Topology/Mayer-Vietoris Sequence

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A Mayer-Vietoris Sequence is a powerful tool used in finding Homology groups for spaces that can be expressed as the unions of simpler spaces from the perspective of Homology theory.

DefinitionEdit

If X is a topological space covered by the interiors of two subspaces A and B, then

 

is an exact sequence where  . There is a slight adaptation for the reduced homology where the sequence ends instead

 

ExamplesEdit

Consider the cover of   formed by 2-discs A and B in the figure.

 
  covered by 2-discs A and B

The space   is homotopy equivalent to the circle. We know that the homology groups are preserved by homotopy and so   for   and  . Also note how the homology groups of A and B are trivial since they are both contractable. So we know that

 

This means that   since   is an isomorphism by exactness.

Consider the cover of the torus by 2 open ended cylinders A and B.

 
How we choose A and B.

ExercisesEdit

(under construction)


Topology
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