<\ell<p/2}\sum_{k=1}^{p-1}\cot\left(\frac{\pi
k}{p}\right)\cot\left(\frac{\pi kq}{p}\right)\sin^{2}\left(\frac{2\pi
k\ell}{p}\right).$

Let us compute the identical sum

$\frac{1}{2p^{2}}\sum_{\ell=0}^{p-1}\sum_{k=1}^{p-1}\cot\left(\frac{\pi
k}{p}\right)\cot\left(\frac{\pi kq}{p}\right)\sin^{2}\left(\frac{2\pi
k\ell}{p}\right);$

this coincides with the sum above because the $\sin^{2}$ term is symmetric
with respect to swapping $\ell$ and $p-\ell$, and when $p$ is even the term
corresponding to $\ell=p/2$ automatically vanishes.

When $p$ is even, notice that every term corresponding to $k=p/2$
automatically vanishes because $\cot(\pi/2)=0$. Further, observe that
$\sum_{\ell=0}^{p-1}\sin^{2}\left(\frac{2\pi k\ell}{p}\right)=p/2$ whenever
$0<k<p$ and $k\neq p/2$: this follows by the double-angle formula
$\sin^{2}(x)=\frac{1}{2}-\frac{1}{2}\cos(2x)$ and the fact that
$\sum_{\ell=0}^{p-1}\cos(4\pi k\ell/p)=0$, because this latter expression is
the real part of the sum $0=1+\zeta+\cdots+\zeta^{p-1}$. Here $\zeta=e^{4\pi
ik/p}\neq 1$ because $0<k<p$ and $k\neq p/2$.

Combining these two facts, we may write

$\lambda_{I}(L(p,q),\varnothing)=\frac{1}{4p}\sum_{k=1}^{p-1}\cot\left(\frac{\pi
k}{p}\right)\cot\left(\frac{\pi kq}{p}\right),$

which is a Dedekind sum traditionally denoted $s(q;p)$.

Finally, we have $\lambda_{CW}(L(p,q))=s(q;p)$, by [Walker, Proposition 6.3];
in comparing to the reference, note that Walker’s $L_{p/q}$ is what we call
$L(-p,q)$, and hence $\lambda_{CW}(L(p,q))$ is written in Walker’s book as
$\lambda(-L_{p/q})=-\lambda(L_{p/q})$, as desired.

The computation for the case $[w]\neq 0\in H_{1}(Y;\mathbb{F}_{2})$ is
similar, with the relevant $\rho$ invariant being

$\rho(\text{ad}_{\ell})=\frac{4}{p}\sum_{k=1}^{p-1}\cot\left(\frac{\pi
k}{p}\right)\cot\left(\frac{\pi kq}{p}\right)\sin^{2}\left(\frac{\pi
k\ell}{p}\right),$

for odd $0<\ell<p$, so that

$\lambda_{I}(L(p,q),w)=\frac{1}{p^{2}}\sum_{j=0}^{p/2-1}\sum_{k=1}^{p-1}\cot\left(\frac{\pi
k}{p}\right)\cot\left(\frac{\pi kq}{p}\right)\sin^{2}\left(\frac{\pi
k(2j+1)}{p}\right).$

This simplifies to the same quantity $s(q;p)$. The argument is nearly
identical: eliminate the terms with $k=p/2$, and sum over $j$ to get rid of
the final factor; here one uses that when $\zeta$ is a $p$’th root of unity
for $p$ even we have $\zeta+\zeta^{3}+\cdots+\zeta^{2p-1}=0$. ∎

######  Remark .

Using the computations in [Helle], together with further computations of the
rho invariants for abelian connections on spherical manifolds, one can also
verify the conjecture for any 3-manifold of the form $SU(2)/\Gamma$ (with
$w=\varnothing$), where $\Gamma$ is a finite subgroup of $SU(2)$. We will not
carry out this computation in detail here. ∎

##  Appendix A Stratified-smooth spaces

In this appendix, we construct the model for singular chains $C_{*}(X)$ we use
in the main body of the text, where $X$ is a smooth manifold, which we will
call _geometric chains_ ; this complex should be functorial for smooth maps
and have homology groups canonically isomorphic to singular homology. Let us
briefly outline the desired construction.

This geometric chain complex should be free on a certain generating set, the
_nondegenerate, nontrivial probes_ , where these probes are something
analogous to $(P,\phi)$ a compact oriented smooth manifold with corners $P$
with a smooth map $\phi$ to $X$; the boundary map should be analogous to the
passage to $\partial P$ in the case of smooth manifolds with corners.

For our purposes, these probes must be more general than smooth manifolds with
corners: they must include the moduli spaces of instantons
$\breve{M}^{+}_{\zeta}(\alpha,\beta)$ discussed in Section 3.4; in Definition
3.4, we named the kind of object we obtain a _stratified-smooth manifold_.
When proving the excision property, one needs to be able to take transverse
intersections, which leads to the more general notion of _stratified-smooth
spaces_ below, roughly the kind of space obtained by taking transverse
intersections of stratified-smooth manifolds.

The first two sections of this appendix develop the bare minimum of
foundations and orientation theory of stratified-smooth manifolds needed to
define geometric chains. In Section 5 of the main text, we also need a certain
blowup construction on these stratified-smooth spaces, which we set up in
Appendix A.3, and carry out some necessary computations. In Appendix A.4 we
construct the geometric chain complex and verify an appropriate version of the
Eilenberg-Steenrod axioms, and therefore verify that the resulting homology
groups are canonically isomorphic to singular homology. In the final section,
Appendix A.5, we perform a truncation procedure on the geometric chain
complex; this is used in Section 4 to ensure we only need to construct moduli
spaces within a certain dimension range.

###  A.1 Foundations of stratified-smooth spaces

Smooth manifolds with corners should be special cases of stratified-smooth
spaces, so first we must decide on a preferred definition. There are a number
of inequivalent definitions (see [Joyce] for a survey). The more general
definitions have more complicated boundary operators, which cannot be defined
solely in terms of the codimension-1 strata. This is harder to work with
effectively when defining and computing the boundary of a stratified-smooth
space; we prefer to be more restrictive.

With eyes towards the moduli spaces in the main text, we instead use a version
of manifolds with corners and stratified-smooth spaces where one is stratified
by a set of faces, which need not be connected, whose closures are all again
manifolds with corners (resp. stratified-smooth spaces). To do so, we record
the stratification with an appropriate kind of partially ordere set.

######  Definition .

A graded poset is a finite partially ordered set $(\Delta,<)$ equipped with a
‘dimension’ function $d:\Delta\to\mathbb{Z}_{\geq 0}$, which is order-
preserving in the sense that $s<t$ implies $d(s)<d(t)$.

We say that the largest integer in the image of $d$ is the dimension of
$\Delta$. ∎

The simplest example of a graded poset, and one which will be very useful for
us, is the cube poset $\\{0,1\\}^{n}$ for some natural number $n$. Writing
elements as functions $s:\\{1,\cdots,n\\}\to\\{0,1\\}$, the ordering has
$s\leq t$ if $s(i)\leq t(i)$ for all $i$, and the dimension function is
$d(s)=\sum s(i)$.

For the objects of interest, the graded poset $\Delta$ will have the property
that the sub-poset

$\Delta_{\geq s}=\\{t\in\Delta\mid t\geq s\\}$

is always isomorphic to a cube poset $\\{0,1\\}^{\dim X-d(s)}$, with dimension
function shifted up by $d(s)$.

Partially ordered sets are relevant because the natural relation between the
different strata of a stratified space is a partial ordering. Below, we use a
definition of stratified space which explicitly records the relevant partially
ordered set.

######  Definition .

A stratified space $(X,\Delta)$ is a pair of a poset $\Delta$ and a
topological space $X$ equipped with, for each $s\in\Delta$, a locally closed
subset $X_{s}\subset X$; these subsets are required to satisfy the property
$\overline{X_{s}}=\bigcup_{t\leq s}X_{t}.$

An isomorphism $(X,\Delta_{X})\cong(Y,\Delta_{Y})$ of stratified spaces is a
pair $(f,\bar{f})$, where $\bar{f}:\Delta_{X}\to\Delta_{Y}$ is a poset
isomorphism and $f$ is a homeomorphism with $f(X_{s})=Y_{\bar{f}(s)}$. ∎

Next we record the naive version of a smooth version of stratified spaces,
where one merely equips each stratum with a smooth structure.

######  Definition .

A naive stratified-smooth space $(X,\Delta)$ of dimension $n$ is a pair of a
graded poset $\Delta$ and a second-countable Hausdorff space $X$ stratified by
$\Delta$, where each stratum $X_{s}$ is given the structure of a smooth
manifold of dimension $d(s)$.

A naive stratified-smooth map $f:X\to M$ from $X$ to a smooth manifold $M$ is
a continuous function for which each $f_{s}:X_{s}\to M$ is smooth; we say
$m\in M$ is a regular value if it is a regular value for all maps
$f_{s}:X_{s}\to M$.

An isomorphism $(f,\bar{f})$ between two naive stratified-smooth spaces
$(X,\Delta_{X})$ and $(Y,\Delta_{Y})$ is an isomorphism of stratified spaces
so that the induced maps between strata $f_{s}:X_{s}\to Y_{\bar{f}(s)}$ are
all diffeomorphisms. ∎

Let us discuss one standard example of a naive stratified-smooth space, as
well as two important constructions which will be used in the definition of
the local models below.

######  Example .

The space $[0,\infty)^{n}$ can be given the structure of a naive stratified-
smooth space. The stratification is by the cube poset $\Delta=\\{0,1\\}^{m}$;
the faces are products of $\\{0\\}$ and $(0,\infty)$. More precisely, write

$F(i)=\begin{cases}\\{0\\}&i=0\\\ (0,\infty)&i=1\end{cases}$

Then the face associated to the stratum $s\in\\{0,1\\}^{m}$ is the subspace

$(0,\infty)^{s}:=\prod_{i=1}^{n}F(s(i));$

that is, the $i$’th factor is $\\{0\\}$ if $s(i)=0$ and is $(0,\infty)$ if
$s(i)=1$. ∎

######  Example .

Let $(X,\Delta)$ be a naive stratified-smooth space, and let $U\subset X$ be
an open subset. Then $U$ also inherits the structure of a stratified-smooth
space, taking $\Delta(U)\subset\Delta$ to be those $s$ for which $U\cap
X_{s}\neq\varnothing$ with strata $U_{s}=U\cap X_{s}$. ∎

######  Example .

Let $(X,\Delta)$ be a naive stratified-smooth space of dimension $n$, and let
$\psi:X\to\mathbb{R}^{\ell}$ be a stratified-smooth map with zero as a regular
value. Write $\Delta(\psi)\subset\Delta$ for the sub-poset

$\Delta(\psi)=\\{s\in\Delta\mid\psi^{-1}(0)\cap X_{s}\neq\varnothing\\},$

with grading function $d_{\psi}(s)=d(s)-\ell$. Then
$(\psi^{-1}(0),\Delta(\psi))$ is a naive stratified-smooth space of dimension
$n-\ell$. ∎

We combine these into one type of model space; a stratified-smooth space will
be a naive stratified-smooth space locally isomorphic to one of the model
spaces.

######  Definition .

Let $V\subset[0,\infty)^{m}\times\mathbb{R}^{n}$ be an open subset containing
zero, and let $\psi:V\to\mathbb{R}^{\ell}$ be a naive stratified-smooth map
with zero as a regular value. Then $Z(\psi)=\psi^{-1}(0)$ carries the natural
structure of a naive stratified-smooth space of dimension $m+n-\ell$. We call
such a naive stratified-space $Z(\psi)$ a cutout locus. ∎

Notice that the assumption $0\in V$ implies that $\Delta(V)=\\{0,1\\}^{m}$;
every face has a point near $0$ and hence a point in $V$. Thus the
stratification poset only changes when passing to $\Delta(\psi)$.

Before moving on to the definition of stratified-smooth space, we make a brief
observation on the combinatorics of this poset $\Delta(\psi)$. The following
argument is inspired by [SeSm, Section 2c], which we follow again when
discussing orientations later.

######  Lemma .

The poset $\Delta(\psi)\subset\\{0,1\\}^{m}$ is an upward-closed sub-poset;
that is, if $s\in\Delta(\psi)$ and $t\in\\{0,1\\}^{m}$ has $t>s$, then
$t\in\Delta(\psi)$ as well.

###### Proof.

Suppose that $Z(\psi)_{s}\subset(0,\infty)^{s}\times\mathbb{R}^{n}$ is
nonempty.111111Recall that here $s\in\\{0,1\\}^{m}$ and the exponent means the
$i$’th factor is either $\\{0\\}$ or $(0,\infty)$, depending on whether $s(i)$
is $0$ or $1$. Because $Z(\psi)_{s}$ is the preimage of a regular value, it is
a smooth manifold of codimension $\ell$ inside $(0,\infty)^{s}$. Choose a
small sphere
$c:S^{\ell-1}\to\big{(}(0,\infty)^{s}\times\mathbb{R}^{n}\big{)}\setminus
Z(\psi)_{s}$ which has linking number one with $Z(\psi)_{s}$, so that in
particular $\psi c:S^{\ell-1}\to\mathbb{R}^{\ell}-\\{0\\}$ is not null-
homotopic.

It follows that in fact
$c:S^{\ell-1}\to\big{(}[0,\infty)^{m}\times\mathbb{R}^{n}\big{)}\setminus
Z(\psi)$ is not null-homotopic (for otherwise $\psi c$ would be null-
homotopic, which it is not).

