εq q " 0. To determine the second term, first note, 1 pG,εq˚ai0 " p´1q |pG,εq|ąp p1 pGp,εpq˚ai0 q b...
‚ The proof for the case G " H is similar.
‚ The argument is the same if ℓ ą p.
Write F p " ra i0,..., a ir s, G p " rb j0,..., b js s. If the simplices are not compatible, both members of the previous equality are zero. The only cases giving compatible simplices are one of the following cases: p˚q a it ă e ă a it`1 with t P t0,..., r´1u, p˚q e ă a i0 and e P ∆ p.
The argument is similar for ℓ ě p. Proposition 11.3. Let X be a filtered space and let XˆR endowed with the product filtration and a perversity p. Denoting also by p the induced perversity on X, there exists a linear map Θ : r Np pXˆR; Rq Ñ r N˚´1 p pXˆR; Rq, such that (11.4) Θ˝δ`δ˝Θ " pI 0˝p rq˚´id.
Proof of Theorem D. This is a direct consequence of the equalities pr˝I 0 " pr˝I 1 " id and Hp pI 0˝p rq " id, proved in (11.4).
The following result is used in Section 14.
Proposition 11.4. Let pX, pq be a paracompact perverse space. Let S ℓ be the unit sphere of R ℓ`1. The canonical projection p X : S ℓˆX Ñ X, pz, xq Þ Ñ x, induces an isomorphism H Proof. It suffices to use an induction on ℓ with the decomposition S ℓ " S ℓ ztNorthpoleu Y S ℓ ztSouthpoleu (see Theorem C and Theorem D).

Blown-up intersection cohomology of a cone. Theorem E.
In this section, X is an n-dimensional compact filtered space and we represent the open cone as the quotientcX " Xˆr0, 8r{Xˆt0u, whose apex is denoted by w. The formal dimension ofcX is n`1 relatively to the conical filtration, pcXq i "cX i´1 if i ě 1 and pcXq 0 " twu. The purpose of this section is to prove the following proposition, cf. also [ Theorem E. Let X be a compact filtered space. Consider the open cone, cX " Xˆr0, 8r{Xˆt0u, equipped with the conical filtration and a perversity p. We also denote by p the perversity induced on X. The following properties are verified for any commutative ring, R. ‚ If ∆ " ∆ 0˚¨¨¨˚∆n`1 is a regular simplex, of formal dimension n`1, we define a regular simplex, of formal dimension n, by p ∆ " ∆ 1˚¨¨¨˚∆n`1. Its filtration is characterized by p ∆ i " ∆ i`1, for each i P t0,..., nu. ‚ Let σ : ∆ σ " ∆ 0˚¨¨¨˚∆n`1 ÑcX be a regular simplex ofcX. Since σp p ∆ σ q Ă Xˆs0, 8r, we define the restriction ‚ For each regular simplex ofcX, σ : ∆ σ " ∆ 0˚¨¨¨˚∆n`1 ÑcX, the image of a point x P ∆ σ can be written as, σpxq " rσ 1 pxq, σ 2 pxqs PcX " Xˆr0, 8r{Xˆt0u.
Associated to the simplex σ, there is the following regular simplex ofcX, cσ : ∆ cσ " ptpu˚∆ 0 q˚¨¨¨˚∆ n`1 ÑcX, defined by cσpp1´tqp`txq " rσ 1 pxq, tσ 2 pxqs. Moreover, if one considerŝ σ : p ∆ σ Ñ Xˆs0, 8rãÑcX as a filtered simplex of the cone, then cσ is a face of cσ. The truncation of a cochain complex is defined for all positive integers s by (12.1) pτ ďs Cq r " where ZC s means the R-module of cocycles whose degree is s.
Proposition 12.1. Let pcX, pq be a perverse space over the cone of the compact space X and let pXˆs0, 8r, pq be the induced perverse space. The correspondence defined above induces a cochain map, f : τ ďppwq r Np pXˆs0, 8r; Rq Ñ τ ďppwq r Np pcX; Rq.
Let ω P r N˚pcX; Rq and τ : ∆ τ Ñ Xˆs0, 8r a regular simplex. We denote by cτ : ∆ cτ " tpu˚∆ τ ÑcX the cone over τ defined above. Notice Ą ∆ cτ " ctpuˆĂ ∆ τ. Let v 0 be the apex of the cone over the component of filtration degree 0 of a filtered simplex. The cone ctpu having two vertices p and v 0, the cochain ω cτ decomposes into Proposition 12.2. Let pcX, pq be a perverse space with X compact and let pXˆs0, 8r, pq be the induced perverse space. The correspondence defined above induces a cochain map, g : τ ďppwq r Np pcX; Rq Ñ τ ďppwq r Np pXˆs0, 8r; Rq.
The proof follows the pattern of that of Proposition 12.1; we leave it to the reader.
Specify the compositions f˝g and g˝f.
We decompose ω cσ P r N˚p∆ cσ q as in the formula (12.4) and set: where v 0 is the apex of the cone over the component filtration of degree 0 and λ ∆0 the sum of 0-cochains on ∆ 0.
Proposition 12.3. Let pcX, pq be a perverse space with X compact.
Let us study the behavior of H towards the differentials. Let σ : ∆ σ " ∆ 0∆ n`1 ÑcX be a regular simplex and ω P r N˚pcX; Rq. Let us start from the equality (12.4) and calculate the differential in eliminating terms having a zero image by H.
Replacing in (12.7) and developing the definition of H, we get: On the other hand, the quantity δ˝H can be written Using (12.5), the sum of the two expressions can be reduced to: (b) As in the proof of Proposition 12.1, we are reduced to consider a singular stratum T ofcX, a p-allowable cochain ω P r N˚pcXq, of degree less than or equal to ppwq, and a regular simplex, σ : ∆ σ ÑcX, with σp∆ σ q X T ‰ H. Let ℓ " codimc X T P t1,..., n`1u. Notice that cσp∆ cσ q X T ‰ H. Thus, according to the definition of perverse degree (cf. Definition 2.5), we have where the equality uses (2.5) and (12.4). We develop this expression by distinguishing two cases.
The determination of the cohomology of a cone follows from the properties of f, g and H.
Proof of Theorem E. (a) From Propositions 12.1, 12.2 and 12.3, we deduce that the map g : τ ďppwq r Np pcX; Rq Ñ τ ďppwq r Np pXˆs0, 8r; Rq is a quasi-isomorphism. We know from Theorem D, that the inclusion I 1 : X Ñ Xˆs0, 8r induces a quasi-isomorphism. It remains to prove I1˝g " ι˚. The involved stratifications on Xˆs0, 8r are different in these two quasi-isomorphisms but the cohomologies are the same (see subsection 3.1).
The proof of the topological independence Theorem of Section 14 is based on the following excision theorem.
12.2. Relative cohomology. Let X be a filtered space and Y Ă X a subspace endowed with the induced stratification. Consider a perversity p on X. We also denote by p the induced perversity on Y. The complex