4
votes
1answer
113 views

0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented ...
1
vote
0answers
42 views

Natural isomorphism between locally finite homology and homology of one-point compactifcation of a forward tame ANR

Let $X,Y$ be a locally compact, separable, metric ANR that is forward tame, which means that for some closed subset $V\subseteq X$ such that $\overline{X\smallsetminus V}$ is compact a proper map ...
0
votes
0answers
178 views

Question on Steenrod realizability problem

René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact ...
15
votes
2answers
941 views

Simple curves on non-orientable surfaces.

Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...
19
votes
2answers
2k views

Does this approach for the Poincare conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincare's fundamental question." The author gave a talk on it today at my institution. ...
9
votes
6answers
2k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
7
votes
4answers
498 views

Realizing complexes with bases as cellular complexes

This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it. Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...
38
votes
4answers
2k views

Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...