Now for $s<t$, we may homotop $c$ into
$c^{\prime}:S^{\ell-1}\to(0,\infty)^{t}\times\mathbb{R}^{n}$ by an arbitrarily
small homotopy; taking this homotopy sufficiently small, its image will not
meet $Z(\psi)$, as $Z(\psi)$ is closed and the domain is compact. In
particular, $\psi c^{\prime}$ is homotopic to $\psi c$ as a map to
$\mathbb{R}^{n}\setminus\\{0\\}$, and therefore $\psi c^{\prime}$ is not null-
homotopic; thus $c^{\prime}$ is not null-homotopic in
$\big{(}(0,\infty)^{t}\times\mathbb{R}^{n}\big{)}\setminus Z(\psi)_{t}$. In
particular, $Z(\psi)_{t}$ is nonempty, which is what we wanted to show. ∎

######  Remark .

It follows that $\Delta(\psi)$ is a _cubical poset_ , which means that for
each $s\in\Delta(\psi)$, we have $\Delta(\psi)_{\geq
s}\cong\\{0,1\\}^{m}_{\geq s}\cong\\{0,1\\}^{m-d(s)}$; the poset of strata
lying above a given stratum is isomorphic to the cube poset of appropriate
dimension. Soon after we define the boundary operator, we will see that this
observation underlies the relation $\partial^{2}=0$.∎

We now define the geometric objects of interest.

######  Definition .

A _stratified-smooth space_ is a naive stratified-smooth space $(X,\Delta)$ so
that, for each $x\in X$, there exists an open subset $x\in U\subset X$ and a
cutout locus $Z(\psi)$ so that $U$ is isomorphic to $Z(\psi)$ as naive
stratified-smooth spaces. We call a choice of open set $U$ and isomorphism
$U\cong Z_{\psi}$ a cutout chart for $x$. ∎

Note that in particular we do not have an ‘atlas’ of such charts in any sense;
we merely demand that they exist.

######  Example .

In [Joyce], a sequence of increasingly more strict notions of manifolds with
corners are defined, and $\langle N\rangle$-manifolds defined there give
examples of stratified-smooth spaces. Our Definition 3.4 of stratified-smooth
manifold is very closely related to Joyce’s definition $\langle
N\rangle$-manifold, though ours is a weaker object, as the smooth structure on
the top stratum of a stratified-smooth manifold does not extend up to the
boundary.

In particular, the simplex $\Delta^{k}$ is a stratified-smooth space for all
$k$, as are all products of simplices. ∎

######  Example .

This is a more restrictive definition than it may seem at first, because of
the combinatorial nature of an isomorphism between naive stratified-smooth
spaces. This is best examplified by the teardrop, defined in [Joyce, Remark
2.11] as an example of a naive topological manifold with corners which does
not satisfy any of the more stringent definitions of manifold with corners.
His teardrop is the set

$\\{(x,y)\mid x\geq 0,\;y^{2}\leq x^{2}-x^{4}\\},$

which has three strata: a corner stratum, one boundary stratum, and one open
stratum. The stratification poset $\Delta$ can be identified with
$\\{0,1,2\\}$ with the total order $0<1<2$.

This is not locally isomorphic as a naive stratified-smooth space to
$[0,\infty)^{2}$ near the corner point. A neighborhood of the corner point in
the teardrop has _one_ (disconnected!) codimension-1 stratum, while a
neighborhood of $0$ in $[0,\infty)^{2}$ has two.

In fact, by Lemma A.1 above, the teardrop is not a stratified-smooth space: if
it were, $\Delta_{\geq 0}=\\{0,1,2\\}$ would be isomorphic to a cubical poset,
which it is not.

A picture of the teardrop with its three strata.

∎

Now that we have a definition of stratified-smooth space, we can define its
boundary. Part of the reason we choose the combinatorial language in the
definition of stratified-smooth space is that it makes the definition of
boundary very simple, using the following lemma; notice that this lemma fails
for the boundary stratum of the teardrop.

######  Lemma .

Let $X$ be a stratified-smooth space of dimension $n$, and let $X_{s}$ be a
stratum of $X$. Then $\overline{X_{s}}$ naturally carries the structure of a
stratified-smooth space of dimension $d(s)$.

###### Proof.

The underlying topological space is of course the closure
$\overline{X_{s}}\subset X$ with the subspace topology. Because
$\overline{X_{s}}=\cup_{t\leq s}X_{t}$ we see that $\overline{X_{s}}$ carries
the natural structure of a stratified space with smooth structure on each
stratum, with face poset $\Delta_{\leq s}$.

We need to construct cutout charts for this subspace. If $x\in
X_{t}\subset\overline{X_{s}}$, choose a cutout chart for $X$ near $x$. To
spell out all this data: choose an open set $x\in U\subset X$, an open subset
$V\subset[0,\infty)^{m}\times\mathbb{R}^{n}$, a naive stratified-smooth map
$\psi:V\to\mathbb{R}^{\ell}$ with zero as a regular value, an isomorphism of
posets $\overline{\phi}:\Delta(U)\to\Delta(\psi)$, and $\phi:U\cong Z_{\psi}$
an isomorphism of naive stratified-smooth spaces lifting $\overline{\phi}$.
Then $\overline{\phi}(s)\in\\{0,1\\}^{m}$; we abbreviate this to
$\overline{s}$.

Write $[0,\infty)^{\overline{s}}$ for the product of $\\{0\\}$ and
$[0,\infty)$, with a $0$ in the $i$’th coordinate if $\overline{s}(i)=0$ and
$[0,\infty)$ otherwise. Write

$V_{\leq\overline{s}}=V\cap\left([0,\infty)^{\overline{s}}\times\mathbb{R}^{n}\right),$

and $\psi_{\leq\overline{s}}$ for the restriction of $\psi$ to this subspace.

Then $\phi$ and $\overline{\phi}$ restrict to give an isomorphism of naive
stratified-smooth spaces between $U\cap\overline{X_{s}}=U_{\leq s}$ and
$Z(\psi_{\leq\overline{s}})$. This gives the desired cutout chart on
$\overline{X_{s}}$. ∎

This is enough to define the boundary of a stratified-smooth space.

######  Definition .

Let $X$ be a stratified-smooth space. The boundary of $X$ is the stratified-
smooth space

$\partial X=\bigsqcup_{\begin{subarray}{c}s\in\Delta\\\ d(s)=\dim
X-1\end{subarray}}\overline{X_{s}}.\qed$

Next, we will use Lemma A.1 to obtain an understanding of $\partial^{2}X$.

######  Proposition .

The boundary $\partial^{2}X$ is isomorphic to $\\{0,1\\}\times S$ for some
stratified-smooth space $S$ of dimension $\dim X-2$.

###### Proof.

First we show that for all $s\in\Delta$ with$d(s)=\dim X-2$, we have
$\Delta_{\geq s}\cong\\{0,1\\}^{2}$; in particular, for each such $s$, there
are exactly two $t$ with $d(t)=\dim X+1$ and $s<t$. This follows by picking a
point $x\in X_{s}$ and a cutout chart $x\in U\subset X$ with $U\cong
Z_{\psi}$. Then we have

$\Delta_{\geq s}=\Delta(U)_{\geq s}\cong\Delta(Z_{\psi})_{\geq
s}\cong\\{0,1\\}^{2};$

the first isomorphism follows because if $s<t$ we have $X_{t}\cap
U\neq\varnothing$ (because $X_{s}\subset\overline{X_{t}}$ and $U$ is open),
the second by the definition of isomorphism of stratified-smooth spaces, and
the last by Lemma A.1 and the discussion in Remark A.1.

Next, observe that because

$\partial{\overline{X}_{t}}=\bigsqcup_{\begin{subarray}{c}s\in\Delta\\\ s<t\\\
d(s)=\dim X-2\end{subarray}}\overline{X_{s}},$

we have

$\partial^{2}X=\bigsqcup_{\begin{subarray}{c}t,s\in\Delta\\\ s<t\\\
d(s)+1=d(t)=\dim X-1\end{subarray}}\overline{X_{s}}.$

Because there are exactly two such $t$, this disjoint union is isomorphic to

$\partial^{2}X=\\{0,1\\}\times\bigsqcup_{\begin{subarray}{c}s\in\Delta\\\
d(s)=\dim X-2\end{subarray}}\overline{X_{s}},$

as desired. ∎

We move on to foundations of smooth maps and fiber products.

######  Definition .

Let $(X,\Delta)$ and $(Y,\Delta^{\prime})$ be stratified-smooth spaces, and
let $B$ be a smooth manifold without boundary. We say $f:X\to B$ is
stratified-smooth if $f$ is continuous, smooth on each stratum, and satisfies
the following property: for each $x\in X$ there exists a cutout chart $x\in
U\cong Z(\psi)\subset V$ and an extension of $f\phi:Z(\psi)\to B$ to a map
$F:V\to B$ which is continuous and smooth on each stratum.

Given two stratified-smooth maps $f:X\to B$ and $g:Y\to B$ we say $f$ and $g$
are stratified-transverse if their restrictions to any pair of strata $X_{s}$
and $Y_{s^{\prime}}$ are transverse. ∎

Then as desired the fiber product of two stratified-smooth spaces is again
stratified-smooth. This includes, for instance, preimages of regular values of
stratified-smooth functions to smooth manifolds (or more generally,
intersection with submanifolds, hence in particular a generic zero set of a
section of a vector bundle).

######  Proposition .

If $f$ and $g$ are stratified-transverse maps as in the above definition,
there is a natural stratified-smooth space structure on the fiber product
$X\times_{B}Y$, whose underlying topological space is the usual fiber product.

###### Proof.

The face poset of $X\times_{B}Y$ is a subset

$\Delta_{B}\subset\Delta\times\Delta^{\prime},$

consisting of those pairs $(s,s^{\prime})$ so that
$X_{s}\times_{B}Y_{s^{\prime}}$ is nonempty, equipped with the grading
$d_{B}(s,s^{\prime})=d(s)+d^{\prime}(s^{\prime})-\dim B$. That the ordering
induced from $\Delta\times\Delta^{\prime}$ coincides with the closure-
inclusion ordering amounts to the relation
$\overline{X_{s}\times_{B}Y_{s^{\prime}}}=\overline{X_{s}}\times_{B}\overline{Y_{s^{\prime}}}.$
The forward inclusion is obvious, while the reverse is not; still, it follows
along lines analogous to the proof of Lemma A.1.

Pick $(x,y)\in X_{s}\times_{B}Y_{s^{\prime}}$. Write $k=\dim X-d(s)$ and
$k^{\prime}=\dim Y-d^{\prime}(s^{\prime})$; we have open sets

$V\subset[0,\infty)^{k}\times\mathbb{R}^{d(s)+m},\;\;\;V^{\prime}\subset[0,\infty)^{k^{\prime}}\times\mathbb{R}^{d^{\prime}(s^{\prime})+n}$

and open neighborhoods

$x\in U\subset X,\;\;\;y\in U^{\prime}\subset Y,$

and maps $\psi:V\to\mathbb{R}^{m}$ and
$\psi^{\prime}:V^{\prime}\to\mathbb{R}^{n}$, with isomorphisms of naive
stratified-smooth spaces

$\phi:Z(\psi)\cong V,\;\;\;\phi^{\prime}:Z(\psi^{\prime})\to V^{\prime}$

sending $x$ and $y$ to zero, respectively.

Further, by the definition of stratified-smooth map, we may assume $V$ and
$V^{\prime}$ are equipped with extensions $F$ and $G$ of $f$ and $g$, which
are only defined on $Z(\psi)$ and $Z(\psi^{\prime})$. We suppose these charts
are chosen sufficiently small that the images of $F$ and $G$ lie in a single
chart of $B$; for convenience, we thus suppose $B=\mathbb{R}^{b}$.

Then consider the open set

$\overline{U}=U\times
U^{\prime}\subset[0,\infty)^{k}\times\mathbb{R}^{d(s)+m}\times[0,\infty)^{k^{\prime}}\times\mathbb{R}^{d^{\prime}(s^{\prime})+n},$

and the map $\xi=(\psi,F-G,\psi^{\prime}):\overline{U}\to\mathbb{R}^{m+b+n}$.
Because the restriction of $F-G$ to the zero set of $(\psi,\psi^{\prime})$ is
precisely $f-g$, the zero set of $\xi$ is the fiber product
$Z(\psi)\times_{\mathbb{R}^{b}}Z(\zeta)$. Further, because $\psi$ and
$\psi^{\prime}$ have zero as a regular value, and $F-G$ has zero as a regular
value on the zero locus of $\psi$ and $\zeta$, it follows that $\xi$ has zero
as a regular value on each stratum as well.

Thus $U\times_{B}U^{\prime}=Z(\xi)$, and $(\phi,\phi^{\prime})$ restricts to
give a homeomorphism $Z(\xi)\cong V\times_{B}V^{\prime}$ which sends strata to
strata and is a diffeomorphism on each stratum. We have constructed the
necessary cutout chart. ∎

This fiber product operation is associative: if we have maps $f:X\to A$,
$g:Y\to A\times B$ and $h:Z\to B$ satisfying the appropriate transversality
requirements, then there is a canonical diffeomorphism of stratified-smooth
spaces

$X\times_{A}(Y\times_{B}Z)\cong(X\times_{A}Y)\times_{B}Z.$

Further, using the description of its face poset in the previous argument, we
may explicitly describe its boundary.

