Tuesday, August 14, 2012

A simple homotopic trick

Here is a simple fact, which seems to be well known to   homotopy theorists.  It might come in handy.

 Suppose that  we have two fibrations

$$ f:Y\to B,\;\; \pi:E\to B, $$

such that  $E$ is contractible.  The pullback to $Y$ of the fibration $\pi: E\to B$  via the map $f$ is a new fibration $g: X\to Y$.   Then the homotopy fiber  $Z$ of $Y\to B$ is homotopic to  $X$.

 Indeed we have a fibration $F:X\to E$ with t homotopy fiber $Z$.  Since $E$ is contractible we  deduce that $X$  is homotopic to $Z$.

This is particularly useful when $X\to Y$ is a principal  $G$-bundle classified by a map $f: Y\to BG$.  The map $f$ can be assumed to be a fibration. The homotopy fiber of $f$ is then $X$.   From the Leray-Serre spectral sequence we obtain a spectral sequnce  converging to the cohomology of $Y$ with $E_2$-term

$$E_2^{p,q}= H^p\bigl(\;BG, \tilde{H}^q(X)\;\bigr), $$

where $\tilde{H}^q(X)$ denotes a local system on $BG$ with stalk $H^q(X)$.
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