Excision theorem

Excision theorem In algebraic topology, una branca della matematica, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space {stile di visualizzazione X} and subspaces {stile di visualizzazione A} e {stile di visualizzazione U} tale che {stile di visualizzazione U} is also a subspace of {stile di visualizzazione A} , the theorem says that under certain circumstances, we can cut out (excise) {stile di visualizzazione U} from both spaces such that the relative homologies of the pairs {stile di visualizzazione (Xsetminus U,Asetminus U)} in {stile di visualizzazione (X,UN)} sono isomorfi.

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

Contenuti 1 Teorema 1.1 Dichiarazione 1.2 Proof Sketch 2 Applicazioni 2.1 Eilenberg–Steenrod Axioms 2.2 Mayer-Vietoris Sequences 3 Guarda anche 4 Riferimenti 5 Bibliography Theorem Statement If {displaystyle Usubseteq Asubseteq X} are as above, lo diciamo {stile di visualizzazione U} can be excised if the inclusion map of the pair {stile di visualizzazione (Xsetminus U,Asetminus U)} in {stile di visualizzazione (X,UN)} induces an isomorphism on the relative homologies: {stile di visualizzazione H_{n}(Xsetminus U,Asetminus U)cong H_{n}(X,UN)} The theorem states that if the closure of {stile di visualizzazione U} is contained in the interior of {stile di visualizzazione A} , poi {stile di visualizzazione U} can be excised.

Often, subspaces that 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.

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 in {stile di visualizzazione (X,UN)} to get another chain consisting of "più piccola" semplice, and continuing the process until each simplex in the chain lies entirely in the interior of {stile di visualizzazione A} or the interior of {displaystyle Xsetminus U} . Since these form an open cover for {stile di visualizzazione 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 {stile di visualizzazione H_{n}(X,UN)} , poi, this says all the terms contained entirely in the interior of {stile di visualizzazione 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 {stile di visualizzazione U} 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.[1] See also Homotopy excision theorem References ^ See Hatcher 2002, p.149, for example Bibliography Joseph J. Rotman, Introduzione alla topologia algebrica, Springer-Verlag, ISBN 0-387-96678-1 Allen Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002. Categorie: Homology theoryTheorems in topology