Excision theorem

The excision theorem states that removing a suitable subspace from a pair of spaces induces an isomorphism on their relative homology groups.

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Relative homology: A homology theory that assigns groups to a pair of spaces, measuring features present in one space relative to a subspace. The theorem's conclusion compares the relative homology groups before and after excision.
Topological pair: An ordered pair of topological spaces consisting of a space and one of its subspaces. Excision begins with a pair and replaces it by a smaller pair.
Local homology: Homology groups that describe the local structure of a space near a point or subset. Excision reduces local homology to a smaller neighborhood around the point.
Homotopy invariance: The principle that homotopic maps induce the same maps on homology groups. It preserves homology through deforming maps, rather than removing a subset from a pair.
Excision in singular homology: The singular-chain form of excision, in which subdivision shows that chains can be represented away from a removable region. This chain-level argument is the standard proof mechanism for the theorem.
Singular chain complex: A chain complex generated by continuous maps from standard simplices into a topological space, with boundary maps given by face restrictions. The proof tracks how excision changes chains and boundaries.
Homology of a sphere: The singular homology groups of a sphere, concentrated in degrees zero and its dimension. Excision supports standard computations of sphere homology from simpler pieces.
Homotopy equivalence: A pair of maps between spaces whose composites are homotopic to the respective identity maps. It gives homology isomorphisms by comparing whole spaces up to deformation.
Singular homology: A homology theory built from formal sums of continuous maps from simplices into a topological space. The classical excision theorem is a structural result about singular homology.
Quotient space: A topological space formed by identifying specified points of another space as equivalent. Under suitable conditions, relative homology can be computed using a quotient by the subspace.