Inalgebraic topology, a branch ofmathematics, theexcision theorem is a theorem aboutrelative homology and one of theEilenberg–Steenrod axioms. Given a topological space${\displaystyle X}$ and subspaces${\displaystyle A}$ and${\displaystyle U}$ such that${\displaystyle U}$ is also a subspace of${\displaystyle A}$, the theorem says that under certain circumstances, we can cut out (excise)${\displaystyle U}$ from both spaces such that therelative homologies of the pairs${\displaystyle (X\setminus U,A\setminus U)}$ into${\displaystyle (X,A)}$ are isomorphic.

This assists in computation ofsingular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.

If${\displaystyle U\subseteq A\subseteq X}$ are as above, we say that${\displaystyle U}$ can beexcised if the inclusion map of the pair${\displaystyle (X\setminus U,A\setminus U)}$ into${\displaystyle (X,A)}$ induces an isomorphism on the relative homologies:

${\displaystyle H_n(X\setminus U,A\setminus U)\cong H_n(X,A)}$

The theorem states that if theclosure of${\displaystyle U}$ is contained in theinterior of${\displaystyle A}$, then${\displaystyle U}$ can be excised.

Often, subspaces that do not satisfy this containment criterion still can be excised—it suffices to be able to find adeformation retract of the subspaces onto subspaces that do satisfy it.

The proof of the excision theorem is quite intuitive, though the details are rather involved. The idea is to subdivide the simplices in a relative cycle in${\displaystyle (X,A)}$ to get another chain consisting of "smaller" simplices (this can be done usingbarycentric subdivision^[1]), and continuing the process until each simplex in the chain lies entirely in the interior of${\displaystyle A}$ or the interior of${\displaystyle X\setminus U}$. Since these form an open cover for${\displaystyle X}$ and simplices arecompact, we can eventually do this in a finite number of steps. This process leaves the original homology class of the chain unchanged (this says the subdivision operator ischain homotopic to the identity map on homology).In the relative homology${\displaystyle H_n(X,A)}$, then, this says all the terms contained entirely in the interior of${\displaystyle U}$ can be dropped without affecting the homology class of the cycle. This allows us to show that the inclusion map is an isomorphism, as each relative cycle is equivalent to one that avoids${\displaystyle U}$ entirely.

The excision theorem is taken to be one of theEilenberg–Steenrod axioms.

TheMayer–Vietoris sequence may be derived with a combination of excision theorem and the long-exact sequence.^[2]

The excision theorem may be used to derive the suspension theorem for homology, which says${\displaystyle {\tilde{H}}_n(X)\cong {\tilde{H}}_{n+1}(SX)}$ for all${\displaystyle n}$, where${\displaystyle SX}$ is thesuspension of${\displaystyle X}$.^[3]

If nonempty open sets${\displaystyle U\subset {\mathbb{R}}^n}$ and${\displaystyle V\subset {\mathbb{R}}^m}$are homeomorphic, thenm =n. This follows from the excision theorem, the long exact sequence for the pair${\displaystyle ({\mathbb{R}}^n,{\mathbb{R}}^n-x)}$, and the fact that${\displaystyle {\mathbb{R}}^n-x}$ deformation retracts onto a sphere.In particular,${\displaystyle {\mathbb{R}}^n}$ is not homeomorphic to${\displaystyle {\mathbb{R}}^m}$ if${\displaystyle m\ne n}$.^[4]

• Homotopy excision theorem

• Joseph J. Rotman,An Introduction to Algebraic Topology, Springer-Verlag,ISBN 0-387-96678-1
• Allen Hatcher,Algebraic Topology. Cambridge University Press, Cambridge, 2002.

• Homology theory
• Theorems in topology

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