Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category …

Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my quest …

Given a connected finite graph G with degree at least 2 at each vertex, what are the conditions G needs to assume in order to attach 2-cells so that the CW- complex is a closed c …

Suppose we have a finite connected graph $G$, I want to add 2 -cells to $G$ so that the 2 cells have boundaries of length 4 (squares) and so that $G$ is the 1 skeleton of a surfac …

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 …

Let $M$ be a topological $n$-manifold, closed and connected (not necessarily oriented): When does $M$ not admit (up to homotopy-type) a CW-structure with a single $n$-cell? By cla …

Let $X$ be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a …

I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like …

From time to time, I pretend to be an algebraic topologist. But I'm not really hard-core and some of the deeper mysteries of the subject are still ... mysterious. One that came u …

In the appendix of Allen Hatcher's book "Algebraic Topology", a CW complex is defined to be a space iteratively constructed by attaching $n$-cells onto an $(n-1)$ skeleton. There …

Let $X$ be a finite CW complex and $x_0$ a point in $X$. My question is then just: Is $X-{x_0}$ still homotopy equivalent to a finite CW complex?

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or tha …

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and pro …

Let $X$ be a CW-complex with one 0-cell two 1-cells three 2-cells no cells in dimensions 3 or higher. Is it always true that $\pi_2(X)\ne 1$?

Let me preface this question by saying that I am not an algebraic topologist. Motivation. I was looking with a colleague at the homotopy type of a family of posets and we were abl …