All Questions

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...

When singular homology is defined using cubes instead of simplices it is important to factor out the degenerate cubes in the course of building the singular chain complex. If you omit this step it is ...

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...

Suppose that we have a regular CW-complex $X$. I want to define the incidence matrix of $k$-skeleton of $X$ with respect to the $k-1$ skeleton and I wonder what might go wrong in this case. If it's ...

In the slides Characteristic Classes of Surface Bundles and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology? Could the coefficient be an ...

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 ...