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.

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.