Mayer-Vietoris
We often find in the literature on intersection homology that the Mayer–Vietoris property (or the equivalent Excision Formula) is stated to hold for intersection homology, using the following argument:
Mutatis mutandis, the classical proof for singular homology applies.
This reasoning appears, for example, in:
In fine, the assertion is true, although the situation is so different between the singular case and the intersection case that
some additional explanation is not out of place. This has been done, for example, in:
The proof of the Mayer–Vietoris technique in the context of singular homology, associated to an open cover
𝒰
={U,V} of X, essentially uses these two tools:
MV1 – For any singular chain η of X there exists an integer k tal que Sdk(η) is a 𝒰-singular chain.
MV2 – Any 𝒰-singular chain ξ possesses a decomposition ξ = ξU + ξV where ξU is a singular chain of U
and ξV is a singular chain of V.
Notice that this decomposition is not unique.
In the context of intersection chains, property MV-1 still holds, replacing singular chains with intersection chains.
The second property also remains valid when singular chains are replaced by allowable chains.
However, we need to work specifically with intersection chains. This is more delicate, since it does not hold in full generality.
It may happen that ξU and ξV are not intersection chains, while ξ is. In other words, ∂ξ may be allowable, while
∂ξU and ∂ξV are not.
In fact, property MV-2 still applies, but we need to provide an explicit method to construct such a decomposition.
The method developed in the last two references mentioned above involves a precise description of the allowable simplices that
are not intersection simplices. Each such simplex σ contains a bad face τ that detects this phenomenon.
Let us now consider an intersection chain ξ, and let τ1 ,…,τm be the bad faces of the simplices in ξ.
The chain ξ1 consists of all simplices in ξ that contain τ1 , and the chains ξ2 ,…,ξm
are defined in the same way. They are intersection chains. The chain made up of intersection simplices is denoted by ξ0 ,
it is also an intersection chain.
Applying property MV-1, we may assume that each of the chains ξ0 , ξ1 ,…,ξm is entirely contained in either U or V. Since
ξ = ξ0 + ξ1 + ··· +ξm
we obtain property MV-2.
In the context of singular homology, we can replace “singular chains” with “singular simplices” in properties MV-1 and MV-2.
However, we cannot do the same in the context of intersection homology: we cannot replace “intersection chain” with “intersection simplex,” since an intersection chain is not necessarily a sum of intersection simplices.
