16
$\begingroup$

Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$.

In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the classifying space $X\to BG$ classify principal $G$-bundles on $X$.

In the literature I have encountered spaces $K(G,V,n)$, where $G$ is a group and $V$ is a finite dimensional $G$-representation, called generalised (or sometimes twisted) Eilenberg MacLane spaces. These spaces are determined up to homotopy by the property that

$\pi_{i}(K(G,V,n)) = \begin{cases} G,\ \ i=1,\\ V,\ \ i=n,\\ 0,\ \ \text{else}. \end{cases} $

My first question is, what do generalised Eilenberg MacLane spaces classify? Am I correct in thinking that they represent cohomology in the local system determined by $G$ and $V$?

My second question is, what does it mean to localise with respect to generalised Eilenberg MacLane spaces?

By this I am thinking of a Bousfield localisation on the model category of spaces in which a local equivalence is declared to be a map $f:X\to Y$ which induces a weak equivalence $$ f^{*}:\text{Map}(Y, K(G,V,n))\to \text{Map}(X,K(G,V,n)), $$ and a space $Z$ is local if any local equivalence $f:X\to Y$ induces a weak equivalence $$ f^{*}:\text{Map}(Y, Z)\to \text{Map}(X,Z). $$

Specifically, I would like to know what information is isolated by performing these localisations for a given group $G$ and representation $V$.

$\endgroup$
18
$\begingroup$

To answer your first question, take a look at the reference

Gitler, Samuel, Cohomology operations with local coefficients, Am. J. Math. 85, 156-188 (1963). ZBL0131.38006.

In particular, Theorem 7.18 in Chapter III does what you want.

To paraphrase, let $\mathcal{V}$ be the local system of groups on your Eilenberg--Mac Lane space $K(G,V,n)$ determined by the representation $G\to \operatorname{Aut}(V)$. Given pointed connected CW-complexes $X$ and $Y$ and a homomorphism $\alpha:\pi_1(X)\to \pi_1(Y)$ between their fundamental groups, let $[X,Y]_\alpha$ denote the set of pointed homotopy classes of pointed maps $f:X\to Y$ such that $f_*=\alpha:\pi_1(X)\to \pi_1(Y)$. Then there is a (natural in an appropriate sense) isomorphism $$ [X,K(G,V,n)]_\alpha \cong H^n(X;\alpha^*\mathcal{V}). $$ The isomorphism is given as in the untwisted case by pulling back an appropriately defined fundamental class $\iota\in H^n(K(G,V,n);\mathcal{V})$.

$\endgroup$
  • $\begingroup$ Thank you for the answer, the reference is well appreciated. Do you have any pointers on the second part of my question? What is obtained by such localisations? Precisely, what class of spaces is local in this sense? $\endgroup$ – Patrick Elliott May 24 at 13:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.