×

Topological invariants of stratified spaces. (English) Zbl 1108.55001

Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-38585-1/hbk). xii, 259 p. (2007).
Poincaré duality is the fundamental symmetry property of the homology of manifolds. This property is crucial in manifold classification theory because it enables the definition of the signature - a fundamental bordism invariant. A method of R. Thom, inspired by Hirzebruch’s signature theorem, constructs characteristic homology classes using the bordism invariance of the signature. These classes are not generally homotopy invariants and thus are fine enough to distinguish manifolds within the same homotopy type.
In this book, a class of singular spaces is considered, namely spaces having points whose neighborhoods are not Euclidean, but which can be decomposed into manifolds. Imposing that some conditions are satisfied when these manifolds intersect, such a decomposition is a stratification. Triangulated spaces and algebraic varieties are singular spaces that can be stratified. An important fact is that for these spaces Poincaré duality in ordinary homology does not hold. In the book, the construction of these invariants for stratified singular spaces is presented, as well as some methods for their computation.
Well written and with modest prerequisites concerning (co)homology theory, simplicial complexes and some basic notions of differential topology, the book is accessible to graduate students. Also, it is useful for the research mathematician wishing to learn about intersection homology and the invariants of singular spaces.
The structure of the book is the following:
Chapter 1: sheaves, complexes of sheaves, cohomology with coefficients in a sheaf (in a complex of sheaves), truncation functors.
Chapter 2: triangulated categories, localization of categories, derived functors.
Chapter 3: construction of the Verdier dualizing complex of a space, Verdier duality functor and Verdier self-dual sheaves.
Chapter 4: stratified pseudomanifolds and their intersection homology, topological invariance of intersection homology, generalized Poincaré duality on singular spaces.
Chapter 5: Chern and Pontrjagin classes, surgery theory for manifolds, Hirzebruch \(L\)-classes of a manifold.
Chapter 6: Witt bordism, signature and \(L\)-class of spaces having no strata of odd dimension, Whitney stratifications, Witt spaces, Novikov additivity.
Chapter 7: definition of \(t\)-structures, gluing of \(t\)-structures, perverse \(t\)-structures.
In Chapter 8 some methods for the computation of the characteristic classes arising from self-dual sheaves are described. The behavior of these classes under stratified maps is studied. A detailed proof of the Cappell-Shaneson decomposition theorem for self-dual sheaves on a space with only even-codimensional strata is given. Then all signature- and \(L\)-class-formulae are obtained.
In Chapter 9 the generalized Poincaré-Verdier self-duality on singular spaces that do not satisfy the Witt condition (assumed in Chapter 8) is studied. The \(L\)-class of non-Witt spaces is constructed and the behavior of these classes under stratified maps is analyzed.
The last Chapter presents an introduction to Cheeger’s method of recovering Poincaré duality for singular Riemannian spaces, using \(L^{2}\) differential forms on the top stratum.

MSC:

55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
55N33 Intersection homology and cohomology in algebraic topology
57N80 Stratifications in topological manifolds
PDFBibTeX XMLCite