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.

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 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.