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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 $\...
Mark Grant's user avatar
  • 35.9k
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 $G$ ...
Janusz Przewocki's user avatar
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 ...
Tyrone's user avatar
  • 5,596
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 ...
Romeo's user avatar
  • 2,734
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 ...
Aaron Bergman's user avatar