nLab
generalized Eilenberg-Mac Lane space

Contents

Contents

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

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

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

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

Last revised on June 12, 2018 at 17:45:10. See the history of this page for a list of all contributions to it.