# Mayer-Vietoris homology sequence

*This article defines a long exact sequence of homology groups, for topological spaces or pairs of topological spaces*

*This fact is related to*: excisive triads

## Contents

## Definition

### Unreduced version

This statement holds, not just for homology with integer coefficients, but for homology with coefficients in any ring or module as long as the same ring or module is used consistently for all spaces.

Suppose is a topological space, and and are subsets of such that the union of the interiors of and cover (Note that in particular this condition is satisfied if and are open subsets whose union is ). Then we get a long exact sequence of homology:

where the maps are as follows. Let be the inclusions from to and be the inclusions from into .

Then the map from the homology of is:

And the map from is:

The sequence ends at:

Note that by adopting the convention that s are zero for negative (a convention that can also be viewed as following from the definition) we can consider the sequence as going on forever.

This statement holds, not just for homology with integer coefficients, but for homology with coefficients in any ring or module.

### Reduced version

This statement holds, not just for homology with integer coefficients, but for homology with coefficients in any ring or module as long as the same ring or module is used consistently for all spaces.

Suppose is a topological space, and and are subsets of such that the union of the interiors of and cover and the intersection of and is nonempty. Then we get a long exact sequence of reduced homology:

Note that for positive dimensions, reduced and unreduced homology coincide, so the sequence looks exactly the same in positive dimension. The difference in dimension zero can be attributed to the removal of a piece that looks like .

Note that by adopting the convention that s are zero for negative (a convention that can also be viewed as following from the definition) we can consider the sequence as going on forever.

## Related facts/methods

### Formulation involving closed subsets

Suppose and are closed subsets of a topological space such that is empty and is path-connected. Then, we apply the Mayer-Vietoris homology sequence setting and .

### The Seifert-van Kampen theorem

A fact that has a somewhat similar flavor to the Mayer-Vietoris sequence is the Seifert-van Kampen theorem, where we use open subsets and whose union is , and whose intersection is a path-connected space. While the Mayer-Vietoris sequence is used to compute homology, the Seifert-van Kampen theorem is used to compute the fundamental group, which is the first homotopy group. These results are often used together, in conjunction with the Hurewicz theorem, to determine whether a topological space is -connected for a given positive integer value of .

## Extensions and applications

### General approach behind applications

The general approach is to select and as open subsets such that are all homotopy-equivalent to *easier* spaces. This is typically done by choosing so that they admit strong deformation retractions to the *easier* spaces. In the case of manifolds, for instance, we may want these to admit strong deformation retractions to manifolds of smaller dimensions.

### Application to algebraic topology constructions

Construction | Relation with homology of original space | Proof/explanation | Brief description of how the Mayer-Vietoris homology sequence is used |
---|---|---|---|

suspension | homology groups shift by one: | homology for suspension | We take and both contractible and deformation retracts to (a space homeomorphic to) . |

double mapping cylinder | ? | ? | ? |

## Interpretation in different homology theories

### For singular homology

For singular homology, the Mayer-Vietoris homology sequence can be viewed as the long exact sequence of homology of a short exact sequence of chain complexes, namely:

where denotes the barycentric subdivision operator. Since is homotopic to the identity map, the homologies of this are the homologies of the original chain complexes.

The rough idea is that by subdividing sufficiently, we can make sure that each simplex goes either entirely within or entirely within .