######  Proposition .

The fiber product $X\times_{B}Y$ has boundary

$\partial(X\times_{B}Y)=(\partial X)\times_{B}Y\sqcup X\times_{B}\partial Y.$

###### Proof.

The left-hand side is

$\bigsqcup_{\begin{subarray}{c}s\in\Delta_{X},t\in\Delta_{Y}\\\
d(s)+d^{\prime}(t)-\dim B=\dim X+\dim Y-\dim
B-1\end{subarray}}\overline{X_{s}}\times_{B}\overline{Y_{t}};$

now $d(s)+d^{\prime}(t)=\dim X+\dim Y-1$ if and only if one of $\dim X-d(s)$
and $\dim Y-d^{\prime}(t)$ is one and the other is zero; this means this is
precisely

$\left(\bigsqcup_{\begin{subarray}{c}s\in\Delta_{X},t\in\Delta_{Y}\\\
d(s)=\dim X\\\ d(t)=\dim
Y-1\end{subarray}}\overline{X_{s}}\times_{B}\overline{Y_{t}}\right)\sqcup\left(\bigsqcup_{\begin{subarray}{c}s\in\Delta_{X},t\in\Delta_{Y}\\\
d(s)=\dim X-1\\\ d(t)=\dim
Y\end{subarray}}\overline{X_{s}}\times_{B}\overline{Y_{t}}\right).$

Because the disjoint union $\bigsqcup_{d(s)=\dim X}\overline{X_{s}}$ is $X$
itself, while $\bigsqcup_{d(s)=\dim X-1}\overline{X_{s}}=\partial X$, the
above sum simplifies to

$\partial X\times_{B}Y\sqcup X\times_{B}\partial Y,$

as desired. ∎

In addition to stratified-smooth spaces defined by equations (such as the
general fiber products described above), we will also find need in Section 5.2
for stratified-smooth spaces defined by _inequalities_ ; we record the
relevant statement below.

######  Proposition .

If $(X,\Delta)$ is a stratified-smooth space and $L:X\to\mathbb{R}$ is
stratified-smooth, then for a regular value $c$ of $L$, the set $X^{\leq
c}=L^{-1}(-\infty,c]$ is a stratified-smooth space with boundary

$\partial X^{\leq c}=L^{-1}(c)\sqcup(\partial X)^{\leq c}.$

###### Proof.

The stratification poset for $X^{\leq c}$ is denoted $\Delta^{\leq c}$, and is
a sub-poset of $\Delta\times\\{-1,0\\}$; we say $(s,-1)\in\Delta^{\leq c}$ if
$L^{-1}(c)\cap X_{s}$ is nonempty, while $(s,0)\in\Delta^{\leq c}$ if
$L^{-1}(-\infty,c)\cap X_{s}\neq\varnothing$. Once we verify that $X^{\leq c}$
is indeed a stratified-smooth space, this gives the stated boundary formula.

The notation for $\Delta^{\leq c}$ chosen so that $d(s,0)=d(s)$ and
$d(s,-1)=d(s)-1$. It is straightforward to see that this is a stratified
topological space; the strata $(X^{\leq c})_{(s,0)}$ are open subsets of
$X_{s}$, hence carry the natural structure of smooth manifolds, whereas the
strata $(X^{\leq c})_{(s,-1)}$ are given as regular values of a smooth map
$L:X_{s}\to\mathbb{R}$, hence also in a natural way smooth manifolds. So
$X^{\leq c}$ is at least a naive stratified-smooth space; the stated boundary
relation follows immediately from the definition of $\Delta^{\leq c}$ and its
dimension function.

The existence of cutout charts for $x\in X^{\leq c}_{(s,0)}$ follows
immediately by restricting cutout charts for $x\in X_{s}$. The interesting
case is $x\in X^{\leq c}_{(s,-1)}$. In this case, one starts by picking a
cutout chart for $x\in X_{s}$; that is, there is a choice of open set
$V\subset[0,\infty)^{m}\times\mathbb{R}^{n}$, a stratumwise-smooth map
$\psi:V\to\mathbb{R}^{\ell}$, and an isomorphism $Z(\psi)\cong U$ of naive
stratified-smooth spaces, where $U$ is an open neighborhood of $x$. By the
assumption that $L$ is stratified-smooth, we may assume $L$ extends to a
function on $V$, which we denote by the same symbol.

We may extend this to a map
$\hat{\psi}:V\times[0,\infty)\to\mathbb{R}\oplus\mathbb{R}^{\ell}$ defined by

$\hat{\psi}(v,s)=(\psi(v),L(v)+s-c).$

For $(s,v)$ to be in the zero set of this map means precisely that $\psi(v)=0$
and that $L(v)\leq c$. The map $\pi:Z(\hat{\psi})\to Z(\psi)$ is a topological
embedding, and a smooth embedding on each stratum; composing this with the map
$\varphi:Z(\psi)\cong U$ gives us the desired stratified-smooth isomorphism
from $Z(\hat{\psi})$ to $U_{\leq c}$, as desired. ∎

###  A.2 Orientations on stratified-smooth spaces

In this section, we will define and analyze the notion of an _oriented_
stratified-smooth space. Briefly, this is an orientation on the top strata
which is consistent with the orientations induced by some cutout chart. We
will also use cutout charts to define the boundary orientation.

The first of these requires that we first understand the orientation on a
cutout chart; for the second, we must verify that the induced boundary
orientation does not depend on the choice of cutout chart.

######  Construction .

To set conventions: suppose $f:X\to B$ is a smooth map between oriented smooth
manifolds and $b\in B$ is a regular value. Given $x\in f^{-1}(b)$, we write
$T_{x}Z$ for the tangent space to $f^{-1}(b)$; write $di:T_{x}Z\to T_{x}X$ for
the differential of the inclusion map. If $s$ is a splitting of the surjective
map $df_{x}:T_{x}X\to T_{b}B$, then we orient $T_{x}Z$ by the assumption that
$(di,s):T_{x}Z\oplus T_{b}B\to T_{x}X$ is an orientation-preserving
isomorphism.

Now let $\psi:[0,\infty)\times\mathbb{R}^{n}\to\mathbb{R}^{m}$ be a continuous
map which, on each stratum, is smooth and has zero as a regular value. Write
$\psi^{\circ}$ for its restriction to $(0,\infty)\times\mathbb{R}^{n}$, and
write $\psi^{\partial}$ for its restriction to
$\mathbb{R}^{n}\cong\\{0\\}\times\mathbb{R}^{n}$, equipped with the boundary
orientation (here, negative the standard orientation).

Then the zero sets $Z(\psi^{\circ})$ and $Z(\psi^{\partial})$ are canonically
oriented, being regular level sets of smooth maps between oriented smooth
manifolds. ∎

For convenience in what follows below, it is worth mildly generalizing the
definition of cutout chart. Instead of frontloading the half-line coordinates
and demanding that the cutout function $\psi$ be defined on
$V\subset[0,\infty)^{m}\times\mathbb{R}^{n}$ an open subset, we allow $V$ to
be an open subset of any product of copies of $[0,\infty)$ and $\mathbb{R}$ in
any order. This is convenient primarily for easy discussion of products (where
we would otherwise have to shuffle coordinates to move the half-line
coordinates to the front, introducing a sign). When discussing the orientation
on the boundary of $Z(\psi)$, one first passes to the boundary (with its
standard boundary orientation) and then passes to the regular zero set of
$\psi^{\partial}$ on the corresponding oriented boundary strata.

This in hand, we can define an oriented stratified-smooth space.

######  Definition .

Let $(X,\Delta)$ be a stratified-smooth space. An _orientation_ of
$(X,\Delta)$ is a choice of orientation for each top stratum so that for each
$x\in X$, there _exists_ a cutout chart $(U,V,\psi,\phi)$ around $x$ so that
the induced diffeomorphism on the top stratum, $\phi:U^{\circ}\to
Z(\psi)^{\circ}$, is orientation-preserving. ∎

In understanding this definition, a useful example to keep in mind is the
$\theta$-graph: there is a continuous function on $[0,1]^{2}$ (which, on each
stratum, is smooth and has zero as a regular value) whose zero set on the
boundary is $\\{1/2\\}\times\\{0,1\\}$ but on the interior is three arcs
running between these points. An orientation on this stratified-smooth space
orients two of the arcs in one direction and one arc in the opposite
direction. For a given cutout chart, only one of these six possible
orientations will be oriented compatibly with that cutout chart; but the
definition only asks that _some_ cutout chart be oriented compatibly.

${\bullet}$${\bullet}$${>}$${>}$${<}$ ${\bullet}$${\bullet}$${>}$${<}$${>}$
${\bullet}$${\bullet}$${<}$${>}$${>}$

${\bullet}$${\bullet}$${<}$${<}$${>}$ ${\bullet}$${\bullet}$${<}$${>}$${<}$
${\bullet}$${\bullet}$${>}$${<}$${<}$

The six ways to make the $\theta$-graph into an oriented stratified-smooth
space.

${\bullet}$${\bullet}$${<}$${<}$${<}$ ${\bullet}$${\bullet}$${>}$${>}$${>}$

The two orientations of the top strata which do not define an oriented
stratified-smooth space.

There is no harm in the fact that this connected stratified-smooth space
admits six distinct orientations. That the two ‘orientations’ in which all
three arcs go the same direction are not compatible with any cutout chart is
shown by the following lemma, which is the same argument as in [SeSm, Section
2c]. This lemma will be the foundation of our proof that the boundary of a
1-dimensional stratified-smooth space is zero points (counted with sign), and
that the boundary orientation is well-defined.

######  Lemma .

Suppose $\psi:[0,\infty)\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is continuous
and smooth with zero as a regular value on each stratum. Suppose furthermore
that $(\psi^{\partial})^{-1}(0)=\\{0\\}$, oriented negatively. Write
$\psi_{t,r}:\\{t\\}\times D^{n}_{r}\to\mathbb{R}^{n}$ for the restriction to a
slice at time $t$ and the ball of radius $r>0$ around the origin. For
sufficiently small $t,r$, the map $\psi_{t,r}$ has zero as a regular value and
$\psi_{t,r}^{-1}(0)$ is a finite collection of points whose orientations sum
to one.

###### Proof.

To briefly remark on the signs, remember that $\psi^{\partial}$ has domain
$\\{0\\}\times\mathbb{R}^{n}$ with the boundary orientation, which is negative
the standard orientation; thus $\psi_{0,r}^{-1}(0)$ is a single point,
oriented oppositely to $(\psi^{\partial})^{-1}(0)$. In particular,
$\psi_{0,r}^{-1}(0)$ is a single _positively-oriented_ point.

By the Sard lemma, for a generic set of $t,r\in(0,\infty)^{2}$, the given map
has zero as a regular value and has no zeroes on the boundary of $D^{n}_{r}$.
Because $D^{n}_{r}$ is compact, the desired count is finite. Further, for
$r,t$ sufficiently small, the subset $S^{n-1}_{r}\times[0,t]$ does not meet
$\psi^{-1}(0)$ (by continuity of $\psi$ and its behavior on the boundary).
Write

$\overline{\psi}_{t,r}=\psi i_{t,r}:S^{n-1}\to\mathbb{R}^{n}\setminus 0$

for the composite of the inclusion
$S^{n}_{r}\hookrightarrow\\{t\\}\times\mathbb{R}^{n}$ and the map $\psi$. Then
the degree of $\overline{\psi}_{t,r}$ is precisely the number of intersection
points of $\\{t\\}\times D^{n}_{r}$ with $\psi^{-1}(0)$.

By assumption, $\deg\overline{\psi}_{0,r}=1$ for all $r$. Further, because
$S^{n-1}_{r}\times[0,t]$ does not meet $\psi^{-1}(0)$, the maps
$\overline{\psi}_{t,r}$ are homotopic for all sufficiently small $t\geq 0$ and
$r>0$. Thus the degree of $\overline{\psi}_{t,r}$ for small $t>0$ is also
equal to one, giving the desired result. ∎

We may understand these counts intrinsically. Suppose $X$ is an oriented
1-dimensional stratified-smooth space. We classify the connected components of
the open stratum of $X$ as three types: components which are equal to their
own closure are called _isolated_. Otherwise, either
$\overline{C}=C\cup\\{x\\}$ or $\overline{C}=C\cup\\{x,y\\}$. In either case,
$C\cong\mathbb{R}$; if both ends of $C$ accumulate (and accumulate to the same
point) we say $C$ is _heteroclinic_ , whereas if exactly one end of $C$
accumulates to $x$ we say $C$ is homoclinic. In the latter case, we say $C$
_leaves $x$_ or _approaches $x$_ depending on whether or not $x$ is an
accumulation point of the negatively-oriented end of $C$ or the positively-
oriented end, respectively.

######  Lemma .

In the situation of Lemma A.2, for sufficiently small $t,r>0$ the count
$\\#\psi_{t,r}^{-1}(0)$ is the number of homoclinic components of
$Z(\psi^{\circ})$ entering $0$ minus the number of homoclinic orbits of
$Z(\psi^{\circ})$ leaving $0$.

###### Proof.

This amounts to counting the contribution of each component to the count
$\psi_{t,r}^{-1}(0)$. Consider a neighborhood of $[0,\varepsilon]\times
D^{n}_{r}$ of $0$ with $\varepsilon,r>0$ sufficiently small that

$Z(\psi)\cap\partial\left([0,t]\times
D^{n}_{r}\right)=Z(\psi)\cap\left(\\{0,t\\}\times D^{n}_{r}\right),$

and so that this intersection is transverse. Write
$Z^{\prime}(\psi)=Z(\psi)\cap\left([0,t]\times D^{n}_{r}\right)$; this is a
compact 1-dimensional stratified-smooth space. Now consider what happens to
the components of $Z(\psi)$ as we pass to $Z^{\prime}(\psi)$:

  * •

An isolated component of $Z(\psi)$ either intersects $Z^{\prime}(\psi)$
trivially, or in an arc whose boundary lies on $\\{t\\}\times D^{n}_{r}$.

  * •

A heteroclinic component intersects with $Z^{\prime}(\psi)$ to give two arcs
running from $0$ to $\\{t\\}\times D^{n}_{r}$ (one oriented forward, the other
oriented backward) and a collection of arcs running from $\\{t\\}\times
D^{n}_{r}$ back to $\\{t\\}\times D^{n}_{r}$.

