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- Excision_theorem abstract "In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology—given topological spaces 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,A) and (X \ U,A \ U) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute. Or, in many cases, it allows the use of induction. Coupled with the long exact sequence in homology, one can derive another useful tool for the computation of homology groups, the Mayer–Vietoris sequence.More precisely, if X, A, and U are as above, we say that U can be excised if the inclusion map of the pair (X \ U,A \ U ) into (X, A) induces an isomorphism on the relative homologies Hq(X,A) to Hq(X \ U,A \ U ). The theorem states that if the closure of U is contained in the interior of A, then U can be excised. Often, subspaces which do not satisfy this containment criterion still can be excised—it suffices to be able to find a deformation 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 (X,A) to get another chain consisting of "smaller" simplices, and continuing the process until each simplex in the chain lies entirely in the interior of A or the interior of X \ U. Since these form an open cover for X 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 to the identity map on homology). In the relative homology Hq(X,A), then, this says all the terms contained entirely in the interior of 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 U entirely.In the axiomatic approach to homology, the theorem is taken as one of the Eilenberg–Steenrod axioms.".
- Excision_theorem wikiPageExternalLink ATpage.html.
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- Excision_theorem wikiPageLength "3031".
- Excision_theorem wikiPageOutDegree "15".
- Excision_theorem wikiPageRevisionID "634739525".
- Excision_theorem wikiPageWikiLink Algebraic_topology.
- Excision_theorem wikiPageWikiLink Category:Homology_theory.
- Excision_theorem wikiPageWikiLink Category:Theorems_in_topology.
- Excision_theorem wikiPageWikiLink Chain_homotopy.
- Excision_theorem wikiPageWikiLink Closure_(topology).
- Excision_theorem wikiPageWikiLink Compact_space.
- Excision_theorem wikiPageWikiLink Deformation_retract.
- Excision_theorem wikiPageWikiLink Eilenberg–Steenrod_axioms.
- Excision_theorem wikiPageWikiLink Exact_sequence.
- Excision_theorem wikiPageWikiLink Homotopy_category_of_chain_complexes.
- Excision_theorem wikiPageWikiLink Homotopy_excision_theorem.
- Excision_theorem wikiPageWikiLink Interior_(topology).
- Excision_theorem wikiPageWikiLink Long_exact_sequence.
- Excision_theorem wikiPageWikiLink Mathematics.
- Excision_theorem wikiPageWikiLink Mayer–Vietoris_sequence.
- Excision_theorem wikiPageWikiLink Relative_homology.
- Excision_theorem wikiPageWikiLink Singular_homology.
- Excision_theorem wikiPageWikiLinkText "Excision Theorem".
- Excision_theorem wikiPageWikiLinkText "Excision theorem".
- Excision_theorem wikiPageWikiLinkText "Excision".
- Excision_theorem wikiPageWikiLinkText "excision map".
- Excision_theorem wikiPageWikiLinkText "excision theorem".
- Excision_theorem wikiPageWikiLinkText "excision".
- Excision_theorem wikiPageWikiLinkText "excisive".
- Excision_theorem hasPhotoCollection Excision_theorem.
- Excision_theorem wikiPageUsesTemplate Template:Reflist.
- Excision_theorem subject Category:Homology_theory.
- Excision_theorem subject Category:Theorems_in_topology.
- Excision_theorem hypernym Theorem.
- Excision_theorem type Theorem.
- Excision_theorem comment "In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology—given topological spaces 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,A) and (X \ U,A \ U) are isomorphic.".
- Excision_theorem label "Excision theorem".
- Excision_theorem sameAs Thxc3xa9orxc3xa8me_dexcision.
- Excision_theorem sameAs משפט_הקיצוץ.
- Excision_theorem sameAs m.04l_b1.
- Excision_theorem sameAs Q3526998.
- Excision_theorem sameAs Q3526998.
- Excision_theorem sameAs 切除定理.
- Excision_theorem wasDerivedFrom Excision_theorem?oldid=634739525.
- Excision_theorem isPrimaryTopicOf Excision_theorem.