Let $M$ be the underlying topological manifold of a smooth complex projective surface. Assume $\pi_1(M)=\{0\}$ and $\pi_2(M)\neq \mathbb{Z}^2$.
Is there a Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<4$?
$\begingroup$
$\endgroup$
2
-
5$\begingroup$ I think you can exclude d=1,3 from the euler characteristic. At least if this is required to be a fibration of manifolds. $\endgroup$– Thomas RotCommented Sep 7, 2020 at 20:35
-
1$\begingroup$ Why is the question changed dramatically? $\endgroup$– Thomas RotCommented Sep 8, 2020 at 20:15
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
I don't think so. (There are the exceptions you mention: the two $S^2$ bundles over $S^2$.) By Thomas's comment, let's assume that d = 2.
If M is a fibration of a manifold with finite complexes for base and fiber, then the base and fiber are Poincare duality spaces (Gottlieb, PAMS 1979). Hence in this dimension, they are surfaces. Since your M is simply connected, the base must be a 2-sphere. But then you can conclude from the long exact sequence of the fibration that the fiber is a 2-sphere as well.
answered Sep 7, 2020 at 23:25
-
$\begingroup$ we don't assume that the fiber is a CW complex $\endgroup$– user164740Commented Sep 8, 2020 at 5:51
-
$\begingroup$ @JoeT What are the circumstances where you might have such a fibration (ie with non-CW complex fiber)? Maybe you could give some more context for your question. $\endgroup$ Commented Sep 8, 2020 at 12:38
-
$\begingroup$ I don't know, I just got curious if there could be such a fibration (seeing as the fiber is strictly speaking not a part of the data of a fibration) $\endgroup$– user164740Commented Sep 8, 2020 at 12:58