Excision theorem
Encyclopedia
In algebraic topology
, a branch of mathematics
, the excision theorem is a useful 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
.
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the excision theorem is a useful theorem about relative homology
Relative homology
In algebraic topology, a branch of mathematics, the 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...
—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
Singular homology
In algebraic topology, a branch of mathematics, 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....
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
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of U is contained in the interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
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
Deformation retract
In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- 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
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, 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
Eilenberg-Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common...
.