Mayer-Vietoris sequence

Let X is a topological spaceMathworldPlanetmath, and A,BX are such that X=int(A)int(B), and C=AB. Then there is an exact sequencePlanetmathPlanetmathPlanetmath of homology groups:


Here, i* is induced by the inclusions i:BX and j* by j:AX, and * is the following map: if x is in Hn(X), then it can be written as the sum of a chain in A and one in B, x=a+b. a=-b, since x=0. Thus, a is a chain in C, and so represents a class in Hn-1(C). This is *x. One can easily check (by standard diagram chasing) that this map is well defined on the level of homologyMathworldPlanetmathPlanetmath.

Title Mayer-Vietoris sequence
Canonical name MayerVietorisSequence
Date of creation 2013-03-22 13:14:52
Last modified on 2013-03-22 13:14:52
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 6
Author bwebste (988)
Entry type Definition
Classification msc 55N10