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(First posted on math.SE, with no answers.)

That is:

For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?

The discussion at this math.SE question has narrowed it down to the following two cases: either

  • $n = k-1$, in which case $k = 2, 4, 8$, realized by the complex, quaternionic, and octonionic Hopf fibrations, or
  • $n = 3k-3$, in which case $k \ge 4$ is even.

I am moderately confident that the second case doesn't occur, but don't know how to rule it out.

Here's what I can show about it: such a submersion gives rise to a smooth fiber bundle $F \to S^{4k-3} \to S^k$ where $F$ is a smooth frameable closed manifold of dimension $3k-3$. Taking homotopy fibers gives a map $\Omega S^k \to F$ whose homotopy fiber is $(4k-5)$-connected, hence which induces an isomorphism on homotopy and on cohomology up to degree $4k-5$.

This determines the cohomology of $F$ as a ring: $F$ has the cohomology of $S^{k-1} \times S^{2k-2}$. When $k = 2$ Mike Miller showed that $F$ must in fact be homeomorphic to $S^1 \times S^2$ and then gets a contradiction from looking at homotopy groups. When $k \ge 4$ we also know that $F$ is simply connected.

Aside from knowing whether it's possible to rule out the last case, I'd also be interested in a simpler argument that $k$ must be even. The argument I gave passes through both the topological Poincaré conjecture and Adams' solution to the Hopf invariant $1$ problem...

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In most cases $\pi_{n+k}(S^k)$ is a finite group, so that the homotopy fiber of any map $S^{n+k}\to S^k$ is rationally equivalent to $\Omega S^k\times S^{n+k}$ and therefore has homology in arbitrarily high dimensions and cannot be a manifold.

The only exceptions with $n>0$ have $n=k-1$.

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    $\begingroup$ Thanks! That's more straightforward than I thought; I guess the appearance of $n = k - 1$ was a clue I should've taken more seriously. $\endgroup$ – Qiaochu Yuan May 26 '15 at 4:02
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I want to add that having a fiber bundle with total space a sphere is very restrictive even if you don't assume that the base is a sphere too. This has been studied. I will exclude the obvious cases of the fiber or the base being points. The starting point is a fairly elementary observation due to Whitehead and Spanier that for a fiber bundle $F\to E\to B$ where all spaces are connected finite dimensional CW complexes, if the fiber $F$ is contractible in $E$ then $F$ must be an $H$-space. This of course restricts things a great deal and by playing with rational cohomology it's easy to see that if $E$ is a sphere (which can only be odd dimensional) the fiber must rationally be an odd dimensional sphere too. Browder later showed that $F$ can only be homotopy $S^1, S^3, S^7$. This implies that the base is a homotopy $\mathbb{CP}^n$, a homology $\mathbb{HP}^n$ or a homotopy $S^8$ respectively. This yields the result when one assumes that the base is a sphere as a special case.

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