  * •

A homoclinic component intersects with $Z^{\prime}(\psi)$ to give one arc
running from $0$ to $\\{t\\}\times D^{n}_{r}$ (oriented forward if the
component leaves $x$, and backward if it approaches $x$) and a handful of arcs
running from $\\{t\\}\times D^{n}_{r}$ to $\\{t\\}\times D^{n}_{r}$.

Now $\\#\psi_{t,r}^{-1}(0)$ is the same as the oriented sum of points in the
intersection of $Z^{\prime}(\psi)$ with the submanifold $\\{t\\}\times
D^{n}_{r}$. The intersection number of this submanifold with any arc which
runs from $\\{t\\}\times D^{n}_{r}$ back to itself is zero, while the
intersection with an arc running from $0$ to $\\{t\\}\times D^{n}_{r}$ is $+1$
and the intersection with an arc running $\\{\varepsilon\\}\times D^{n}_{r}$
to $0$ is $-1$. This means isolated components contribute nothing to the
count; heteroclinic components contribute $1-1=0$ to the count; and homoclinc
components contribute either $1$ or $-1$ to the count, depending on whetherh
they leave $x$ or approach $x$. ∎

In particular, this explains why only six of the eight apparent orientations
of the $\theta$-graph are compatible with a cutout chart. What’s more, this
description of the signed count is _intrinsic to the stratified space_ : if
$Z(\psi)$ is 1-dimensional, we may determine whether or not $\partial Z(\psi)$
is compatibly oriented with the orientation on the interior by investigating
the orientation on each arc, without ever passing to a cutout chart.

######  Corollary .

If $Z(\psi_{i})$ are 1-dimensional and $\phi:Z(\psi_{1})\to Z(\psi_{2})$ is an
isomorphism of stratified-smooth spaces which is orientation-preserving on the
interior, then $\phi$ is orientation-preserving on the boundary, as well.

We can now use this to confirm that the boundary orientation may be
intrinsically defined in general (so is independent of the choice of a cutout
chart).

######  Lemma .

Suppose $\psi_{i}:[0,\infty)\times\mathbb{R}^{n_{i}}\to\mathbb{R}^{m_{i}}$ are
a pair of continuous maps which are smooth and have zero as a regular value on
each stratum; further suppose $\phi:Z(\psi_{1})\to Z(\psi_{2})$ is an
isomorphism of stratified-smooth spaces which is orientation-preserving on the
top stratum. Then $\phi$ is also orientation-preserving on the boundary
strata.

###### Proof.

We reduce to the 1-dimensional case. If $(X,\Delta)$ is a naive stratified-
smooth space, and $f:X\to\mathbb{R}^{m}$ is a naive stratitified-smooth map,
we say $f$ is _special_ if $f^{-1}(0)$ is a stratified-smooth space (that is,
it admits cutout charts). Notice that if $\phi:X\to Y$ is an isomorphism of
naive stratified-smooth spaces, and $f:Y\to\mathbb{R}^{m}$ is special, then
$f\phi$ is also special; $\phi$ induces an isomorphism
$f^{-1}(0)\cong(f\phi)^{-1}(0)$, and the property of being a stratified-smooth
space — that is, admitting cutout charts — is preserved under isomorphisms of
naive stratified-smooth spaces.

Now consider the situation in the statement of the lemma. Pick $x\in\partial
Z(\psi_{1})$; we aim to show that $d\phi_{x}$ is orientation preserving as a
map $T_{x}\partial Z_{1}\to T_{\phi(x)}\partial Z_{2}$. Passing to a smaller
open neighborhood of $x$ if necessary, choose a stratified-smooth function
$f:Z(\psi_{2})\to\mathbb{R}^{n_{2}-m_{2}}$ so that $f$ induces an isomorphism
$T_{x}\partial Z_{2}\to\mathbb{R}^{n_{2}-m_{2}}$. Explicitly, one may choose
$f$ to be a linear map restricted to
$Z\subset[0,\infty)\times\mathbb{R}^{n_{2}}$. Because this function extends to
a cutout chart, it follows that $f^{-1}(0)\subset Z(\psi_{2})$ again carries
the structure of a stratified-smooth space, so $f$ is special. Call this zero
locus $Z(\psi_{2},f)$.

We thus have a stratification-preserving homeomorphism which is a
diffeomorphism on each stratum $\phi_{f}:Z(\psi_{1},f\phi)\to Z(\psi_{2},f)$.
Further, because $\phi$ was orientation-preserving and the construction
defining orientations of zero sets is natural, $\phi_{f}$ is orientation-
preserving on the top (1-dimensional) stratum and is orientation-preserving on
the boundary stratum if and only if $\phi$ is orientation-preserving at $x$.

But by Corollary A.2, we see that $\phi_{f}$ is indeed orientation-preserving
on the boundary; thus $\phi$ is orientation-preserving at an arbitrary
boundary point, as desired. ∎

Now we are prepared to define the boundary of oriented stratified-smooth
spaces and check the basic properties.

######  Definition .

Let $(X,\Delta)$ be an oriented stratified-smooth space. Its _boundary_ is the
disjoint union

$\partial X=\bigsqcup_{d(s)=\dim X-1}\overline{X_{s}},$

where each $X_{s}$ is oriented with the _boundary orientation_ induced by a
choice of cutout chart at an arbitrary point $x\in X_{s}$. ∎

Note that this boundary orientation is well-defined by the previous lemma: it
is independent of the choice of cutout chart. Because an orientation on a
stratified-smooth space is induced from an orientation on
$[0,\infty)^{m}\times\mathbb{R}^{n}$ via a cutout chart, the boundary
operation on stratified-smooth spaces shares many formal properties with the
usual boundary operator on oriented smooth manifolds with corners.

######  Proposition .

Let $(X,\Delta)$ be an oriented stratified space. Then the double boundary
$\partial^{2}(X,\Delta)$ supports an orientation-reversing involution. More
precisely,

$\partial^{2}(X,\Delta)\cong\bigsqcup_{d(s)=\dim
X-2}\overline{X_{s}}\sqcup-\overline{X_{s}}.$

###### Proof.

We saw that

$\partial^{2}X=\bigsqcup_{\begin{subarray}{c}s\in\Delta\\\ d(s)=\dim
X-2\end{subarray}}\overline{X_{s}}\sqcup\overline{X_{s}}$

as unoriented stratified-smooth spaces in the proof of Proposition A.1. We
need to verify that these two copies of $\overline{X_{s}}$ appear with the
opposite orientation.

Because the definition of boundary orientation is local, it suffices to verify
this claim for a given codimension-2 face of $Z(\psi)$. But the (iterated)
boundary orientation on a given stratum of $Z(\psi)$ is induced in a uniform
way by the (iterated) boundary orientation on the corresponding stratum of
$[0,\infty)^{m}\times\mathbb{R}^{n}$, and the stated property holds for
Euclidean space with corners. ∎

We have an analogue of the usual result that the boundary of a compact
oriented 1-manifold consists of a finite number of points whose signs sum to
zero. The proof is essentially identical to [M, Lemma 6.3] and to the argument
opening [SeSm, Section 2c], and it is an immediate application of Lemma A.2.

######  Proposition .

Let $(X,\Delta)$ be a compact connected oriented 1-dimensional stratified-
smooth space. Then $\partial(X,\Delta)$ is a finite set of points which are
canonically oriented with a sign $\pm 1$; the sum of these signs is zero, and
hence $\partial(X,\Delta)$ supports an orientation-reversing diffeomorphism.

###### Proof.

First, $\Delta$ is a finite set, so there are finitely many 0-dimensional
strata. Each bottom stratum is a closed set, hence compact because $X$ is
compact. Thus the 0-dimensional strata in $\Delta$ form a compact smooth
manifold of dimension 0.

Now each zero-dimensional stratum has a cutout chart
$(U_{s},V_{s},\psi_{s},\phi_{s})$. If we delete each neighborhood $U_{s}$,
what remains is a compact 1-dimensional manifold with boundary
$X^{\prime}\subset X$, and it is well-known that the boundary of a
1-dimensional manifold with boundary consists of zero points, counted with
sign.

To be more precise, $\partial X^{\prime}=\sqcup_{s}\partial U_{s}$. Then to
conclude, one needs to verify that the signed count of points in $\partial
U_{s}$ agrees with the signed count of points in $X_{s}$; but (choosing the
$U_{s}$ appropriately) this is precisely the content of Lemma A.2. ∎

We conclude with a discussion of orientations on fiber products.

######  Proposition .

If $X$ and $Y$ are oriented stratified-smooth spaces and $f:X\to B$ and
$g:Y\to B$ are stratified-smooth maps to an oriented smooth manifold $B$
without boundary, so that $f$ and $g$ are stratified-transverse, then
$X\times_{B}Y$ has the natural structure of an oriented stratified-smooth
space and

$\partial(X\times_{B}Y)=(\partial X)\times_{B}Y\sqcup(-1)^{\dim X+\dim
B}X\times_{B}(\partial Y).$

###### Proof.

Recall that the fiber-product of smooth manifolds is oriented with ‘$B$ in the
middle’. Precisely, at the level of vector spaces, suppose $f_{i}:V_{i}\to W$
are transverse. Write $I_{j}=\text{im}(f_{j})$ and $K_{j}=\text{ker}(f_{j})$;
additionally write $I_{1}\cap I_{2}=I_{12}$.

Split $I_{1}\cong I_{1}^{\prime}\oplus I_{12}$ while $I_{2}\cong I_{12}\oplus
I_{2}^{\prime}$. Further split

$V_{1}\cong K_{1}\oplus I_{1}\cong K_{1}\oplus I_{1}^{\prime}\oplus I_{12}$

and

$V_{2}\cong I_{2}\oplus K_{2}\cong I_{12}\oplus I_{2}^{\prime}\oplus K_{2}.$

Lastly, we may now split

$W\cong I_{1}^{\prime}\oplus I_{12}\oplus I_{2}^{\prime}.$

If $V_{1},V_{2},$ and $W$ are oriented, choose orientations on
$I_{1}^{\prime},I_{2}^{\prime},I_{12},K_{1},K_{2}$ arbitrarily so that the
three above isomorphisms are orientation-preserving; there are four ways to do
so, say

$\displaystyle(o_{I_{1}^{\prime}},o_{I_{2}^{\prime}},o_{I_{12}},o_{K_{1}},o_{K_{2}}),$
$\displaystyle(-o_{I_{1}^{\prime}},o_{I_{2}^{\prime}},-o_{I_{12}},o_{K_{1}},-o_{K_{2}}),$
$\displaystyle(o_{I_{1}^{\prime}},-o_{I_{2}^{\prime}},-o_{I_{12}},-o_{K_{1}},o_{K_{2}}),$
$\displaystyle(-o_{I_{1}^{\prime}},-o_{I_{2}^{\prime}},o_{I_{12}},-o_{K_{1}},-o_{K_{2}}).$

Now orient $V_{1}\times_{W}V_{2}\cong K_{1}\oplus I_{12}\oplus K_{2}$ so that
this isomorphism is orientation-preserving; this defines the same orientation
on the fiber product in all four cases above, so is well-defined. Applying
this construction to the appropriate tangent spaces gives rise to our fiber
product orientation on $X\times_{B}Y$.

The cutout charts constructed in the proof of Proposition A.1 are not
orientation-preserving. However, we can analyze the difference from the
expected orientation.

Suppose the cutout chart used for $X$ in the above proposition is
$\psi:V\to\mathbb{R}^{\ell}$, and write $b=\dim B,y=\dim Y$; a straightforward
and tedious linear algebra argument shows that the orientation of the cutout
chart in Proposition A.1 disagrees with the fiber product orientation by a
factor of $(-1)^{\ell b+\ell y+by}$, and in particular this is consistent over
the entire cutout chart.

It follows that the fiber product orientation on $X\times_{B}Y$ does indeed
give the fiber product the structure of an oriented stratified-smooth space;
one may simply flip the orientation on the cutout charts constructed before,
if necessary, to make them orientation-preserving.

The desired boundary orientation follows from a tedious definition chase and
the above sign computation; notice that it agrees with the expected boundary
orientations on a fiber product when this is a transverse fiber product of
smooth manifolds with boundary. ∎

######  Remark .

Suppose the maps $X\to B$ and $Y\to B$ are stratumwise submersions. Then the
coordinate-swapping diffeomorphism $\phi:X\times_{B}Y\to Y\times_{B}X$ has
sign $(-1)^{(x-b)(y-b)}$, with $x=\dim X$ and so on. Because this is computed
in the top stratum, this result follows from the corresponding fact for smooth
manifolds, which is straightforward from the definition of fiber product
orientation. ∎

We conclude with a similar computation for the final proposition of Appendix
A.1.

######  Proposition .

Suppose $(X,\Delta)$ is an oriented stratified-smooth space and
$L:X\to\mathbb{R}$ a stratified-smooth map, with $c$ a regular value. Then
$X^{\leq c}$ is also an oriented stratified-smooth space, with boundary

$\partial X^{\leq c}=(-1)^{\dim X-1}L^{-1}(c)\sqcup(\partial X)^{\leq c},$

where $L^{-1}(c)$ is oriented as a regular zero set.

###### Proof.

The orientations on the top strata are inherited as an open subset of $X$.
Recall that we orient zero sets $Z(\psi)$ for $\psi:V\to\mathbb{R}^{\ell}$ by
the rule that $TZ\oplus\mathbb{R}^{\ell}\cong TV$.

It is straightforward linear algebra and definition-pushing to see that the
cutout charts constructed in Proposition A.1 are orientation-preserving with
respect to the existing orientation on the top stratum of $X$; this amounts to
saying that

$(TZ\oplus\mathbb{R}^{\ell})\oplus\mathbb{R}\cong TV\oplus\mathbb{R}$

is orientation-preserving, where the map is upper-triangular and the identity
on the bottom-right entry, and we know that the top-left entry is orientation-
preserving.

