Subject: Question about rational homology spheres
Date: Mon, 14 Oct 2002 12:09:31 -0400 (EDT)
From: Piotr Hajlasz
I have a question about rational homology spheres. I work with analysis,
and I know almost nothing about topology, however it turned out that the
following question is very important for my recent paper.
QUESTION 1. Let $X$ be an $n$-dimensional smooth compact orientable
manifold without boundary. What is a necessary and sufficient condition
for the existence of a smooth map $f:S^n-> X$ of nonzero degree?
I believe that the answer is the following:
CONJECTURE. Such mapping exists if and only if $X$ is a rational homology
sphere.
I have some evidence for this conjecture. First of all it is a
necessary condition. $X$ has to be a rational homology sphere - it easily
follows form the Poincare duality (as I work in analysis, it was easier
for me to see that the Poincare duality for de Rham cohomology implies
that the de Rham cohomologies of $X$ have to be the same as those for the
sphere, which means $X$ is a rational homology sphere).
As concerns the other implication I can only prove it for simply connected
rational homology spheres, so assume now that $X$ is a simply connected
rational homology sphere. As I understand the Hurewicz theorem
modulo the Serre class of torsion groups implies that the Hurewicz
homomorphism in dimension $n$ is the isomorphism mod the class of torsion
groups. Since the Hurewicz homomorphism is defined by the degree it
follows that one can even find the map of degree $1$ from $S^n$ onto $X$.
On the other hand the examples of non simply connected rational homology
spheres that I know (lens spaces, Poincare homological dodakedral sphere)
have the property that the mapping $f:S^n-> X$ of nonzero degree exists.
If the answer to the question below is in the positive, then the above
proof for simply connected rational homological spheres implies the
nonsimply connected case as well.
QUESTION 2. Is the universal cover of a rational homology sphere a
rational homology sphere?
Piotr Hajlasz
Department of Mathrmatics
University of Michigan