# Classifying the homotopy classes of lifts

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?

• 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 Feb 14, 2022 at 15:57
• 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. Feb 14, 2022 at 16:10