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Let $\pi: E \to B$ be a fiber bundle of (topological or differentiable) manifolds. Denote by $[B, E]_{\pi}$ the set of all homotopy classes of sections of the bundle, i.e

\begin{align} [B, E]_\pi &= \{\sigma: B \to E \ | \ \pi\sigma = \text{id}_B \}/\sim \\ \sigma \sim \sigma' &\iff \exists H: I \times B \to E \ | \ H_0 = \sigma, H_1 = \sigma', \pi H_t = \text{id}_B \end{align}

Is it known how to calculate such set? With "calculate" I mean to reduce the computation of it to the computation of something more known, as the homology/cohomology/homotopy groups of $E$, $B$ or some combination of them.

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    $\begingroup$ Try H. Baues's Obstruction Theory if the more modern texts don't have what you need. In $\S$ 6 he constructs a spectral sequence derived from a Moore-Postnikov decomposition which seems to be what you are asking for. $\endgroup$
    – Tyrone
    May 30, 2020 at 7:39
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    $\begingroup$ The keyword is indeed obstruction theory, but in practice this will be very difficult. E.g. if B=S^n\times S^k, with fiber S^k, you are asking for a computation of the homotopy groups of spheres. $\endgroup$
    – Thomas Rot
    May 30, 2020 at 8:26
  • $\begingroup$ @ThomasRot That's indeed my goal: reduce the computation to something well known (or well studied, at least, as the homotopy groups of spheres) $\endgroup$
    – CNS709
    May 30, 2020 at 10:06
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    $\begingroup$ The machinery is the same, but the computations are obviously going to be simpler. As Thomas points out, the keyword is 'obstruction theory'. If you haven't seen it before then there are better starting places than Baues's book. For instance, Davis-Kirk's 'Lecture Notes in Algebraic Topology' has a good introductory chapter on the subject, as does Hatcher's book. Arkowitz's 'Introduction to Homotopy Theory' constructs Moore-Postnikov factorisations. These are all good starting points you should consult first, but they don't treat the more advanced topics like non-simple fibrations. $\endgroup$
    – Tyrone
    May 30, 2020 at 11:22
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    $\begingroup$ A good introduction to obstruction theory is the first few chapters of Mosher and Tangora's book. $\endgroup$ May 30, 2020 at 11:31

1 Answer 1

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The relevant (simple) obstruction theory mentioned in the comments is contained in G. W. Whitehead's Elements of Homotopy Theory, Section VI.6. There an answer to the more general lifting problem is obtained under certain conditions, one of which is that the fibre $F$ is $q$-simple for certain values of $q$ (meaning $\pi_1(F)$ acts trivially on $\pi_q(F)$). The answer is then given in terms of cohomology with local coefficients.

In the non-simple case, you might check the papers of P. Olum (or the work of H. J. Baues citeed in the comments), but the answer is likely to be complicated.

However, you mentioned in the comments that you are interested in the case of the projectivized tangent bundle $$ \mathbb{R}P^{n-1} \to PTS^n \to S^n $$ of the $n$-sphere. A section of this bundle is also known as a line field on $S^n$. It is well known that there exists a line field on a closed manifold $N$ if and only if the Euler characteristic $\chi(N)$ is zero (this is proved as Theorem 2.3 in https://arxiv.org/abs/1612.04073, for example). Hence there exist line fields on $S^n$ if and only if $n$ is odd.

In the case $n$ odd, the fibre $\mathbb{R}P^{n-1}$ is not $(n-1)$-simple (the acion of $\pi_1(\mathbb{R}P^{n-1})=\mathbb{Z}/2$ on $\pi_{n-1}(\mathbb{R}P^{n-1})=\mathbb{Z}$ is non-trivial) so the standard obstruction theory does not apply to classify the sections. However, the more general problem of classifying line fields on a closed $n$-manifold $N$ up to homotopy, i.e. classifying sections of the projectivized tangent bundle $$ \mathbb{R}P^{n-1} \to PTN \to N $$ up to vertical homotopy, has been attacked by Koschorke in

Koschorke, Ulrich, Homotopy classification of line fields and of Lorentz metrics on closed manifolds, Math. Proc. Camb. Philos. Soc. 132, No. 2, 281-300 (2002). ZBL0994.57024.

There it is claimed (in Example 1.7) that the number of line fields on $S^n$ up to homotopy is given by $$ \ell(S^n) = \begin{cases} 0 & n\mbox{ even}\\ 1 & n\equiv 1(4)\\ 2 & n\equiv 3(4), n\ge7\\ \infty & n=3. \end{cases} $$

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  • $\begingroup$ I was wondering about the infinitely many line fields on $S^3$, but this is clear: Since $S^3$ is parallelizable, homotopy classes of line fields are just homotopy classes of maps $S^3\to \mathbb{R}P^2$, and $\pi_3(\mathbb{R}P^2)\cong \pi_3(S^2)\cong \mathbb{Z}$. $\endgroup$
    – Mark Grant
    Jun 4, 2020 at 10:12
  • $\begingroup$ Yes, this works for $n=1,3,7$ ($S^n$ parallizable) and it gives one example for type (together with $n$ even). $[S^1, RP^0] = e, [S^7, RP^6] = [S^7, S^6] = \pi_1^s = Z_2$. I will definitively look at that paper for the other cases, thank you. I was hoping there'd be more "classical methods" $\endgroup$
    – CNS709
    Jun 4, 2020 at 14:33

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