In computing the sign on the boundary stratum $L^{-1}(c)$, the additional
factor arises because boundaries are oriented with an outward-normal-first
convention, while zero sets place the $d/dt$ factor as the final basis
element; comparing them requires commuting a factor of $\mathbb{R}$ across
$TL^{-1}(c)$, introducing the stated sign. ∎

###  A.3 The real blowup of a stratified-smooth space

In Section 5 of the main text, we will find ourselves in the following
situation.

Suppose we have a stratified-smooth space $P$, a vector bundle $q:E\to M$ over
a smooth manifold, and a stratified-smooth map $\psi:P\to E$ transverse to the
zero section. We will want to construct, in a canonical way, a stratified-
smooth space $P^{\prime}$ with a map $\psi^{\prime}:P^{\prime}\to S(E)$, where
$S(E)$ is the unit sphere bundle of $E$. Over the locus
$P\setminus\psi^{-1}(0)$, the map $\psi/\|\psi\|$ gives a canonical such lift;
however, this lift cannot be extended over the locus where $\psi=0$.

To remedy this, we can replace our stratified-smooth space with its _real
blowup_ (or simply ‘blowup’) along $\psi=0$; the definition of this space
furnishes it with a canonical such lift.

In this section, we first define this blowup construction and verify that the
result is again a stratified-smooth space. After that, we define an
equivalence relation on stratified-smooth spaces with maps to $M$ so that a
stratified-smooth space is equivalent to any of its real blowups. We conclude
by verifying some technical facts which we will make use of in Sections
5.3–5.5.

######  Construction .

Let $P$ be an oriented stratified-smooth space. Let $M$ be a closed smooth
manifold equipped with an oriented vector bundle $q:E\to M$, and a stratified-
smooth map $\psi:P\to E$ transverse to the zero section. Give $E$ a fiberwise
metric; its only purpose is to pick out a sphere bundle of $E$.

We define $B(\psi)\subset P\times[0,\infty)\times S(E)$ to be the subset given
by

$B(\psi)=\\{(p,t,v)\mid\psi(p)=tv\\}.$

Both maps $\pi_{1}:B(\psi)\to P$ and $\pi_{3}:B(\psi)\to S(E)$, given by
projection to first and last coordinate, will be important to us.

Let us give $B(\psi)$ the structure of a stratified-smooth space.

If $P$ has face poset $\Delta$, then call $B(\psi)$’s face poset
$\Delta_{B}(\psi)\subset\Delta\times\\{-1,0\\}$. To be explicit about this, we
have a subset $\Delta_{Z}(\psi)\subset\Delta$ given by the faces on which
$\psi^{-1}(0)\cap P_{s}\neq\varnothing$. Then $B(\psi)$’s face poset is

$\Delta_{B}(\psi)=\Delta_{Z}(\psi)\times-1\sqcup\Delta\times 0,$

with grading function $d_{\psi}(s,-1)=d(s)-1$ and $d_{\psi}(s,0)=d(s)$, and
with order induced from $\Delta\times\\{-1,0\\}$ with the lexicographic order.

For any $s\in\Delta$, the strata $(s,0)$ are given by

$B(\psi)_{s,0}=\pi_{1}^{-1}\left(P_{s}\setminus\psi^{-1}(0)\right).$

Being the inverse image of locally closed subsets by a continuous map, these
are locally closed; a point in this stratum takes the form
$\left(p,\|\psi(p)\|,\psi(p)/\|\psi(p)\|\right)$ for $p\in P_{s}$, and we give
$B(\psi)_{s,0}$ the unique smooth structure so that $\pi_{1}:B(\psi)_{s,0}\to
P_{s}\setminus\psi^{-1}(0)$ is a diffeomorphism.

On the other hand, if $s\in\Delta_{Z}(\psi)$, then
$B(\psi)_{s,-1}=\pi_{1}^{-1}(Z(\psi)_{s});$ a point in this stratum takes the
form $(p,0,v)$ for $p\in\psi^{-1}(0)\cap P_{s}=Z(\psi)_{s}$ and any $v\in
S(E_{q\psi(p)})$. In this case, $\pi_{3}$ gives $B(\psi)_{s,-1}$ the structure
of a sphere bundle over $Z(\psi)_{s}$, canonically identified with the
pullback of the sphere bundle $E\to M$ under $f|_{Z(\psi)_{s}}$. We give
$B(\psi)_{s,-1}$ the unique smooth structure so that this identification is a
diffeomorphism.

Because

$\dim Z(\psi)_{s}=d(s)-\text{rank}(E),$

and the sphere bundle has fiber of dimension $\text{rank}(E)-1$, we see that
$\dim B(\psi)_{s,0}=d(s)-1=d_{\psi}(s,0)$.

It is straightforward to confirm that

$\displaystyle\overline{B(\psi)_{s,-1}}=\bigcup_{\begin{subarray}{c}t\in\Delta_{Z}\\\
t\leq s\end{subarray}}B(\psi)_{t,-1},$
$\displaystyle\overline{B(\psi)_{s,0}}=\bigcup_{\begin{subarray}{c}i\in\\{-1,0\\}\\\
t\in\Delta_{Z}\\\ t\leq s\end{subarray}}B(\psi)_{t,i}.$

Because we have an isomorphism of naive stratified-smooth spaces
$\pi_{1}:B(\psi)_{\geq s,0}\cong P_{\geq s}$, a cutout chart around any
$(p,r,v)\in B(\psi)_{s,0}$ is induced by a choice of cutout chart around $p\in
P_{s}$ (simply lift it along the isomorphism $\pi_{1}$).

Cutout charts around $(p,0,v)\in B(\psi)_{s,-1}$ are more intricate. Suppose
$(p,0,v)\in B(\psi)_{s,-1}$. Pick a cutout chart

$\big{(}U\subset P,V,f:V\to\mathbb{R}^{k},\phi:U\to Z(f)\big{)}$

for $p\in P_{s}$. We may choose $U$ small enough so that $\psi(U)$ lands
inside a trivializing chart for the vector bundle $E$; trivializing the bundle
and setting $m=\dim M,e=\text{rank}(E)$, we may write $\psi$ as a map

$\psi:Z(f)\to\mathbb{R}^{m}\times\mathbb{R}^{e}.$

By definition of stratified-smooth map, we may choose this cutout chart so
that the section $\psi$ extends to a map
$\widehat{\psi}=(\sigma,g):V\to\mathbb{R}^{m}\times\mathbb{R}^{e}$.

Now choose a small chart $j:\mathbb{R}^{e-1}\to S^{e-1}\cong E_{q\psi(p)}$
around $v$. Recall here that $q:E\to M$ is the bundle projection.

Our cutout chart for $B(\psi)$ around $p$ has domain

$V^{\prime}=V\times[0,\varepsilon)\times\mathbb{R}^{e-1}$

and smooth map

$h(x,t,v)=(f(x),g(x)-tj(v))\in\mathbb{R}^{k}\times\mathbb{R}^{e}.$

The map $\phi^{\prime}:h^{-1}(0)\to B(\psi)$ is given by

$(x,t,v)\mapsto(\phi^{-1}(x),t,j(v))\in B(\psi)\subset P\times[0,\infty)\times
S(E);$

this is a homeomorphism onto its image, the set of $(t,x,v)\in B(\psi)$ with
$\|\psi(x)\|<\varepsilon$ and $x\in U$, which is open. Because this map
$\phi^{\prime}$ also induces a bijection between the natural stratifications,
it gives a cutout chart near $(p,0,v)$.

Now orient $B(\psi)$ is oriented so that the map $\pi_{1}:B(\psi)\to P$ is an
orientation-preserving diffeomorphism off of $Z$. It is easy to see the cutout
charts constructed above are orientation-preserving, so that this gives
$B(\psi)$ the structure of an oriented stratified-smooth space. ∎

In the situation above, $(P,\phi)$ and $(B(\psi),\phi\pi_{1})$ are both
stratified-smooth spaces equipped with stratified-smooth maps to $M$; if $P$
is connected, then $B(\psi)$ is as well. For the sake of our technical
arguments in the main text, we wish to construct an equivalence relation on
connected stratified-smooth spaces mapping to $M$ so that these are considered
equivalent. To get there — and to define the geometric chain complex in the
next section — we need a handful of definitions.

######  Definition .

Let $M$ be a smooth manifold. The set of _stratified-smooth probes_ into $M$,
written $\mathsf{P}(M)$, is the set of pairs $(P,\phi)$, where $P$ is a
compact connected oriented stratified-smooth space, $\phi:P\to M$ is a
stratified-smooth map, and we consider $(P,\phi)=(Q,\eta)$ if there exists an
orientation-preserving isomorphism $f:P\to Q$ with $\eta=\phi f$. This set
carries an involution, $\iota(P,\phi)=(-P,\phi)$, orientation-reversal of $P$.
We call a probe which is fixed under this involution _trivial_.

We say that a probe $(P,\phi)$ is _degenerate_ if $\dim P>0$ and, for all
strata $P_{s}$ of dimension $d(s)>0$ and all $p\in P_{s}$, we have

$\text{rank}(d\phi_{s}:T_{p}P_{s}\to M)<d(s).$

We write $\mathsf{D}(M)\subset\mathsf{P}(M)$ for the set of degenerate probes
in $M$; this subset is $\iota$-invariant. ∎

######  Remark .

There are two other notions of degenerate probe in the literature, defined
using: the _small image_ condition of [LipGH] and the _small rank_ of [FMMS2].
The first says that a probe is small rank if its image can be covered by the
image of a probe of smaller dimension. For the latter, the ‘small rank’
condition is that $d\phi$ has non-maximal rank on the top stratum. Both
authors then say that a degenerate probe is a probe of small image/rank whose
boundary is the union of a trivial probe and one of small image/rank.

Our definition of ‘degenerate probe’ includes fewer probes (for instance, the
projection map $[0,1]^{2}\to[0,1]$ is not degenerate according to our
definition, whereas it is for both of these authors), and in particular our
$C_{*}^{gm}(M)$ will usually be supported in infinitely many degrees; for
these authors, it is supported in degrees $*\in[0,\dim M+1]$. However, when
analyzing the notion of collapse-equivalence below, it is difficult to work
with the condition that the boundary be a sum of a small rank probe and a
trivial probe. The definition chosen above is better suited for studying
collapse-equivalence. ∎

We will now define _collapse-equivalence_ of stratified-smooth spaces (though
we will only apply it under the assumption that the given stratified-smooth
spaces are connected), and then verify that $(P,\phi)$ and
$(B(\psi),\phi\pi_{1})$ are collapse-equivalent.

######  Definition .

Let $(P,\Delta)$ be a stratified-smooth space. We say a _stratified-smooth
subspace of codimension $k$_ is a stratified-smooth space $(Z,\Delta_{Z})$, an
order-preserving injection $\overline{j}:\Delta_{Z}\to\Delta$ with
$d(\overline{j}(s))=d_{Z}(s)+k$, and a topological embedding $j:Z\to P$ so
that $j$ restricts on each stratum $Z_{s}$ to a smooth embedding

$j:Z_{s}\to P_{\overline{j}(s)}.$

We identify $Z$ with its image $j(Z)$ in the following discussion.

Let $(P,\phi)\in\mathsf{P}(M)$. We say that a closed subset $P_{SL}\subset P$
is a _small locus_ if $P_{SL}=P_{\operatorname{codim}>0}\cup P_{D}$, where
$P_{\operatorname{codim}>0}$ is the union of finitely many codimension $k>0$
stratified-smooth subspaces of $P$ (not necessarily compact), and
$P_{D}\subset\partial P$ is the union of degenerate strata of positive
codimension.

Suppose there exist small loci $P_{SL}\subset P,Q_{SL}\subset Q$, and an
orientation-preserving diffeomorphism of stratified smooth spaces
$f:P\setminus P_{SL}\to Q\setminus Q_{SL}$ so that $f\phi=\eta$. Then we say
$(P,\phi)$ and $(Q,\eta)$ are collapse-equivalent, and write this as
$(P,\phi)\sim_{C}(Q,\eta)$. ∎

This is indeed an equivalence relation, and plays well with the notion of
degeneracy and boundary operators.

######  Lemma .

Collapse-equivalence gives an equivalence relation $\sim_{C}$ on
$\mathsf{P}(M)$. If $(P,\phi)$ is degenerate, then so is $(Q,\eta)$. If
$(P,\phi)\sim_{C}(Q,\eta)$ then $\iota(P,\phi)\sim_{C}\iota(Q,\eta)$ and
$\partial(P,\phi)\sim_{C}\partial(Q,\eta)+(R,\sigma_{D})$, where
$(R,\sigma_{D})$ is degenerate.

###### Proof.

Reflexivity and symmetry are obvious, as is the claim about orientation-
reversal.

For transitivity, observe that the union of two small loci is again a small
locus; if $(P,\phi)\sim_{C}(Q,\eta)\sim_{C}(R,\psi)$ and we have small loci
$P_{SL}\subset P,$ $Q^{1}_{SL},Q^{2}_{SL}\subset Q$ and $R_{SL}\subset R$ so
that

$\displaystyle f:(P\setminus P_{SL},\phi)\cong(Q\setminus Q^{1}_{SL},\eta)$
$\displaystyle g:(Q\setminus Q^{2}_{SL},\eta)\cong(R\setminus R_{SL},\psi),$

Each of these small loci is written as a union
$P_{SL}=P_{\operatorname{codim}>0}\cup P_{D}$ of positive-codimension
subspaces and degenerate strata. Write

$P^{\prime}_{\operatorname{codim}>0}=P_{\operatorname{codim}>0}\cup
f^{-1}\mathopen{}\left(Q^{2}_{\operatorname{codim}>0}\right)\mathclose{}\quad\quad
R^{\prime}_{\operatorname{codim}>0}=R_{\operatorname{codim}>0}\cup
g\mathopen{}\left(Q^{1}_{\operatorname{codim}>0}\right)\mathclose{};$

take the locus $P^{\prime}_{D}$ to be the union of $P_{D}$ and the strata
intersecting $f^{-1}(Q^{2}_{D})$, similarly for $R^{\prime}_{D}$. Notice that
the components meeting $f^{-1}(Q^{2}_{D})$ are degenerate: those which aren’t
already in $P_{D}$ have small rank on the open dense set $(P\setminus
P_{SL})_{s}$ of each positive-dimension stratum, so by semicontinuity of rank
they have small rank on each positive-dimension stratum. So $P^{\prime}_{D}$
and similarly $R^{\prime}_{D}$ are both unions of degenerate strata of $P$ and
$R$.

