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Lifts of $f \colon B \to X$ into a fibration $F \to E \to X$ can be identified with sections of the pullback bundle $f^*(E) \to B$. I want to try to compute the path components $\pi_0(\Gamma(f^*(E))$ of the space of sections of the pullback bundle (a.k.a. lifts of $f$). (I believe the correct topology for the space of sections would be the subspace topology coming from the compact-open topology in the mapping space $\text{Map}(S^n,f^*(E))$).

From obstruction theory we know that obstructions to extending an $(n-1)$-partial vertical homotopy $\overline{f_0}\vert_{B_{n-1}} \simeq \overline{f_1}\vert_{B_{n-1}}$ of sections of $f^*(E) \to B$ lie in $H^n(B,\pi_n(F))$. For simplicity, suppose that $B = S^n$ and we have a vertical homotopy over the $0$-cell of $S^n$. Then, write $d^n(\overline{f_0},\overline{f_1})$ for the element in $\pi_n(F)$ corresponding to the obstruction in $H^n(S^n,\pi_n(F))$.

Any two lifts (a.k.a. sections) of $f$ represent classes $[\overline{f_0}]$ and $[\overline{f_1}]$ in $\pi_n(f^*(E))$. Consider the homotopy long exact sequence $$ \dots \to \pi_{n+1}(S^n) \xrightarrow{\delta} \pi_n(F) \xrightarrow{i_*} \pi_n(f^*(E)) \xrightarrow{p_*} \pi_n(S^n) \to \dots.$$ A sufficient condition for the two lifts to be distinct as lifts is that they are distinct as maps into $f^*(E)$. In this case we can see that $p_*^{-1}\{[1]\}$ contains all the distinct classes of maps which lift $f$, and since $p_*$ is a group homomorphism then one has a bijection $$p_*^{-1}\{[1]\} \leftrightarrow \ker p_* = \text{im} \,i_*,$$ thus giving an injective set-map $$\Phi \colon \text{im}\, i_* \hookrightarrow \pi_0(\Gamma(f^*(E))).$$

My first preliminary question is: under what conditions is this set-map $\Phi$ not surjective? For example, if $i_*$ is injective then it seems that $\Phi$ is surjective. Is the converse true?

Furthermore, suppose that $\Phi$ isn't surjective. This means that we need to distinguish maps $S^n \to f^*(E)$ that are homotopic but not through any homotopy of sections. For any two classes of lifts of $f$ we have that their difference $[\overline{f_0}]- [\overline{f_1}]$ defines an element of $\ker p_*$ and hence of $\text{im}\, i_*$. Once again, if their difference is non-zero in $\pi_n(f^*(E))$ then they will be distinct as lifts. But, if their difference is zero then it seems feasible that the obstruction $d^n(\overline{f_0},\overline{f_1})$ to finding a homotopy of sections would lie within $\ker i_*$. This might lead one to hope that $\ker i_*$ injects into $\pi_0(\Gamma(f^*(E)))$ and that there is a bijection $$\ker i_* \oplus \text{im}\, i_* \leftrightarrow \pi_0(\Gamma (f^*(E))).$$

My second question is if there is such a bijection?

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  • $\begingroup$ Maybe start with the fiber bundle case. One shouldn't expect there to be a very general answer. here is a reasonable first pass with some references in the comments/answer. A good $\endgroup$ Feb 14, 2022 at 15:57
  • $\begingroup$ As for your preliminary question, I think it is not usually true. It's a little early for me so the following might be wrong, hopefully someone corrects me if it is. Take $f$ to be the identity map. Then there is a homeomorphism between maps $g:X \to Y$ and global sections of the trivial bundle $X \times Y \to X$, so take $Y$ to be connected, and any two non-homotopic maps $X \to Y$ (they are in different path components) to show that $i_*$ can't be surjective in this case. $\endgroup$ Feb 14, 2022 at 16:10

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