All Questions
5 questions
7
votes
0
answers
333
views
Positive instances of the Eilenberg-Ganea conjecture with families
The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
11
votes
1
answer
167
views
A group of type F that is an extension of type F-by-type F
Let us first recall that a group of type $F$ is a group admitting a compact classifying space.
Let $K$ and $Q$ be groups of type $F$. Consider the family $\mathcal{G}(K, Q)$ consisting of groups $G$ ...
4
votes
0
answers
240
views
The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group
Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
7
votes
2
answers
537
views
Residually finite + torsion free + finite index = finite complex?
Suppose $G$ is a residually-finite group and $H < G$ a torsion-free subgroup of finite index.
What characterizes such $G$ such that $BH$ is homotopic to a finite complex?
I believe Serre showed ...
28
votes
4
answers
4k
views
Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...