nLab generalized Eilenberg-Mac Lane space

Disambigation

This entry is about ‘generalized Eilenberg-Mac Lane spaces’ as used in Stable Homotopy Theory. The same term is used for a related concept, namely the representing fibration for cohomology with local coefficients. That cohomology is also called twisted cohomology so we have used the term twisted Eilenberg - Mac Lane space for this second use.

Idea

A generalized Eilenberg-Mac Lane space is a topological space with the homotopy type of a Cartesian product of (finitely many) Eilenberg-MacLane spaces.

Accordingly a generalized Eilenberg-Mac Lane spectrum is a spectrum equivalent to a wedge sum of Eilenberg-Mac Lane spectra.

to be expanded… please add if you have the time

Applications

• Generalized Eilenberg-Mac Lane spectra appear as coefficients in an Adams resolution used in the Adams spectral sequence.

Theorem. Let $R$ be a commutative ring with a unit and let $fMod_R$ be the category of finitely generated free $R$-modules. If $H\colon fMod_R\to Spaces$ is a functor such that

• $H(0)$ is a contractible space,
• $H(R)$ is connected and for every $n$ the projections $pr_k \colon R^n\to R$ induce a weak equivalence $H(R^n) \stackrel{\simeq}{\to} H(R)^n$

then the space $H(R)$ is weakly equivalent to a product \prod_{i=1}^{\infty} K(M_i , i)} where $\prod_{i=1}^{\infty} K(M_i , i)}M_i$ is an $R$-module.

Proof. This is the main theorem of [Badzioch]

References

Badzioch, Bernard. “Recognition principle for generalized Eilenberg-Mac Lane spaces.” Cohomological methods in homotopy theory. Birkhäuser, Basel, 2001. 21-26.