Lesson plan
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.
Article summary
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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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