Can anyone give an example of a pointed map $p:(E,e)\to (B,b)$ between connected pointed spaces (reasonably nice, say of the homotopy type of CW complexes) such that $p$ admits a homotopy section but does not admit a pointed homotopy section? That is, there exists a map $s:B\to E$ such that $p\circ s:B\to B$ is freely homotopic to the identity, but there does not exist any pointed map $\sigma:(B,b)\to (E,e)$ such that $p\circ \sigma:(B,b)\to (B,b)$ is pointed homotopic to the identity?
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asked Sep 16, 2021 at 15:29
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1$\begingroup$ Let $B$ be the comb space with its bad basepoint, let $E=\ast$, and let $p$ be the inclusion of the basepoint. $\endgroup$– TyroneCommented Sep 16, 2021 at 16:29
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3$\begingroup$ If $p\circ s$ is freely homotopic to the identity then it is a weak homotopy equivalence, and therefore has a pointed homotopy inverse. Precomposing $s$ with this inverse will give you a pointed homotopy section. So it seems to me that what you are asking for is impossible. $\endgroup$– Gregory AroneCommented Sep 16, 2021 at 18:43
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