1

Let $X\to Y$ be a proper map between pseudo-manifolds such that fibers are $S^2\vee S^2$, is it true that $X\to Y$ is locally trivial $S^2\vee S^2$ bundle?

  • If the map is a submersion, proper and $X$ and $Y$ the answer is yes. See Ehresman lemma. https://en.wikipedia.org/wiki/Ehresmann's_lemma – Tsemo Aristide Jan 31 '16 at 01:59
  • @TsemoAristide If the map is submersion, I think the fibers would also be smooth? I think the map is not a submersion. –  Jan 31 '16 at 02:02
  • 1
    @Tsemo: If the map is a submersion the fibers are manifolds. –  Jan 31 '16 at 02:03
  • I think this is a very believable claim but would have no idea how to prove it. Do you have any simple test cases where this occurs? –  Jan 31 '16 at 02:05
  • 1
    I have a hard time imagining it possible for a manifold to bundle with $S^2 \vee S^2$ fibers over another space, even locally. A simple test obstruction is that for every $x$ in $M$ the relative homology groups $H_*(M, M-x)$ should be the homology of a sphere for every point, but it seems hard to maintain this homogeneity of data between points corresponding to general points on one of the spheres and those at the wedge point. – jxnh Jan 31 '16 at 02:35
  • 1
    That's a good point. You can't possibly have fiber $S^2 \vee S^2$ since then you'd have an open set homeomorphic to $(S^2 \vee S^2) \times \Bbb R^k$, which is not a manifold! –  Jan 31 '16 at 02:45
  • @MikeMiller I was looking at Artin-Mumford's example of non-rational unirational 3-fold. In that case, $X,Y$ are topological spaces associated to singular varieties, where $Y$ is the union of two elliptic curves in $\mathbb{P}^2$ the fibers are $\mathbb{P}^1\vee\mathbb{P}^1$. I had thought to post the question with $X,Y$ pseudo-manifolds to reduce confusion, and it turns out I should not drop the "pseudo".. –  Jan 31 '16 at 02:46
  • @MikeMiller Ah.. That's the remark why I shouldn't say manifold! But now I think in that special case, I can check it is a locally trivial bundle, since everything is defined by polynomials. –  Jan 31 '16 at 02:47

0 Answers0