Excision theorem: Difference between revisions
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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]].<ref>See Hatcher 2002, p.149, for example</ref> |
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]].<ref>See Hatcher 2002, p.149, for example</ref> |
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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 ''H<sub>q</sub>''(''X'',''A'') to ''H<sub>q</sub>''(''X |
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 ''H<sub>q</sub>''(''X \ U'',''A \ U '') to ''H<sub>q</sub>''(''X'',''A'') . The theorem states that if the [[Closure (topology)|closure]] of ''U'' is contained in the [[Interior (topology)|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. |
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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 space|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 homotopy|chain homotopic]] to the identity map on homology). In the relative homology ''H<sub>q</sub>''(''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. |
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 space|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 homotopy|chain homotopic]] to the identity map on homology). In the relative homology ''H<sub>q</sub>''(''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. |
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Revision as of 12:51, 14 January 2018
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.[1]
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 \ U,A \ U ) to Hq(X,A) . 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.
See also
References
- ^ See Hatcher 2002, p.149, for example
Bibliography
- Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, ISBN 0-387-96678-1
- Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002.