excision theorem
   HOME

TheInfoList



OR:

In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, a branch of mathematics, the excision theorem is a theorem about
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
and one of the
Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...
. 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 In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.


Theorem


Statement

If U\subseteq A \subseteq X are as above, we say that U can be excised if the inclusion map of the pair (X \setminus U,A \setminus U ) into (X, A) induces an isomorphism on the relative homologies: The theorem states that if the closure of U is contained in the interior of A, then 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 In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformat ...
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 (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 \setminus U. Since these form an open cover for X and simplices are
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, 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 H_n(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.


Applications


Eilenberg–Steenrod Axioms

The excision theorem is taken to be one of the Eilenberg–Steenrod Axioms.


Mayer-Vietoris Sequences

The
Mayer–Vietoris sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due ...
may be derived with a combination of excision theorem and the long-exact sequence.See Hatcher 2002, p.149, for example


See also

*
Homotopy excision theorem In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let (X; A, B) be an excisive triad with C = A \cap B nonempty, and suppose the pair (A, C) is (m-1)-connect ...


References


Bibliography

* Joseph J. Rotman, ''An Introduction to Algebraic Topology'', Springer-Verlag, {{ISBN, 0-387-96678-1 *
Allen Hatcher Allen, Allen's or Allens may refer to: Buildings * Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee * Allen Center, a skyscraper complex in downtown Houston, Texas * Allen Fieldhouse, an indoor sports arena on the Unive ...

''Algebraic Topology.''
Cambridge University Press, Cambridge, 2002. Homology theory Theorems in topology