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Obstruction theory is an intuitive yet powerful method in homotopy theory. It helps study the space of maps (more generally, sections of bundles) inductively on skeletons. For example, it provides a geometric proof of that $K(G,n)$ classifies cohomology [1].

I'd like to learn the full theory, but most textbooks I found (e.g. [2][3][4][5]) only deal with the primary obstructions. In this question, I will lay out what I understand so far, and some theorems I wish to have.

Primary Obstructions

Let $F \to E \to B$ be a fiber bundle. Suppose $B$ has a CW decomposition, and $F$ is $0$-connected, $1$-connected, with nontrivial homotopy groups at levels $m < m' < m'' < \ldots$. Call a section over the $k$-skeleton a $k$-section. Then obstruction theory shows, up to homotopy, that

  1. There's a unique $(m-1)$-section, and there is at least one $m$-section.

  2. The primary obstruction class $\theta$ lives in $H^{m+1}(B, \pi_{m}(F))$, which depends only on the bundle, but not the selected section so far. $\theta = 0$ if and only if there is at least one $m$-section that can be extended to an $(m+1)$-section.

  3. If $\theta = 0$, the space of $(m+1)$-extendable $m$-sections is $H^m(B, \pi_{m}(F))$.

Higher Obstructions?

I'd like to complete the picture above. So let's assume $\theta = 0$, and pick an extendable $m$-section $x \in H^m(B, \pi_m(F))$ (in light of the classification of $(m+1)$-extendable sections above). I wish we have, up to homotopy, that

  1. There's a unique $(m'-1)$-section extending the section $x$, and there is at least one $m'$-section extending $x$.

  2. The next obstruction class $\theta'$ lives in $H^{m'+1}(B, \pi_{m'}(F))$, which depends on both the bundle $E$ and the selected $m$-section $x$.

  • Question 1: Are statements 4 and 5 correct?

If they are correct, some questions naturally arise:

  • Question 2: Do we know how to compute or approximate $\theta'(E,x)$ explicitly? Is $\theta'(E,-)$ a linear map (or does it have any structure)?

  • Question 3: How do we classify $(m'+1)$-extendable $m'$-sections? Do we have an analogue statement that $\theta'(E,x)=0$ if and only if the space of $(m'+1)$-extendable $m'$-sections is $H^{m'}(B, \pi_{m'}(F))$?

  • Question 4: Use the classification in question 2 to pick an $m'$-section. What's the next obstruction and classification of sections?

  • Question 5: Unified theory for all obstructions at all $m < m' < m'' < \ldots$.

Reference

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    $\begingroup$ I found Norman Steenrod's Topology of Fibre Bundles to be a good source for learning the basics of obstruction theory. $\endgroup$ Commented Jan 24 at 14:41
  • $\begingroup$ These questions get relatively tricky to navigate and some of them are not so simple. Eg: question 2 about the map being linear does have a positive answer if $m'$ is small, roughly up to twice $m$, but not in general. Question 3 is not quite true, but instead: if you fix an $m'$-extendable $(m'-1)$-section, then $\pi_0$ of the space of extensions to an $(m'+1)$-extendable $m'$-section is the specified cohomology group. (You also need to be careful about local systems throughout.) $\endgroup$ Commented Jan 24 at 19:03
  • $\begingroup$ I don't know a great reference for you, but I believe that the most comprehensive answer would be to (a) reduce to the version where $B = |K|$, the realization of a simplicial object, and (b) then apply Bousfield's paper on homotopy spectral sequences and obstructions (link.springer.com/article/10.1007/BF02765886) to the corresponding space of sections, which I believe then has a cosimplicial structure. $\endgroup$ Commented Jan 24 at 19:06
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    $\begingroup$ There is a very abstract and technical book "Obstruction Theory -- On Homotopy Classification of Maps" by Hans Baues that's probably worth mentioning (maybe later users will find this question when they are interested in other questions about obstruction theory). $\endgroup$ Commented Jan 25 at 15:07
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    $\begingroup$ @JeffStrom Thanks! I did check that book from the library, and it claims to solve the classification of extension problem in chapter 3 with a spectral sequence derived from a sequence of cofibration maps (e.g. CW cell attachments). However, I find it very difficult to understand, and it is not clear to me why it has solved it. Its result does not seem as clean as suggested in Mark Grant's answer either. $\endgroup$ Commented Jan 25 at 15:40

1 Answer 1

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I sympathise with you: the literature on obstruction theory is notoriously scattered and hard to navigate. That said, I think you might find answers to your questions in Chapter IX of

Whitehead, George W., Elements of homotopy theory, Graduate Texts in Mathematics. 61. Berlin-Heidelberg-New York: Springer-Verlag. XXI, 744 p. DM 69.00; $ 38.00 (1978). ZBL0406.55001.

In particular, Theorem 7.5 in that chapter seems to give an attractive answer to your Question 2. Without wading through Whitehead's notation to check, I guess what it says in the notation of the question is the following. The first non-trivial k-invariant of $F$ is a map $\theta:K(\pi_m(F),m)\to K(\pi_{m'}(F),m'+1)$, which corresponds to a cohomology operation $\theta:H^m(-;\pi_m(F))\to H^{m'+1}(-;\pi_{m'}(F))$. Now if $x\in H^m(B;\pi_m(F))$ is an $m$-section, then $\theta'(E,x)=\theta(x)\in H^{m'+1}(B;\pi_{m'}(F))$. (Actually, I guess if this is literally true it might require that $\pi_1(B)$ acts trivially on $\pi_*(F)$.)

Generally speaking though, it's worth learning the Postnikov tower approach if you're serious about computing higher obstructions. The monograph

Thomas, Emery, Seminar on fiber spaces. Lectures delivered in 1964 in Berkeley and 1965 in Zürich. Berkeley notes by J. F. McClendon, Lecture Notes in Mathematics. 13. Berlin-Heidelberg-New York: Springer Verlag. iv, 45 p. (1966). ZBL0151.31604.

might be a good place to start. Then look at papers from around the same period by Thomas, Hermann, Mahowald and others for further applications.

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  • $\begingroup$ Thanks for both references! Thomas's deals with basic Postnikov tower theory, but I guess the relevant thing here is to construct the Postnikov tower $F_i$ for the fiber $F$, and address the lifting problem incrementally along the sequence $F = F_\infty \to \ldots \to F_{i+2} \to F_{i+1} \to F_{i} \to \ldots$ (here the arrows can be described by the $k$-invariants). I think what you said is correct, according to Whitehead's treatment. $\endgroup$ Commented Jan 24 at 14:49
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    $\begingroup$ However, I really wish to see this picture extended to all cases: Denote $m^{(3)} = m'''$, $m^{(4)} = m''''$, and so on. Then I would hope that the $l$-th nontrivial $k$-invariant $\theta^{(l)}: K(\pi_{m^{(l)}}(F), m^{(l)}) \to K(\pi_{m^{(l+1)}}(F), m^{(l+1)}+1)$ maps the chosen-so-far $m^{(l)}$-section (classified as elements in $H^{m^{(l)}}(B; \pi_{m^{(l)}}(F))$) to the next obstruction. Unfortunately, Whitehead postpones such a discussion into his never-existent Volume II, and thus his treatment stops at $l = 2, 3$. Hopefully someone can negate or confirm this, and provide a reference. $\endgroup$ Commented Jan 24 at 14:49

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