Abstract
In this chapter, we introduce intersection homology from a chain-theoretic perspective, as originally developed by Goresky–MacPherson
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- 1.
Recall that a formal linear combination ζ =∑ σζ σ ⋅ σ of singular k-simplices in X is a locally finite k-chain if for each x ∈ X there is an open neighborhood U x of x in X such that the set
$$\displaystyle \begin{aligned}\{ \zeta _{\sigma} \mid \zeta _{\sigma} \neq 0, \sigma^{-1}(U_x) \neq \emptyset \}\end{aligned}$$is finite.
- 2.
In the notations of Figure 2.4, σ = AB, \({\hat {\sigma }}=C\), and D X(σ) = EC ∪ CF.
- 3.
We leave it as an exercise for the reader to formulate and prove the corresponding Mayer–Vietoris result for intersection homology groups; it is a simple adaptation of the analogous result in simplicial/singular homology.
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Maxim, L.G. (2019). Intersection Homology: Definition, Properties. In: Intersection Homology & Perverse Sheaves. Graduate Texts in Mathematics, vol 281. Springer, Cham. https://doi.org/10.1007/978-3-030-27644-7_2
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