3
votes
1answer
278 views

Maps between classifying spaces

Let $G$ be a discrete group and let $BG \simeq K(G,1)$ be its classifying space. Let $H$ be a topological group with classifying space $BH$. In case $H$ is also discrete, it was pointed out in the ...
11
votes
1answer
432 views

The semidihedral group of order 16 and ko

Let $\mathcal{A}(1)$ denote the subalgebra of the $\mathrm{mod}\ 2$ Steenrod algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. The cohomology with $\mathbf{F}_2$ coefficients of the ...
20
votes
2answers
669 views

Is super-vector spaces a “universal central extension” of vector spaces?

Is there some sense in which the category $sVect$ of super-vector spaces is the "maximal non-trivial extension" of $Vect$ as a symmetric monoidal category? Is the $\mathbb Z/2$ that shows up in the ...
15
votes
3answers
2k views

Why is BG infinite dimensional for G finite ?

If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$ then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This can be found, for example, there: ...
6
votes
2answers
794 views

Transfer homomorphisms with coefficients

In group cohomology, for $H$ a finite-index subgroup of $G$ and $M$ a $G$-module, there is a transfer (or corestriction) map $Cor : H^* (H;M) \to H^*(G;M)$. In homotopy theory, there is a transfer ...
1
vote
2answers
418 views

Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...