Then $P^{\prime}_{SL}=P^{\prime}_{\operatorname{codim}>0}\cup P^{\prime}_{D}$
is a small locus, as is
$R^{\prime}_{SL}=R^{\prime}_{\operatorname{codim}>0}\cup R^{\prime}_{D}$.
Finally, $fg$ gives an isomorphism $(P\setminus
P^{\prime}_{SL},\phi)\cong(R\setminus R^{\prime}_{SL},\psi)$. This proves
transitivity of $\sim_{C}$.

As implicitly mentioned above, degeneracy off of a small locus implies
degeneracy on the whole probe, so collapse-equivalence preserves degeneracy.

The claim about boundaries is also clear: for all strata not contained in the
small locus, the isomorphism $f:P\setminus P_{SL}\to Q\setminus Q_{SL}$
descends to an isomorphism $\overline{P_{s}}\setminus
P_{SL}\to\overline{Q_{f(s)}}\setminus Q_{SL}$. Any strata contained in the
small locus are degenerate, by definition of small locus. So $f$ provides a
collapse-equivalence between the non-degenerate parts of $\partial P$ and the
non-degenerate parts of $\partial Q$, and the remaining pieces are collapse-
equivalent and form the chain $(R,\sigma_{D})$ mentioned in the statement of
the proposition. ∎

Henceforth, we write $\mathsf{Q}(M)=\mathsf{P}(M)/\sim_{C}$. We write
$\mathsf{D}_{Q}(M)=\mathsf{D}(M)/\sim_{C}\subset\mathsf{Q}(M)$.

######  Lemma .

Let $\phi:P\to M$ be an oriented stratified-smooth probe equipped with a lift
$\psi:P\to E$ of $\phi$ to a section of a vector bundle $q:E\to M$, which is
stratified-transverse to the zero section. Then the real blowup $B(\psi)$
satisfies the following properties:

  1. (i)

$B(\psi)$ is an oriented stratified-smooth probe in $S(E)$ via the projection
map $\pi_{3}:B(\psi)\to S(E)$, with boundary relation

$\partial(B(\psi),\pi_{3})=(B(\psi|_{\partial P}),\pi_{3})-(-1)^{\dim
Z_{\psi}}(Z_{\psi}\times_{B}S(E),\pi_{2}).$

  2. (ii)

When composed with the projection map $q:S(E)\to M$, the blowup defines a
probe $(B(\psi),q\pi_{3})=(B(\psi),\phi\pi_{1})$ in $M$ which is collapse-
equivalent to $(P,q\psi)$.

###### Proof.

The boundary relation may be ready off from the description of the cutout
charts.

We defined collapse-equivalence so that this would be true. The projection map
$\pi_{1}:B(\psi)\to P$ is an isomorphism away from $Z=\psi^{-1}(0)\subset P$
(a small locus), and $(\pi^{-1}(Z),\phi\pi_{1})$ is a subset of $\partial
B(\psi)$ which factors through the fiber bundle projection $\pi_{1}^{-1}(Z)\to
Z$ with positive-dimensional fibers. It follows that every face of
$(\pi^{-1}(Z),\phi\pi_{1})$ is small rank, and hence this is a degenerate
probe. Thus $(B(\psi),\phi\pi_{1})$ is collapse-equivalent to $(P,\sigma)$. ∎

In the main text, we will need a variation on this which identifies two
stratified-smooth probes related to this blowup construction. We include this
construction and lemma in this section, as it’s in the spirit of the work
carried out above.

######  Construction .

Let $(P,\sigma)$ and $(Q,\eta)$ be transverse stratified-smooth probes mapping
submersively to $B$, equipped with lifts $\psi:P\to E$ and $\phi:Q\to E$
lifting $\sigma$ and $\eta$, respectively (where $E\to B$ is an oriented
vector bundle of rank $\text{rk}(E)>1$). Assume further that
$(p,t,q)\mapsto(1-t)\psi(p)-t\phi(q)$ is transverse over $P\times[0,1]\times
Q$ to the zero section.

We write $Z_{\psi\phi}\subset P\times_{B}([0,1]\times Q)$ for the zero set of
$(p,t,q)\mapsto(1-t)\psi(p)-t\phi(q)$; by the assumption that this is a
transverse zero set, $Z_{\psi\phi}$ is an oriented stratified-smooth space. If
$P$ and $Q$ are furthermore equipped with maps to $A$ and $C$, respectively,
$Z_{\psi\phi}$ defines a probe in $A\times C$.

There is another way to use these maps to construct a stratified-smooth probe
in $A\times C$. We write $B(\psi)$ and $B(\phi)$ for the blowups, considered
as probes in $A\times S(E)$ and $S(E)\times C$, and consider the probe
$B(\psi)\times_{S(E)}B(\phi)$ in $A\times C$. This fiber product is clearly
transverse over blowup strata (which map submersively onto the fibers of
$S(E)$), and as will be seen in the proof below, on the open strata the fact
that this is a transverse fiber product is equivalent to the statement that
$(1-t)\psi-t\phi$ is transverse to zero. ∎

In Section 5, we will see stratified-smooth chains of the form $Z_{\psi\phi}$
naturally appear from analytic considerations, whereas stratified-smooth
chains of the form $B(\psi)\times_{S(E)}B(\phi)$ appear in the same context
from algebraic considerations. To square these two perspectives, we will need
to see that these two stratified-smooth chains are collapse-equivalent.

######  Lemma .

In the situation of the above construction, the fiber product
$B(\psi)\times_{S(E)}B(\phi)$ is collapse-equivalent to $Z_{\psi\phi}$ as
probes in $A\times C$. If $B$ and $E$ are both even-dimensional, then this
collapse-equivalence is orientation-preserving (otherwise there is an overall
sign).

###### Proof.

One should intuit this claim as follows. On the open stratum where
$\psi,\phi\neq 0$, the space $B(\psi)\times_{S(E)}B(\phi)$ is diffeomorphic to
the subset of the fiber product $P\times_{B}Q$ of those pairs $(p,q)$ so that
$\psi(p)$ and $\phi(q)$ are positive scalar multiples of one another. On the
other hand, supposing $\psi(p),\phi(q)$ are both nonzero, there exists a
unique $t$ with $(1-t)\psi(p)=t\phi(q)$ if and only if the two vectors
$\psi(p)$ and $\phi(q)$ are positive scalar multiples of one another.

Indeed, this suggests a reasonably simple map from one to the other. Points in
the fiber product $B(\psi)\times_{S(E)}B(\phi)$ are sextuples $(p,t,v,q,s,w)$
with

$\sigma(p)=\eta(q),\;\;\;\psi(p)=tv,\;\;\;\phi(q)=sw,\;\;\;v=w.$

We would like to map $F:B(\psi)\times_{S(E)}B(\phi)\to Z_{\psi\phi}$ via

$f(p,t,v,q,s,w)=(p,\frac{t}{s+t},q).$

Notice that because the maps to $A$ and $C$ are defined by projecting to
$P\times Q$, and $F$ preserves the projection to $P\times Q$, the map $F$
preserves the map to $A\times C$.

However, $F$ is not defined everywhere, and is not surjective.

The map $F$ is ill-defined on is the set where $s=t=0$; these are the points
in the blowup corresponding to $\psi=\phi=0$. This is a union of degenerate
strata

$(Z_{\psi}\times_{B}S(E))\times_{S(E)}(S(E)\times_{B}Z_{\phi})=Z_{\psi}\times_{B}S(E)\times_{B}Z_{\phi}$

in the domain. This is degenerate because each stratum takes the form
$Z_{s}\times_{B}S(E)\times_{B}Z_{t}$, which factors through the projection to
$Z_{s}\times_{B}Z_{t}$ with positive-dimensional closed fibers, and hence has
small rank on every stratum.

Next, $F$ misses the codimension $\text{rank}(E)$ subset of $Z_{\psi\phi}$
defined by $t\psi-(1-t)\phi=0$; this condition implies that $\psi=\phi=0$,
because the points in $Z_{\psi\phi}$ have $(1-t)\psi-t\phi=0$ to begin with;
this subset can be identified with $Z_{\psi}\times_{B}[0,1]\times Z_{\phi}$.

Thus $F$ defines an isomorphism away from small loci in the domain and
codomain, which preserves the map to $A\times C$, and hence is a collapse-
equivalence.

What remains is a discussion of orientations. We will freely use global
symbols (such as $M$, or $B$) to mean small open subsets of these as
appropriate; because everything maps submersively to $B$ and $B$ is even-
dimensional, we may effectively ignore it by passing to fibers, so we suppress
it below, writing $E$ for an oriented vector space. We orient $S(E)$ with the
outward-normal-first convention, so that the scaling map $\mathbb{R}_{+}\times
S(E)\cong E-0$ is an orientation-preserving isomorphism.

An open locus in $B(\psi)\times_{S(E)}B(\phi)$ may be identified with

$P^{\langle v\rangle}\times S(E)\times Q^{\langle v\rangle},$

where the superscript indicates that we take a fiber above
$\mathbb{R}^{+}v\subset E$, where $v$ is fixed; in the first case, we orient
this so that the natural locally-defined map $P^{\langle v\rangle}\times
S(E)\cong P$ is an oriented diffeomorphism; in the second case $S(E)\times
Q^{\langle v\rangle}\cong Q$, as is our convention for orientations on the
fiber product. Then an open subset of the fiber product is identified with

$B(\psi)\times_{S(E)}B(\phi)\sim P^{\langle v\rangle}\times S(E)\times
Q^{\langle v\rangle}.$

On the other side, there is a natural identification of $Z_{\psi\phi}$ with
the zero locus of the map $(p,t,q)\mapsto(1-t)\psi(p)-t\phi(q)$ from
$P\times[0,1]\times Q$ to $E$. Using the identifications above near a point
$(p,t,q)$ with $\psi(p)$ and $\phi(q)$ positive scalar multiples of $v\in
S(E)$, we write this as the zero set of a map on $P^{\langle v\rangle}\times
S(E)\times[0,1]\times S(E)\times Q^{\langle v\rangle}$. Further, we identify a
neighborhood of $v\in S(E)$ with a neighborhood of zero in $\langle
v\rangle^{\perp}$, oriented so that $\langle v\rangle\oplus\langle
v\rangle^{\perp}\cong E$. If $\psi(p)=v_{p}$ and $\psi(q)=v_{q}$ with
$(1-t_{0})v_{p}=t_{0}v_{q}$, we can identify of $Z_{\psi\phi}$ near
$(p,t_{0},q)$ as the zero set of the map

$P^{\langle v\rangle}\times\langle v\rangle^{\perp}\times(0,1)\times\langle
v\rangle^{\perp}\times Q^{\langle v\rangle}\to E\quad\text{given
by}\quad(p,w,t,u,q)\mapsto(1-t)w-tu+(t-0-t)v_{p}+(t_{0}-t)v_{q}.$

For now, we ignore the outside terms and focus on the zero set, considered as
a function on $\langle v\rangle^{\perp}\times(0,1)\times\langle
v\rangle^{\perp}$.

The first part of the given expression lies in $\langle v\rangle^{\perp}$,
while the latter lies in $\langle v\rangle$; they both must be zero. Because
$v_{p}$ and $v_{q}$ are positive scalar multiples of one another,
$-(t-t_{0})(v_{p}+v_{q})$ is only zero if $t=t_{0}$, and then the first
equation gives $u=\frac{1-t_{0}}{t_{0}}w$, so by projection to the $w$-factor,
this zero set is diffeomorphic to $\langle v\rangle^{\perp}$.

To justify that this projection is an oriented diffeomorphism, we check that
for fixed $w_{0}$, the map

$(t,u)\mapsto(1-t)w_{0}-tu-(t-t_{0})(v_{p}+v_{q})$

is an oriented diffeomorphism $(0,1)\times\langle v\rangle^{\perp}\to E$
between open subsets. To see this, take the derivative; if $(t_{0},u_{0})$ is
fixed and $\\{u_{2},\cdots,u_{n}\\}$ is an oriented basis for $\langle
v\rangle^{\perp}$, at $(t_{0},u_{0})$ the maps the oriented basis
$\\{t,u_{2},\cdots,u_{n}\\}$ to
$\\{-u_{0}-w_{0}-v_{p}-v_{q},-tu_{2},\cdots,-tu_{n}\\}$; this is equivalent to
the oriented basis $\\{-v,-u_{2},\cdots,-u_{n}\\}$, which is equivalent to
$(-1)^{\text{rank}(E)}$ the standard basis. By the assumption that $E$ has
even rank, this proves the claim that $Z_{\psi\phi}$ is locally oriented
diffeomorphic to $P^{\langle v\rangle}\times\langle v\rangle^{\perp}\times
Q^{\langle v\rangle}$.

Finally, with respect to this description in the domain and codomain, the map
$F$ is isotopic to the identity. Thus $F$ is orientation-preserving, as
desired. ∎

###  A.4 Geometric homology from stratified-smooth spaces

In this section, we use the set of stratified-smooth probes modulo collapse-
equivalence to define a chain complex associated to smooth manifolds without
boundary which is natural under smooth maps. Using a variation on the
Eilenberg-Steenrod axioms, we will verify that the resulting homology theory
is naturally isomorphic to singular homology on closed manifolds (or, more
precisely, on pairs $(X,A)$ which are the interiors of a handlebody-pair).

