suspension isomorphism

Proposition 1.

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

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

where $S\mathit{}X$ stands for the unreduced suspension of $X\mathrm{.}$

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

$$s:{\tilde{H}}_{n+1}(\mathrm{\Sigma }X)\to {\tilde{H}}_n(X),$$

where $\mathrm{\Sigma }\mathit{}X$ isthe 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.