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I've found that discrete Morse theory is very helpful in this context. Here's a link to a nice article by Forman. If you can define a good discrete vector field, it's often possible to drastically red …

It is my understanding that Dennis Johnson defined a `relative weight filtration,' of the mapping class group of an oriented surface. My question is what is this filtration, and how does it relate to …

The moduli space of graphs $MG_n$ is the quotient of Culler-Vogtmann's outer space $X_n$ by the action of $\mathrm{Out}(F_n)$. It can be thought of as the space of metric graphs homotopy equivalent to …

The word problem for the fundamental group of a closed surface is solvable, using Dehn's algorithm. Since any finitely presented group appears as the fundamental group of some closed $4$-manifold, and …

This is not exactly what you asked, but it's certainly not the case that every CW complex has a discrete vector field where the Morse complex has trivial differential. In particular this would imply t …

Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H_{\alpha}(X)$ where $\alpha$ is an ord …

This is not a trivial problem. A favorite example of mine is the case of a knot complement, $S^3\setminus K$. (It is known that these are Eilenberg-Maclane spaces.) If you pick $A$ to be the commutato …

I like to think of Alexander duality in terms of linking numbers of submanifolds (or, in general, k cycles). This is one way to define the pairing you are looking for. In general, consider a $k$-cycle …

Outer automorphisms of free groups have a rational classifying space given by metric graphs, called "outer space," first described in a paper by Culler and Vogtmann. The rational cohomology of Out$(F …

Take the $n$-skeleton. It has trivial rational homology except possibly in degree $n$. Now add enough $n+1$-cells from the $n+1$-skeleton to kill this top homology. You won't have created any $n+1$-di …

Some industrial conveyor belts are hooked up like a Möbius strip, so I've heard, in order to wear evenly on "both" sides. Of course nonorientabilty has got to show up in more fundamental physical wa …