long exact sequence (of homology groups)

If X is a topological spaceMathworldPlanetmath, and A and B are subspacesMathworldPlanetmath with XAB, then there is a long exact sequence:


where i* is induced by the inclusion i:(A,B)(X,B), j* by the inclusion j:(X,B)(X,A), and is the following map: given aHn(X,A), choose a chain representing it. a is an (n-1)-chain of A, so it represents an element of Hn-1(A,B). This is *a.

When B is the empty setMathworldPlanetmath, we get the long exact sequence of the pair (X,A):


The existence of this long exact sequence follows from the short exact sequenceMathworldPlanetmathPlanetmath


where i and j are the maps on chains induced by i and j, by the Snake LemmaMathworldPlanetmath.

Title long exact sequence (of homology groups)
Canonical name LongExactSequenceofHomologyGroups
Date of creation 2013-03-22 13:14:50
Last modified on 2013-03-22 13:14:50
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 55N10
Related topic NChain
Related topic ProofOfSnakeLemma