By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that $\pi_1(Y)$ has a minimal presentation with two generators? How about three generators? Here ar...

As indicated in the comments, the greatest class of rank 2 3-manifolds (including rational homology spheres) are the genus 2 manifolds, which as indicated by Ruberman are double branched covers over links (coming from the hyperelliptic involution of the genus 2 surface which extends over both han...

I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has thrown that theory to the wind. There are quite a few spaces I need to introduce, so please forgive t...

Here are some more possible approaches to show that the cohomology of $\Omega U(n)$ is torsion-free, as a complement to Neil Strickland's answer: In general, a useful tool to compute the cohomology of loop spaces is the Eilenberg-Moore spectral sequence, of which you can find an overview in McCle...

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S \ldots\times_S X}_{n+1 \text{ components}} $$ with face maps just deleting certain components and ...