# Tag: algebraic-topology

## group theory – Does a four-variable analog of the Hall-Witt identity exist?

The Question : 110 people think this question is useful Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: $$[x,y^{-1},z]^y[y,z^{-1},x]^z[z,x^{-1},y]^x=1,$$ which holds for any three elements of every group. $\ldots$ One can think of the Hall-Witt formula as a

## geometry – How to distinguish between walking on a sphere and walking on a torus?

The Question : 151 people think this question is useful Imagine that you’re a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can’t see the difference from this point of view. If you are interested, this question arose while I was

## general topology – Is there a homology theory that counts connected components of a space?

The Question : 161 people think this question is useful It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the corresponding statement would be that $\check{H}_0(X)$ is generated by the quasicomponents of

## general topology – Why can you turn clothing right-side-out?

The Question : 577 people think this question is useful My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a “trick” where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids