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In Ralph Cohen's notes on the topology of fiber bundles (pp.63) he claims that the space of all $G$-equivariant maps from $P$ to $EG$ denoted by Map$^G(P,EG)$ is aspherical, where $EG$ is the total space of the classifying space $BG$ and $P$ is any principal $G$-bundle. Could somebody help me to see why it is true? Is it because $EG$ is aspherical by the definition of universal bundle?

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    $\begingroup$ There is an answer here link, in fact it shows that it is contractible. $\endgroup$ – Goa'uld May 31 '16 at 20:36
  • $\begingroup$ Well, the mapping space has a free action of the gauge group and the point of showing that it is contractible is to identify that the quotient is a classifying space for the gauge group. So being fair, it really has an appropriate title. $\endgroup$ – Goa'uld Jun 2 '16 at 12:20

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