Suppose we have an inclusion of a closed sub-Lie group $H\to G$ and take the space of left cosets of $H$, $G/H$. How is this related to the homotopy cofiber of the inclusion of topological spaces $cof(\iota:H\to G)$, i.e. "coning off" $H$ in $G$. Are they homotopy equivalent?
2026-05-16 21:10:40.1778965840
Connection between spaces of cosets and homotopy cofibers
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I think the following is a counterexample. Consider $H:= \{\pm 1\}\subset G:= U(1)$. Then $G/H = RP^1$, whereas the cone on the inclusion is homotopy equivalent to $S^1\vee S^1$.