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Results tagged with classifying-spaces
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user 318
The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.
3
votes
Accepted
Group completion of a monoid (braid groups)
This is explained in my paper "Group-Completion", local coefficient systems, and perfection, Q. J. Math. (Quillen Memorial Issue) 64 (3) (2013) 795-803. A sufficient condition is for the monoid to be …
- 18.2k
2
votes
fibrations of classifying spaces - Leray Hirsch Theorem converse
If you work with coefficients in a field $\mathbb{F}$, assume that $H^*(BH;\mathbb{F})$ is a free $H^*(BG;\mathbb{F})$-module, and add the assumption that the Serre spectral sequence has a product str …
- 18.2k
7
votes
Accepted
The Image of the Mod 2 Homology of BSp in the Homology of BSO
We have $H^*(BO, \mathbb{F}_2) = \mathbb{F}_2[w_1, w_2, \ldots]$, where the $w_i$ are the Stiefel--Whitney classes. If $f: \mathbb{RP}^\infty \to BO$ classifies the reduced universal real line bundle, …
- 18.2k
6
votes
Accepted
classifying space of a strange category
I think it is equivalent to $BG$. Picking an $x_0 \in X := hom(p_1, p_2)$ we get a map
$$X \to G$$
sending x_0 * g to g. Using this on morphism sets we get a map $F: \mathcal{C} \to G$ which you eas …
- 18.2k