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Intersection homology of regular and cylindrical neighborhoods. (English) Zbl 1092.55004

A filtered space is a space \(X\) together with a collection of closed subspaces \[ \emptyset = X^{-1} \subset X^0 \subset \cdots \subset X^{n-1} \subset X^n = X. \] The associated subsets \(X_i = X^i - X^{i-1}\) are called the strata of \(X\). The number \(n\) is called the (stratified) dimension of \(X\) and \(X^k\), the \(k\) skeleton or codimension \(n-k\) skeleton. For a simplicial complex, or a cellular complex, dimension and codimension are geometric and classical, but this framework allows other stratifications. As defined by M. Goresky and R. MacPherson [Topology 19, 135–165 (1980; Zbl 0448.55004)], a perversity is a sequence of integers \([\bar{p}(0), \bar{p}(1), \bar{p}(2), \ldots]\) such that \(\bar{p}(i) \leq \bar{p}(i+1) \leq \bar{p}(i) + 1\) and such that \(\bar{p}(0) = \bar{p}(1) = \bar{p}(2) = 0\).
The (singular) intersection homology of \(X\) with perversity \(\bar{p}\) is the homology of the chain complex \(IC^{\bar{p}}_i(X)\), which is the submodule of singular \(i\)-chains \(\sigma\colon \Delta^i\to X\) on \(X\) with the property that \(\sigma^{-1}(X_{n-k} - X_{n-k-1})\) is contained in the \(i - k + \bar{p}(k)\) skeleton of \(\Delta^i\), and that the boundary of \(\sigma\) is a linear combination of \(i\)-simplices and \(i-1\)-simplices that satisfy this property. The homology of this complex is denoted \(IH^{\bar{p}}_\ast(X)\) and it may not be a topological invariant of \(X\), depending on the filtration.
The calculation of intersection homology is limited by the need for good homological machinery for computation. In particular, by carefully defining a stratum-preserving mapping and stratum-preserving homotopy, one can define a stratified fibration to be a mapping satisfying the stratified version of the homotopy lifting property. For such fibrations, the author constructs a Leray-Serre spectral sequence. For this, the right notion of a system of local coefficients is needed, and it is developed in §2. In the case of \(X\) a weakly stratified space and there is a cylindrical nearly-stratum preserving deformation retract of a pure subset \(K\), the author has a Leray-Serre spectral sequence with \(E^2\)-term given by the homology of \(K\) in a PL stratified system of coefficients determined by the filtration. This generalizes the author’s work on the computation of intersection Alexander polynomials for non-locally-flat knots.

MSC:

55N33 Intersection homology and cohomology in algebraic topology
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
57N80 Stratifications in topological manifolds
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)

Citations:

Zbl 0448.55004
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References:

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