In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms, the theorem says that under certain circumstances, we can cut out (excise from both spaces such that the relative homologies are isomorphic.

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

Theorem

Statement

can be excised if the inclusion map induces an isomorphism on the relative homologies:

The theorem states that if the closure is contained in the interior can be excised.

Often, subspaces that do not satisfy this containment criterion still can be excisedit suffices to be able to find a deformation retract of the subspaces onto subspaces that do satisfy it.

Proof sketch

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 to get another chain consisting of "smaller" simplices (this can be done using barycentric subdivision and simplices are compact, 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 is chain homotopic entirely.

Applications

Eilenberg–Steenrod axioms

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

Mayer–Vietoris sequences

The Mayer–Vietoris sequence may be derived with a combination of excision theorem and the long-exact sequence.

Suspension theorem for homology

is the suspension.

Invariance of dimension

are homeomorphic, then m = n. This follows from the excision theorem, the long exact sequence.

See also

• Homotopy excision theorem

References

Bibliography

• Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag,

• Allen Hatcher, Cambridge University Press, Cambridge, 2002.