example of cohomology and Mayer-Vietoris sequence

Consider n-dimensional sphere Sn={vRn+1||v|=1}.

Let A={(x0,,xn)Sn|x0>-1/2} and B={(x0,,xn)Sn|x0<1/2}.

Of course both A and B are open (in Sn) and their union is Sn. Furthermore, it can be easily seen, that their intersection can be contracted into ”big circle”, i.e. AB has homotopy typeMathworldPlanetmath of Sn-1. Also both A and B are contractibleMathworldPlanetmath (they are homeomorphicMathworldPlanetmath to Rn via stereographic projection). So, write part of a Meyer-Vietoris sequence (for the cohomologyPlanetmathPlanetmath Hm(X)=Hm(X,G), where G is a fixed Abelian groupMathworldPlanetmath):


Since both A and B are contractible and AB is homotopicMathworldPlanetmathPlanetmath to Sn-1, we have the following short exact sequence:


which shows that Hm(Sn-1) is isomorphicPlanetmathPlanetmathPlanetmath to Hm(Sn) for every n>0 and m>0. So, in order to calculate cohomology groupsPlanetmathPlanetmath of spheres, we only need to know the cohomology groups of S1. And those can be also calculated, if we once again apply previous schema. Note, that in the case of S1 we have that AB has the homotopy type of a discrete space with two points. Therefore all their cohomology groups are trivial, except for H0 (which can be easily calculated to be equal to H0(*)H0(*), where * is a one-pointed space).

This schema can be used for other spaces like the torus (which can be also calculated from Kunneth’s formula).

Title example of cohomology and Mayer-Vietoris sequence
Canonical name ExampleOfCohomologyAndMayerVietorisSequence
Date of creation 2013-03-22 19:13:27
Last modified on 2013-03-22 19:13:27
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Example
Classification msc 55N33