Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \ …

Let $G$ be a topological group. In some cases, e.g. when $G$ is discrete or when the spaces $G^n$ are locally contractible and the coefficients are discrete, the cohomology of the …

Suppose I have a topological category $\mathcal{C}$ of the following form: The object space consists of just two points $p_1, p_2$. The endomorphism space of $p_1$ contains just th …

Is there an explicit description of the Pontryagin product on the homology of $CP^{\infty}$? Also, what is the homology of the classifying spaces $BU(n)$?

Let $G$ be a nonabelian group, with classifying space $BG$. Motivation: We can compute its homology, $H_\ast(BG)=H_\ast(G)$. It would be nice to see some equivariant computations, …

For a (discrete) monoid $M$, the classifying space $BM$ is the geometric realization of the nerve of the one object category whose hom-set is $M$. (This definition gives the usual …

Let $X$ be a reasonable topological space (let's say it has the homotopy type of a CW complex) and let $G$ be a topological group acting on that space. Let $E_G \rightarrow B_G$ be …

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics …

Suppose we have a group $G$ then one can construct $BG$ and one of the essential part of the construction is the co-unit map. Now suppose we have a ring spectrum $R$, then having a …

Let's assume we start with Chern--Weil theory in the following form: Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfra …

If the following is true, it is probably well-known to the experts. Nevertheless, I could not find a reference for it. Suppose $P$ and $Q$ are $A_{\infty}$-operads in topological …

I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the w …

Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's article in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ bloc …

[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg-Maclane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (or if not a manifold …

Let $G$ be a topological group. The geometric bar construction $BG = B_{\bullet}(pt, G, pt)$ together with $EG = B_{\bullet}(pt,G,G)$ and the map $EG \to BG$ yields the universal p …