# Topology/Mayer-Vietoris Sequence

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.

## Definition[edit | edit source]

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

## Examples[edit | edit source]

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

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.

## Exercises[edit | edit source]

(under construction)