# suspension isomorphism

###### Proposition 1.

Let $X$ be a topological space. There is a natural isomorphism

 $s:H_{n+1}(SX)\rightarrow H_{n}(X),$

where $SX$ stands for the unreduced suspension of $X.$

If $X$ has a basepoint, there is a natural isomorphism

 $s:\widetilde{H}_{n+1}(\Sigma X)\rightarrow\widetilde{H}_{n}(X),$

where $\Sigma X$ is the reduced suspension.

A similar proposition holds with homology replaced by cohomology.

In fact, these propositions follow from the Eilenberg-Steenrod axioms without the dimension axiom, so they hold for any generalized (co)homology theory in place of integral (co)homology.

Title suspension isomorphism SuspensionIsomorphism 2013-03-22 13:25:58 2013-03-22 13:25:58 antonio (1116) antonio (1116) 4 antonio (1116) Theorem msc 55N99 Suspension