1
$\begingroup$

Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups of $X$ (in terms of homology of some spaces).

Summary of Serre's method

Case. $k=n$

By Hurewitz theorem, $\pi_n(X) \simeq H_n(X)$.

Case. $k=n+1$

Serre's method is to use Hurewitz theorem again [1], by producing another simplicial complex $E = E_X$ whose homotopy groups agree with those of $X$, except $\pi_n(E) = 0$. Indeed, with such $E$, Hurewitz theorem guarantees that $\pi_{n+1}(E) \simeq H_{n+1}(E)$. Hence $\pi_{n+1}(X) \simeq H_{n+1}(E)$.

Case. $k>n+1$

Repeat the previous prodecure, with $E$ being $X$, and so on.

Crux

How do we actually construct such $E$? I understand that if we construct $E$ as a fibre product $E \to X$, with fiber $F \simeq K(\pi_{n}(X), n-1)$, then by (writing down) the associated long exact sequence of homotopy groups, it is easy to see that the goal is equivalent to proving that the induced map in the long exact sequence

$$\Phi: \pi_{n}(X) \to \pi_{n-1}(K(\pi_{n}(X), n-1))$$

is an isomorphism.

Question

  1. How to build that bundle / fibre product so that $\Phi$ is an isomorphism?
  2. Can we build such $E$ so that the result is functorial, i.e. each $f: X \to Y$ has $f^E: E_X \to E_Y$ such that the composition

$$ \pi_{n+1}(X) \xrightarrow{b.iso^{-1}} \pi_{n+1}(E_X) \xrightarrow{h.iso} H_{n+1}(E_X) \xrightarrow{f^E} H_{n+1}(E_Y) \xrightarrow{h.iso} \pi_{n+1}(E_Y) \xrightarrow{b.iso} \pi_{n+1}(Y)$$ equals the map $\pi_{n+1}(X) \xrightarrow{f} \pi_{n+1}(Y)$? (Here, $h.iso$ denotes the Hurewitz isomorphism, while $b.iso$ denotes the isomorphism induced from the bundle data constructed in 1.)

Reference

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ If X is (n-1)-connected, there is a classifying map X -> K(pi, n) which induces an isomorphism on pi_n. (This map can be constructed functionally for (n-1)-connected spaces.) Your space E is the homotopy fiber of this map. I think this construction, under the name "Whitehead tower", should appear in many textbook references on homotopy theory. $\endgroup$
    – mme
    Commented Dec 4, 2022 at 0:21
  • 4
    $\begingroup$ The homotopy fiber may be constructed in the category of simplicial sets, yes (which is often more appropriate than the category of simplicial complexes); this too should be available in textbook references that cover simplicial sets, though I don't know which one to suggest you. Your previous post from April suggests you're interested in algorithmic computations, and those computations take place in the category of simplicial sets. Notice that the Kenzo program linked there works with simplicial sets. $\endgroup$
    – mme
    Commented Dec 4, 2022 at 3:00
  • 1
    $\begingroup$ I guess the answer is in mathoverflow.net/questions/5268/functorial-whitehead-tower $\endgroup$
    – user43326
    Commented Dec 5, 2022 at 9:18
  • 1
    $\begingroup$ @user43326: that may answer the functoriality question, but I think item 1 is asking how to construct $E$ algorithmically. (At least, in another thread the OP seemed interested in understanding why homotopy groups of spheres are computable. I too would like to know this!) Perhaps the OP can clarify if that's what they mean by item 1. $\endgroup$
    – HJRW
    Commented Dec 5, 2022 at 12:14
  • 1
    $\begingroup$ @HJRW That would be ideal, the closer to an actual algorithm the better and clearer. $\endgroup$
    – Student
    Commented Dec 5, 2022 at 14:50

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .