nLab suspension isomorphism

Idea

In cohomology of topological spaces/homotopy types, the suspension isomorphism identifies the degree-$n$ reduced cohomology of a pointed space $X$ with the degree-$(n+k)$ cohomology of its $k$-fold suspension, hence of its smash product with the $k$-sphere

$$\tilde H^n(X,E) \simeq \tilde H^{n+k}(S^k \wedge X, E) \,.$$

In particular this serves to express cohomology in negative degree in terms of cohomology in non-negative degree of suspended spaces.

Requiring this to hold in equivariant cohomology theory not just for integer grading and spheres but also for RO(G)-grading and representation spheres leads to the concept of genuine equivariant cohomology represented by genuine G-spectra.

Related concepts

• Freudenthal suspension theorem

• Steenrod square

• cohomology theory

References

Textbook accounts include

• Akira Kono, Dai Tamaki, p. 19 in Generalized cohomology, AMS 2002, esp. chapter 2 (pdf)

In equivariant homotopy theory:

• Peter May, chapter XI of Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)