Cohomology of a fiber bundle with fiber $H$ and base space $BG$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:35:22Z http://mathoverflow.net/feeds/question/112224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112224/cohomology-of-a-fiber-bundle-with-fiber-h-and-base-space-bg Cohomology of a fiber bundle with fiber $H$ and base space $BG$ Xiao-Gang Wen 2012-11-12T21:23:50Z 2012-11-13T02:08:07Z <p>Are there any general results on the (integral) cohomology of fiber bundle, where the fiber is a compact group $H$ (continuous or discrete) and the base space is the classifying space $BG$ of another compact group $G$ (continuous or discrete). Any literature references is much appreciated.</p> <p>Since we have two groups $G$ and $H$. I wonder if the result can be expressed as group cohomology of the two groups.</p> http://mathoverflow.net/questions/112224/cohomology-of-a-fiber-bundle-with-fiber-h-and-base-space-bg/112234#112234 Answer by algori for Cohomology of a fiber bundle with fiber $H$ and base space $BG$ algori 2012-11-13T00:20:16Z 2012-11-13T02:08:07Z <p>A typical example would be the case when $G$ is a subgroup of $H$. Then $(EH\times H)/G$ (diagonal action) is 1. homotopy equivalent to $H/G$, and 2. fibered over $BG=EH/G$ with fiber $H$. Note that this works both in the Lie case and the discrete case but in the latter case what we get is not very interesting since the fiber of our fibration is a potentially infinite discrete space.</p> <p>[upd: There is one thing one can extract from this though: the $i$-th cohomology group of $G$ with coefficients in the infinite product <code>$\Pi_{h\in H}\mathbb{Z}_{(h)}$</code> is $\Pi_{h G\in H/G} \mathbb{Z}_{(hG)}$ when $i=0$ and is 0 otherwise; this may be of some use when $G$, or its index in $H$, is finite.]</p> <p>On the other hand, if $G$ is normal in $H$ one can go a bit further: $BH=EH/H$ is the quotient of $BG=EH/G$ by a free action of $H/G$. So, as above, we construct a fibration over $B(H/G)$ with fiber $BG$ and total space $BH$. If we now take an $H$-module $M$ (i.e., a local system on $BH$) we get the Hochschild-Serre spectral sequence</p> <p>$$E_2^{pq}=H^p(H/G,H^q(G,M))\Rightarrow H^{p+q}(H,M).$$</p> <p>There are lots of references where this is discussed. One could take a look e.g. at the original paper by Hochschild and Serre (Cohomology of group extensions, Transactions AMS 1953).</p>