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Jun 3, 2016 at 0:40 vote accept QSR
Jun 2, 2016 at 8:57 answer added user83633 timeline score: 8
Jun 2, 2016 at 8:29 comment added Mark Grant This seems highly unlikely. Better to think of $f$ as the fibre inclusion in a fibration sequence $M\to M/\Sigma_k\to K(\Sigma_k,1)$ (where the last map classifies the covering). Then you see $F$ as the fibre inclusion in a corresponding fibration of Eilenberg--Mac Lane spaces.
Jun 2, 2016 at 8:07 history asked QSR CC BY-SA 3.0