# Homology for suspension

## Contents

## Statement

In this article, we give the key results relating the homology groups of a topological space and the homology groups of its Suspension (?).

### Version for unreduced homology

This states that:

where and denote the and homology groups. The result also holds for homology groups with coefficients.

Further:

and:

### Version for reduced homology

This states that:

where denotes the reduced homology. Note that for , reduced homology and unreduced homology coincide; for , the unreduced homology has an extra in it. For , the right side is the trivial group, giving that is trivial, so is a path-connected space.

### Category-theoretic version

The isomorphisms between the homology groups of a topological space and its suspension are *natural* isomorphisms between these functors. In particular, if is a continuous map, then we have an induced continuous map . There is a commuting diagram relating the homomorphism on reduced homology between and and the homomorphism on reduced homology between and .

## Facts used

## Proof

Recall that is obtained by taking (where is the closed unit interval) and then identifying all points in with each other and separately identifying all points in with each other. We will call these two points and respectively. We consider the following open subsets and to use for the Mayer-Vietoris homology sequence:

Subset | Concrete description | Has a strong deformation retraction to ... | More explanation |
---|---|---|---|

a point (i.e., it is contractible) | is homeomorphic to the cone space | ||

a point (i.e., it is contractible) | is homeomorphic to the cone space | ||

the factor is a contractible space |

### Proof version with reduced homology

We note that and are open subsets and their union is . Further, because of the strong deformation retraction facts mentioned, *all* reduced homology groups of and are trivial groups and *all* reduced homology groups of are isomorphic to the corresponding reduced homology groups of .

The original Mayer-Vietoris homology sequence reads:

Every third term of this sequence is zero, aand the fragment above simplifies to:

Since this is a long exact sequence, the map is forced to be an isomorphism. This completes the proof.