We first outline the construction of the underlying graded group of our chain
complex.

######  Construction .

Write $\mathsf{Q}_{d}(X)=\mathsf{P}_{d}(X)/\sim_{C}$ for the set of
$d$-dimensional probes in $X$ modulo collapse-equivalence.

Set

$K_{d}(X)=\mathbb{Z}[\mathsf{Q}_{d}(X)]/\mathbb{Z}[\mathsf{Q}_{d}(X)]^{\iota}.$

The purpose of collapsing the $\iota$-invariant subgroup is to set
$\iota(P,\phi)=-(P,\phi)$, so that orientation-reversal corresponds to
negation; however, quotienting by this more naive relation would introduce
2-torsion into the complex, corresponding to probes which support orientation-
reversing isomorphisms. Collapsing the entire $\iota$-invariant subgroup
avoids this issue.

If one writes $\mathsf{T}_{d}(X)=\mathsf{Q}_{d}(X)^{\iota}$ for the ‘trivial
probes’ (those which admit an orientation-reversing isomorphism), then
$K_{d}(X)$ can be identified with the free abelian group on
$(\mathsf{Q}_{d}\setminus\mathsf{T}_{d})/\iota$.

Next, collapse the subgroup of $K_{d}(X)$ spanned by the image of the
degenerate probes. The degenerate probes $\mathsf{D}(X)\subset\mathsf{P}(X)$
define a $\sim_{C}$-invariant subset; we define the quotient by this relation
to be $\mathsf{D}_{Q}(X)\subset\mathsf{Q}(X)$. We set

$C^{gm}_{d}(X)=K_{d}(X)/\langle\mathsf{D}_{Q}(X)\rangle.$

We call this the complex of _geometric chains_ , and an element a geometric
chain. Again, the group $C^{gm}_{d}(X)$ is free abelian: one may give it a
basis in bijection with
$\big{(}\mathsf{Q}_{d}\setminus(\mathsf{T}_{d}\cup(\mathsf{D}_{Q})_{d})\big{)}/\iota.$

Then $C_{*}^{gm}(X)$ is the underlying graded abelian group of what we will
call the geometric chain complex. ∎

To justify the name ‘geometric chain complex’, we should construct a well-
defined boundary operator which squares to zero.

######  Lemma .

The operation $\partial(P,\phi)=\sum_{d(s)=\dim P-1}(\overline{P_{s}},\phi)$
sending a stratified-smooth probe to its geometric boundary gives a well-
defined map $\partial:C_{d}(X)\to C_{d-1}(X)$ with $\partial^{2}=0$.

###### Proof.

To be slightly more precise, recall that the faces $P_{s}$ are not assumed to
be connected, whereas probes are; when we write $(\overline{P_{s}},\phi)$
above, we mean the (finite) sum over its connected components. The compactness
of $P$ guarantees that the stratification poset $\Delta$ is finite, and the
compactness of each $\overline{P_{s}}$ guarantees there are only finitely many
such components, so this is indeed a finite sum. As for orientations, each
$\overline{P_{s}}$ is equipped with its boundary orientation.

It is clear that $\partial$ defines a map
$\mathbb{Z}[\mathsf{P}_{d}(X)]\to\mathbb{Z}[\mathsf{P}_{d-1}(X)]$; because
$\partial\iota(P,\phi)=\iota\partial(P,\phi)$, this decends to a map on
$\mathbb{Z}[\mathsf{P}]/\mathbb{Z}[\mathsf{P}]^{\iota}$.

Before moving on, we explain why the boundary of a degenerate chain is
degenerate, or rather, zero in

$\mathbb{Z}[\mathsf{P}_{d-1}]/\mathbb{Z}[\mathsf{P}_{d-1}]^{\iota}+\mathbb{Z}[\mathsf{D}].$

For $d>1$, this is obvious: a degenerate chain is defined to be a chain of
dimension $d>0$ so that each stratum $P_{s}$ of dimension $d(s)>0$ is
degenerate. When $d=1$, this is slightly more subtle.

First, observe that the assumption ‘$\phi$ has small rank’ means that
$d\phi=0$ on each connected component of a 1-dimensional stratum, and hence
$\phi$ is constant on each such stratum. It follows that $\phi$ is constant on
the closure of each connected component of the 1-dimensional stratum; because
$P$ is connected and the $0$-dimensional strata lie in the closure of the
1-dimensional strata, it follows that $\phi$ is also constant on the
0-dimensional stratum and thus that the constant value $\phi$ takes on
different components all coincide. So $\phi$ is constant. So by Proposition
A.2, the boundary $\partial(X,\phi)$ is an even number of points all mapping
to the same point, whose signed sum is zero. So $\partial(X,\phi)=0$ in
$\mathbb{Z}[\mathsf{P}_{0}]/\mathbb{Z}[\mathsf{P}_{0}]^{\iota}$.

Next, the boundary map mentioned above does _not_ descend naively to a well-
defined map on $\mathbb{Z}[\mathsf{Q}]$, because the boundary of collapse-
equivalent chains need not be collapse-equivalent; by Lemma A.3 all we know is
that if $\phi\sim_{C}\eta$, then $\partial\phi\sim\partial\eta+\sigma_{D}$,
where $\sigma_{D}$ is a degenerate probe.

Still, this is enough to see that $\partial$ descends to an operation on
$C_{*}$. For each equivalence class of nondegenerate probe
$[(P,\phi)]\in\mathsf{Q}$, we choose a representative $(P,\phi)$, and send

$\partial[(P,\phi)]=\sum_{d(s)=\dim P-1}[(\overline{P_{s}},\phi)];$

this is well-defined regardless of the choice of representative because
degenerate chains are identically zero in $C_{*}$, and if
$(P,\phi)\sim_{C}(Q,\eta)$, then
$\partial(P,\phi)\sim\partial(Q,\eta)+(R,\sigma_{D})$, where $(R,\sigma_{D})$
is degenerate.

Finally, the fact that this operation has $\partial^{2}=0$ is Proposition A.2.
∎

So for each smooth manifold without boundary $X$, we obtain a chain complex
$C_{*}^{gm}(X)$ and a corresponding homology group $H_{*}^{gm}(X)$. This
defines a functor on the category of smooth manifolds: if $f:X\to Y$ is a
smooth map, simply set $f_{*}(P,\phi)=(P,f\phi)$. It is obvious both that this
is functorial and that it respects the boundary operator.

We would like to extend this to a functor on the category of manifold pairs,
which enjoys a relative long exact sequence on homology. This requires a
technical lemma.

######  Lemma .

When $i:A\to X$ is an injective immersion, the map $C_{*}^{gm}(A)\to
C_{*}^{gm}(X)$ is injective.

###### Proof.

First we check that this map is injective on $\mathsf{P}$, then on
$\mathsf{P}\setminus\mathsf{D}$, then on $\mathsf{Q}\setminus D_{Q}$.

If $(P,\phi),(Q,\eta)\in\mathsf{P}(A)$ and $f:P\to Q$ is an isomorphism so
that $i\eta=i\phi f$. Because $i:A\to X$ is injective, we also have $\eta=\phi
f$. Thus if $(P,i\phi)=(Q,i\eta)$ then in fact $(P,\phi)=(Q,\eta)$. So the map
$\mathsf{P}(A)\to\mathsf{P}(X)$ is injective. An identical argument shows that
the induced map $\mathsf{Q}(A)\to\mathsf{Q}(X)$ is injective as well.

Suppose $(P,\phi)\in\mathsf{P}(A)$ is such that $(P,i\phi)$ is degenerate.
Because $i\phi$ has small rank on every stratum of $P$ of positive dimension,
and $di$ is injective, $\phi$ also has small rank on every stratum of $P$.
Thus $i_{*}\mathsf{P}(A)\cap\mathsf{D}(X)=i_{*}\mathsf{D}(A)$, and thus the
map $i_{*}$ restricts to a well-defined injective map

$\mathsf{P}(A)\setminus\mathsf{D}(A)\to\mathsf{P}(X)\setminus\mathsf{D}(X).$

Finally, by Lemma A.3, we also have
$i_{*}\mathsf{Q}(A)\cap\mathsf{D}_{Q}(X)=i_{*}\mathsf{D}_{Q}(A)$, and hence
$i_{*}$ gives a well-defined injection

$\mathsf{Q}(A)\setminus\mathsf{D}_{Q}(A)\to\mathsf{Q}(X)\setminus\mathsf{D}_{Q}(X).$

It remains to quotient by the involution, but again by injectivity of $i$, the
probe $(P,\phi)$ in $A$ will admit an orientation-reversing isomorphism if and
only if $(P,i\phi)$ does. So the map

$[\mathsf{Q}(A)\setminus\left(\mathsf{D}_{Q}(A)\cup\mathsf{T}(A)\right)]\to[\mathsf{Q}(X)\setminus\left(\mathsf{D}_{Q}(X)\cup\mathsf{T}(X)\right)]$

is injective; the same is true after quotienting by the (now fixed-point-free)
involution $\iota$, and hence the map $C_{*}^{gm}(A)\to C_{*}^{gm}(X)$ is
injective (it induces an injection on a basis). ∎

Suppose $A\subset X$ is an open submanifold. We may thus define a homology
group

$C_{*}^{gm}(X,A)=C_{*}^{gm}(X)/i_{*}C_{*}^{gm}(A),$

and this is obviously natural for maps of manifold pairs $(X,A)\to(Y,B)$.
Further, because the map $C_{*}^{gm}(A)\to C_{*}^{gm}(X)$ is injective, the
standard homological algebra shows that there exists a long exact sequence

$\cdots\xrightarrow{[-1]}H_{*}^{gm}(A)\to H_{*}^{gm}(X)\to
H_{*}^{gm}(X,A)\xrightarrow{[-1]}\cdots$

While we’re at it, we verify the two next-easiest Eilenberg-Steenrod axioms.

######  Lemma .

For two pairs $(X,A)$ and $(Y,B)$, the natural maps $i:(X,A)\to(X\sqcup
Y,A\sqcup B)$ and $j:(Y,B)\to(X\sqcup Y,A\sqcup B)$ induce an isomorphism

$H^{gm}_{*}(X,A)\oplus H^{gm}_{*}(Y,B)\cong H^{gm}_{*}(X\sqcup Y,A\sqcup B).$

###### Proof.

Because probes $(P,\phi)$ are defined with $P$ connected, a probe in $X\sqcup
Y$ factors through either $X$ or $Y$. Thus there are canonical bijections
$\mathsf{Q}(X\sqcup Y)=\mathsf{Q}(X)\sqcup\mathsf{Q}(Y)$, and the boundary
operator respects the obvious decomposition $C^{gm}_{*}(X\sqcup Y)\cong
C^{gm}_{*}(X)\oplus C_{*}^{gm}(Y)$. The inverse to this decompositon is
$i_{*}\oplus j_{*}$. The result follows. ∎

######  Lemma .

Let $\star$ be the one-point manifold. There is a canonical isomorphism
$C_{*}^{gm}(\star)\cong\mathbb{Z}[0]$, and hence a canonical isomorphism
$H_{*}^{gm}(\star)\cong\mathbb{Z}[0].$

###### Proof.

A probe $(P,\phi)$ to the point has $\phi$ constant; by definition, a probe is
degenerate if $\dim P>0$ and for all positive-dimensional strata, $d\phi$ has
rank less than the dimension of the stratum. Because $\phi$ is constant, this
is tautological, and thus all positive-dimensional probes are zero. Because
there is only one zero-dimensional connected stratified-smooth space,
$C_{*}^{gm}(\star)$ is free abelian on one degree-zero generator; further, the
unique zero-dimensional stratified-smooth probe has a canonical orientation:
the orientation $+1\in\mathbb{R}=\operatorname{det}(\mathbb{R}^{0})$. This
gives a canonical generator of $C_{0}^{gm}(\star)$, and hence a canonical
isomorphism to $\mathbb{Z}[0]$. ∎

The homotopy axiom is only slightly more complicated. The usual proof follows
by constructing a map to $C_{*}(X)\to C_{*}(I\times X)$ which satisfies an
appropriate boundary formula, but we choose to only define $C_{*}^{gm}(X)$ for
open manifolds. Instead, we construct an analogous map to $\mathbb{R}\times
X$.

######  Lemma .

There is a natural map

$\mathsf{Cyl}:C_{*}^{gm}(X)\to C_{*+1}^{gm}(\mathbb{R}\times X);$

writing $i^{0}:X\to\mathbb{R}\times X$ for the inclusion of $\\{0\\}\times X$
and similarly for $i^{1}$, we have the boundary formula

$\partial\mathsf{Cyl}+\mathsf{Cyl}\partial=i^{1}_{*}-i^{0}_{*}.$

Given a smooth homotopy $H:I\times X\to Y$, one may extend this noncanonically
to a smooth map $\hat{H}:\mathbb{R}\times X\to Y$. Then
$\hat{H}_{*}\mathsf{Cyl}$ gives a chain homotopy between $H_{0}$ and $H_{1}$.

###### Proof.

We set $\mathsf{Cyl}(P,\phi)=(I\times P,\text{id}\times\phi)$, where
$I=[0,1]$. Because this operation respects orientation-reversal, collapse-
equivalence, and sends degenerate chains to degenerate chains, it induces a
map on the underlying chain complexes. The boundary formula here is a special
case of the fiber-product boundary formula in Proposition A.2. ∎

We have now verified almost all of the Eilenberg-Steenrod axioms.
Unfortunately, the excision axiom is the most subtle, and an order of
magnitude more difficult to prove.

