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 $\mathrm{\Sigma }\mathit{}X$ 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
Canonical name	SuspensionIsomorphism
Date of creation	2013-03-22 13:25:58
Last modified on	2013-03-22 13:25:58
Owner	antonio (1116)
Last modified by	antonio (1116)
Numerical id	4
Author	antonio (1116)
Entry type	Theorem
Classification	msc 55N99
Related topic	Suspension