We develop “Mayer–Vietoris arguments” that can be used to show the equivalence of two functors on a manifold or stratified space. We apply such arguments to prove an intersection homology version of the Künneth theorem when one factor is a manifold; this includes a detailed construction of the cross product for intersection homology. We also treat intersection homology with coefficients and discuss universal coefficient theorems and their obstructions, including a local torsion-free condition. We show that PL and singular intersection homology are isomorphic on PL stratified spaces, and we prove that intersection homology is stratification-independent when using certain perversities, including the original ones of Goresky and MacPherson. The chapter closes with a proof that the intersection homology of compact pseudomanifolds is finitely generated.

In this appendix we provide background about simplicial and PL (piecewise linear) topology. We discuss simplicial complexes and Euclidean polyhedra; PL spaces and maps; PL subspaces; and cones, joins, and products of PL spaces. We also consider in detail the Eilenberg–Zilber shuffle triangulation of products of simplicial complexes, leading to the simplicial cross product.

We develop intersection cohomology and versions of the cup, cap, and cohomology cross products. We prove all the expected properties about these products, including versions of naturality, commutativity, associativity, existence of units, stability, and properties about the compositions of different products. We also introduce intersection cohomology with compact supports and study its properties.

We show that oriented pseudomanifolds possess fundamental intersection homology classes, and we prove that intersection homology possesses a Poincaré duality given by the cap product with the fundamental class. We also prove Lefschetz duality for pseudomanifolds with boundary. We derive from both of these dualities nonsingular cup product and torsion pairings. We include an expositional survey of intersection pairings and the original approach of Goresky and MacPherson to intersection homology duality using such pairings.

We develop the basic properties of PL and singular intersection homology. This includes the behavior of the intersection homology groups under stratified maps and homotopies and the invariance of intersection homology groups under stratified homotopy equivalences. We introduce relative intersection homology, the long exact sequence of a pair, Mayer–Vietoris sequences, and excision. An important special computation is that of the intersection homology of a cone, which provides a good basic example of an intersection homology computation but also provides a formula that plays an essential role throughout the theory, as all points in pseudomanifolds have neighborhoods that are stratified homotopy equivalent to cones.

Chapter 2 is an introduction to stratified spaces. We begin with filtered spaces and move progressively through more and more constrained classes, including manifold stratified spaces, locally cone-like spaces, the CS sets of Siebenmann, recursive CS sets, and topological and piecewise linear (PL) pseudomanifolds. To facilitate this last definition, we provide some background on PL topology. In the later sections of the chapter, we turn to some more specialized topics, including normalization of pseudomanifolds, pseudomanifolds with boundary, and other more specialized types of spaces, such as Whitney stratified spaces, Thom–Mather stratified spaces, and homotopically stratified spaces. We observe that the class of pseudomanifolds includes many spaces that arise naturally in other mathematical areas, such as singular varieties and orbit spaces of group actions. We also discuss stratified maps between stratified spaces and close with two specialized topics: intrinsic filtrations and products and joins of stratified spaces.

The final chapter provides suggestions for further reading on related topics, including pseudomanifold bordism groups, characteristic classes, intersection spaces, analytic approaches to intersection homology such as L²-cohomology and perverse differential forms, stratified Morse theory, perverse sheaves, and Hodge theory.

We motivate intersection homology theory by discussing how Poincaré duality fails on spaces with singularities. We see that one difficulty is the failure of general position, explaining why generalizations of general position will play an important role in the definition of intersection homology, which is a variant of simplicial or singular homology that recovers a version of Poincaré duality for singular spaces. We also discuss some conventions that will hold throughout the book and provide a quick overview of the difference between GM and non-GM intersection homology. We also provide a chapter-by-chapter outline of the rest of the book.

We introduce “non-GM” intersection homology, which is a version of intersection homology that has better properties for arbitrary perversity parameters, though it agrees with GM intersection homology with certain perversity restrictions. We develop the basic properties of this version of intersection homology, including behavior under stratified maps and homotopies, relative intersection homology, excision, Mayer–Vietoris sequences, cross products, and a new cone formula. We also develop a Künneth theorem for products of stratified spaces, and prove theorems about splitting intersection chains into smaller pieces.

We introduce Witt and IP spaces, which are the spaces on which middle-dimensional intersection cohomology self-pairings are possible. This leads to signature invariants for such spaces, and we demonstrate their basic properties, including Novikov additivity. Using the signature, we provide in detail the Goresky–MacPherson construction of the characteristic L-classes for Witt spaces, which is modeled upon the classical construction for PL manifolds. We also provide a survey of bordism groups and bordism homology theories based on different types of pseudomanifolds.

We introduce (GM) intersection homology, beginning with a discussion of perversity parameters. The treatment of intersection chains begins with the simplicial version, followed by PL (piecewise linear) intersection chains, and then singular intersection chains. As PL homology is less common than simplicial and singular homology, we provide the necessary background. We provide definitions, examples, and the most fundamental properties of intersection homology.