Excisive triad

Definition

An excisive triad is a triple comprising a topological space and two subspaces, such that the union of the interiors of the subspaces is the whole space.

Facts

Mayer-Vietoris homology sequence

Further information: Mayer-Vietoris homology sequence

Given an excisive triad, we can construct an associated Mayer-Vietoris homology sequence. This relates the homologies of the space, the two subspaces, and their intersection.

Excision isomorphism

Further information: Excision isomorphism

Given an excisive triad, we can construct an excision isomorphism, which relates the relative homology of one subspace with the intersection, and the relative homology of the whole subspace and the other space. We can use the excision isomorphism to yield information about the homologies of the spaces themselves, if we combine it with the long exact sequence of homology of a pair.