Suppose $(X,A)$ is a manifold pair, and $B$ is a closed subset of $X$
contained in $A$. We want to show that $H_{*}(X-B,A-B)\to H_{*}(X,A)$ is an
isomorphism. Choose a separating codimension-1 closed submanifold $C$ which is
contained in $A$ and so that $B$ is contained on one side of $C$; write
$X=Y\cup_{C}Z$, with $B\subset Y$.

Given a stratified-smooth probe in $X$ whose boundary lies in $A$, then
‘intersection with $Z$’ should produce a stratified-smooth chain in $X-B$
whose boundary lies in $A-B$. In fact, if $\phi$ is the chain in $X$ and
$\phi_{Y},\phi_{Z}$ are intersections with $Y$ and $Z$ respectively, then
$\phi$ should be homologous to $\phi_{Y}+\phi_{Z}$.

To see why, let us be more precise. Using the Urysohn lemma, choose a smooth
function $f:X\to[-1,1]$ with $B\subset f^{-1}(-1)$ and $A^{c}\subset
f^{-1}(1)$; further suppose that $0$ is a regular value. We write
$Y=f^{-1}[-1,0]$ and $Z=f^{-1}[0,1]$, and $C=f^{-1}(0)$ for the codimension 1
submanifold arising as their common boundary. We write

$W=\\{(t,x)\in[-1,1]\times X\mid t\leq 1-|f(x)|\\}$

for the ‘creased cylinder’, or ‘silo’. It is a manifold with corners, whose
boundary is given by $Y\sqcup Z\sqcup-X$, where $-X$ is $X$ equipped with the
opposite orientation.

Then intersection with $W$ should provide the homology between $\phi$ and
$\phi_{Y}+\phi_{Z}$. This, together with some formal manipulation, should
prove the excision property. There are some additional difficulties involving
transversality, which we will go through in the proof below.

######  Lemma .

If $(X,A)$ is a manifold pair and $B\subset X$ is a closed subset which is
contained in $A$, then the map $i_{*}:H_{*}(X-B,A-B)\to H_{*}(X,A)$ induced by
inclusion is an isomorphism.

###### Proof.

We will show that $i_{*}$ is both injective and surjective.

To start, pick a homology class $[\phi]\in H_{*}(X,A)$; pick a representative

$\phi=\sum[(P_{i},\phi_{i})]\in C_{*}^{gm}(X)$

where $(P_{i},\phi_{i})$ are stratified-smooth probes, with
$\partial\phi=\phi_{A}+\phi_{\text{cancel}}$ given by a sum of two terms:
$\phi_{A}$, a chain whose image lies in $A$, and $\phi_{\text{cancel}}$, a
chain which is zero in $C_{*}^{gm}(X)$. This means that $\phi_{\text{cancel}}$
is the disjoint union of components which are collapse-equivalent to each
other with opposite orientations, and which thus cancel out after identifying
collapse-equivalent chains.

Write $j_{C}:C\hookrightarrow X$ for the inclusion of the submanifold $C$
constructed before this proposition. It need not be the case that the
$\phi_{i}$ are transverse to $j_{C}$. However, by standard transversality
arguments, one may construct a smooth homotopy $H:[0,1]\times X\to X$ so that
$H_{0}=\text{Id}$ and $H_{1}\phi_{i}$ are all transverse to $j_{C}$;
furthermore, one may take $H$ to be arbitrarily close to the constant
homotopy. Using $H$, construct the chain $\phi_{H}=H(I\times\phi)$; this has
boundary

$\partial\phi_{H}=H_{1}\phi-\phi+H(I\times\partial\phi).$

By the assumption that $H$ is arbitrarily small and $\phi_{A}$ has image in
$A$, we also have that $H(I\times\phi_{A})$ has image in $A$ as well as well;
further, $H(I\times\phi_{\text{cancel}})$ still cancels out, so that
$H(I\times\partial\phi)\in C_{*}^{gm}(A)$. If one thus considers $\phi_{H}$ as
an element of $C_{*}^{gm}(X,A)$, this formula shows that $[H_{1}\phi]=[\phi]$.
So we may assume that $[\phi]$ has a representative transverse to $j_{C}$,
hence also to all of $j_{Y},j_{Z},j_{W}$.

Next, set $\phi_{Z}=Z\times_{X}\phi$ and similarly for
$\phi_{C},\phi_{Y},\phi_{W}$. Because

$\partial\phi_{Z}=\phi_{C}+(\partial\phi)_{Z},$

and $C$ is contained in $A$, and $\partial\phi=\phi_{A}+\phi_{\text{cancel}}$
with the first term contained in $A$, we see that $\partial\phi_{Z}$ lies in
$C_{*}^{gm}(A)$; this requires the observation that
$Z\times_{X}\phi_{\text{cancel}}$ still cancels out to give zero (which is
true as $j_{Z}$ is transverse to $\phi_{\text{cancel}}\subset\partial\phi$).
One needs to do similar manipulations with $\phi_{\text{cancel}}$ a handful of
other times in the course of the proof, but the discussion is nearly
identical, so we suppress it from here on.

In particular, $[\phi_{Z}]\in H_{*}(X,A)$ is well-defined. Further, because
$Z\subset X-B$, this in fact defines an element of $H_{*}(X-B,A-B)$. We wish
to show that $[\phi_{Z}]=[\phi]$, so that the map $i_{*}$ is surjective.

To see this, observe that

$\partial(\phi_{W})=\phi_{Y}+\phi_{Z}-\phi+(\partial\phi)_{W}.$

Because $\phi_{Y}$ lies in $C_{*}^{gm}(A)$, and $(\partial\phi)$ lies in
$C_{*}^{gm}(A)$, so too does $(\partial\phi)_{W}$; it follows that in
$C_{*}^{gm}(X,A)$ we have $\partial\phi_{W}=\phi_{Z}-\phi$, and thus
$[\phi]=[\phi_{Z}]$ in $H_{*}(X,A)$, as desired. So $i_{*}$ is surjective.

Injectivity follows essentially the same lines, now using that we may
construct $H$ so small that chains which are already transverse to $j_{C}$
remain transverse to $j_{C}$. ∎

We are now ready to assemble this into the main theorem of this Appendix.

######  Theorem .

The geometric chain complex $C_{*}^{gm}(X)$ is a functorial chain complex on
the category of smooth manifolds and smooth maps. When $X$ is the interior of
a compact manifold, the resulting homology groups are canonically isomorphic
to singular homology $H_{*}(X)$.

###### Proof.

We have now seen that $H_{*}^{gm}(X,A)$ defines a functor on the category
$\mathsf{MP}$ of pairs $(X,A)$, where $X$ is a smooth manifold without
boundary and $A$ is an open submanifold, and that this functor satisfies an
appropriate version of the Eilenberg-Steenrod axioms for this category. Let us
now explain how to modify the usual Eilenberg-Steenrod argument to this
setting.

There is a closely related category, the category $\mathsf{HP}$ of _handlebody
pairs_ $(Y,B)$. Here, $Y$ is a compact manifold (possibly with boundary)
equipped with a handlebody decomposition, where the index of newly attached
handles is monotonically increasing and $B$ is a submanifold (possibly with
boundary) obtained by only attaching some of these handles. We demand that $Y$
is obtained by attaching handles of index no smaller than the largest index of
a handle in $B$. Then $(Y,B)$ carries a natural filtration $(Y^{(k)},B^{(k)}$
of the submanifolds obtained by attaching only handles up to index $k$.

We say that a smooth map $(Y,B)\to(Y^{\prime},B^{\prime})$ is handle-
preserving if it preserves this filtration. There is a functor
$\mathsf{HP}\to\mathsf{MP}$ sending $(Y,B)$ to $(Y^{\circ},B^{\circ})$.

Both $\mathsf{MP}$ and $\mathsf{HP}$ have a natural notion of homotopy; in the
former category, this just means smooth homotopy; in the latter category, we
say $f_{0},f_{1}:(Y,B)\to(Z,C)$ are homotopic if there exists a handle-
preserving smooth map $(I\times Y,I\times B)\to(Z,C)$ interpolating between
them. A cellular approximation-type lemma implies that the corresponding
functor on homotopy categories $\mathsf{hHP}\to\mathsf{hMP}$ is an equivalence
of categories.

Now by passing to $\mathsf{MP}$, the geometric homology functor $H_{*}^{gm}$
defines a functor on the category $\mathsf{hHP}$. If we can show that
$H_{*}^{gm}$ is naturally isomorphic to singular homology on $\mathsf{hHP}$,
the result follows for $\mathsf{hMP}$ and hence $\mathsf{MP}$ as well. But
what remains follows essentially the standard argument; one determines a
canonical identification $H_{*}^{gm}(X^{(k)},A^{(k)})\cong
H_{*}(X^{(k)},A^{(k)})$ inductively on the index $k$, using the Eilenberg-
Steenrod axioms. ∎

######  Remark .

It seems probable that the assumption that $X$ is the interior of a compact
manifold could be dropped, but we will only actually use this theorem for the
manifolds $X=SO(3),S^{2},\star$, so it seems rather pointless to extend the
result more generally. ∎

###  A.5 Truncating the geometric chain complex

In some presentations of geometric homology theory, the chain complex
$C_{*}^{gm}(M)$ is concentrated in degrees $[0,\dim M+1]$. This is a desirable
feature for two reasons. First, the periodic homological algebra of [M,
Appendix A] only works for bounded dg-algebras. Second, this phenomenon
guarantees we only need the moduli spaces
$\breve{M}^{+}_{\zeta}(\alpha,\beta)$ up through a certain dimension range,
which allows us to ignore the bubbling phenomenon.

This is not the case for our model $C_{*}^{gm}(M)$ whenever $\dim M>0$. Pick
any nondegenerate $\phi:P\to M$ of positive degree (for instance, take
$(P,\phi)=(M,1_{M})$ to be the identity probe). Then $(P\times
I^{k},\phi\pi_{1})$ is nondegenerate, as the positive-dimensional stratum
$(P\times\\{0\\},\phi)$ does not have small rank. So $C_{*}^{gm}(M)$ is
nonzero in arbitrarily large degrees. This is because our rather strict notion
of nondegeneracy includes fewer nondegenerate probes than other authors; see
Remark A.3 for more details.

We briefly explain how to correct for this failure.

Let $M$ be a smooth manifold. Set the _truncated geometric chain complex_ to
be

$tC_{k}^{gm}(M)=\begin{cases}C_{k}^{gm}(M)&k<\dim M\\\ C_{k}^{gm}(M)/\partial
C_{k+1}^{gm}(M)&k=\dim M\\\ 0&k>\dim M\end{cases}.$

That is, $tC_{*}^{gm}(M)$ is the complex obtained by collapsing the acyclic
subcomplex

$D_{*}(M)=\begin{cases}0&k<\dim M\\\ \partial C_{k+1}^{gm}(M)&k=\dim M\\\
C_{k}^{gm}(M)&k>\dim M.\end{cases}$

######  Theorem .

The complex $tC_{*}^{gm}(M)$ satisfies the following properties.

  1. (i)

It is concentrated in degrees $[0,\dim M]$.

  2. (ii)

Its homology groups are canonically isomorphic to singular homology.

  3. (iii)

If $f:M\to N$ is a smooth map wth $\dim M\geq\dim N$, then there is an induced
chain map $f_{*}:tC_{*}^{gm}(M)\to tC_{*}^{gm}(N)$.

  4. (iv)

Suppose $(W,f\times g)\in\mathsf{P}(M\times N)$ is a probe, and suppose
$f:W\to M$ is a submersion. Then there is an induced map
$\times_{M}W:tC^{gm}_{*}(M)\to tC^{gm}_{*}(N)$ of degree $\dim W-\dim M$,
sending $(P,\phi)\mapsto(P\times_{M}W,\pi_{2}g)$, and satisfying the boundary
relation

$\partial(P\times_{M}W)=(\partial P)\times_{M}W+(-1)^{\dim P+\dim
W}P\times_{M}(\partial W).$

###### Proof.

It is obvious that this complex is concentrated in degrees $[0,\dim M]$.
Further, because $D_{*}(M)$ is an acyclic complex, the canonical map
$C_{*}^{gm}(M)\to tC_{*}^{gm}(M)$ is a quasi-isomorphism. Because
$H_{*}^{gm}(M)$ was already seen to be canonically isomorphic to singular
homology, so too is $tH_{*}^{gm}(M)$.

The claimed maps are induced by maps on $C_{*}^{gm}$. We have already
discussed the induced map $f_{*}$ on $C_{*}^{gm}$; because $f_{*}$ is a chain
map of degree zero, it obviously has $f_{*}D_{*}(M)\subset D_{*}(N)$, and thus
induces a map on the truncated complexes.

To see that $\times_{M}W$ is well-defined at the level of $C_{*}^{gm}$, one
needs to see that it respects all of: orientation-reversal of probes;
degenerate probes; and collapse-equivalence of probes. It is easy to verify
that it does. The stated boundary relation is given in Proposition A.2. We now
need to verify it descends to $tC$.

Because $f:W\to M$ is a submersion, the map $\times_{M}W$ has non-negative
degree. If $\deg(\times_{M}W)=\dim W-\dim M>0$ it is obvious that
$F(W)D_{*}(M)\subset D_{*}(N)$ for degree reasons. If $\deg F(W)=0$, then
$\dim W=\dim M$; for $f:W\to M$ is a submersion it must be on all boundary
strata as well, but because boundary strata have dimension less than $\dim M$,
it follows that $\partial W=\varnothing$. It follows that $F(W)$ is a chain
map of degree zero and any such map descends to $tC^{gm}_{*}$ as desired. ∎

As a corollary to Theorem A.5(iii), if $G$ is a Lie group, then
$tC^{gm}_{*}(G)$ is a dg-algebra and if $G$ acts on $M$ then $tC^{gm}_{*}(M)$
is a dg-module.

## References