Articles of classifying spaces

How to compare the K-theory and singular cohomology of a classifying space

In their foundational paper “Vector bundles and homogeneous spaces,” Atiyah and Hirzebruch show, among many other things, that for $G$ a compact, connected Lie group, the K-theory of the classifying space $BG$, taken as an inverse limit of the K-theory of a family of compact approximations of $B_n G$, is isomorphic to the completion of […]

Is the classifying space $B^nG$ the Eilenberg-MacLane space $K(G, n)$?

Question: How should we interpret and understand the classifying space $B^nG$? Is that Eilenberg-MacLane space $K(G,n)$? What one can learn about $BG$ follows the basic: A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ (i.e. a topological space for which all its homotopy groups are trivial) […]

Is there a classifying space for covering maps?

It is often said that a sheaf on a topological space $X$ is a “continuously-varying set” over $X$, but the usual definition does not reflect this because a sheaf is not a continuous map from $X$ to some “space of sets”. (Such a space must have a proper class of points!) However, I recently had […]

Proof that classifying spaces for discrete groups are the Eilenberg-MacLane spaces

A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ by a free action of $G$. The claim is that if $G$ is a discrete group then $EG/G$ is an Eilenberg-MacLane space, so that the fundamental group of $EG/G$ is $G$ and all higher homotopy groups are […]

Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?

I’m looking for a reference for the following result: If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ is trivial. The proof supposedly uses homotopy theory and classifying spaces. I’m not very familiar with either, so I don’t know where to […]

canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization of the nerve. [Martin Brandenburg.] The classifying space $BM$ of a monoid $M$ is (by definition) the classifying space of the corresponding category.[Martin […]

Group structure on Eilenberg-MacLane spaces

How do we put a group structure on $K(G,n)$ that makes it a topological group? I know that $\Omega K(G,n+1)=K(G,n)$ and since we have a product of loops this makes $K(G,n)$ into a H-space. But what about being a topological group?

Why is the cohomology of a $K(G,1)$ group cohomology?

Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular cohomology of $K(G,1)$, and it is a theorem that this is isomorphic to the group cohomologies $H^*(G, \mathbb{Z})$. According to one of […]