Excision theorem

In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A , the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X \setminus U,A \setminus U ) into (X, A) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.

In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A , the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X \setminus U,A \setminus U ) into (X, A) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.

Invariance of dimension

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

References

Bibliography

Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, Allen Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002.

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