Let M be a manifold and $B_p(1,M)$ a ball of radius 1 and center p in M. Let $F:B_p(1,M)\to \mathbb{R}^n$ a map such that $F(\partial B_p(1,M))\subset B_0(1+\epsilon,\mathbb{R}^n)\ …

I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in wh …

Suppose we have a homology theory such that the associated Euler characteristic of one object divides that of another. What can we infer from this?

Hi everyone, Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it. I was wondering if someo …

The question is simple: What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)? I'm not sure what th …

Hi there, I want to ask about the 2-cycle of K3 surface. As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators. Is there any topological way to figure ou …

Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety. Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero? Background If $X$ is a Kaehle …

As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem: A weak homotopy equivalence induces isomorphisms of the corres …

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200): Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian cate …

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$: a simplex in $K$ consists of fi …

Background I am currently working on the homology of some modulispace and there exists a much simpler chaincomplex with the same homology. It is a quotient of a bisimplicial compl …

Let $Y$ be a contractible finite simplicial 2-complex. Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$). Is $X$ contractible? (Equivalently, is …

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow H^{k}(X …

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 sm …

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direc …