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Theorem: ''Let $EG \to BG$ be a principal bundle such that for each finite CW $X$, the transformation $\eta_X$ from $[X;BG]$ to the set of isomorphism classes of principal bundles is a bijection. Then …

What you should take as a model is the homotopy quotient $EG \times_G BG$. From the homotopy sequence of the fibration $EG \times_G BG \to BG$ (projection on first factor), you get that $EG \times_G …

Recall the Chern-Weil homomorphism $Sym^{\ast} \mathfrak{g}^{\vee} \to H^{\ast}(BG; \mathbb{R})$ for each Lie group $G$. If $G$ is compact, it is an isomorphism. See Dupont, Curvature and Characterist …

The argument in the quoted paper is a bit too sketchy. An actual proof will have two parts: Let $f:X \to Y$ be map. Assume that for each $y \in Y$, $\tilde{H}^{\ast} (f^{-1}(y);\mathbb{Q})=0$ (this …

This is a question about general vector bundles, not tangent/normal bundles. So let $X$ be a finite complex, and $V,W \to X$ be two real vector bundles such that $V \oplus W$ is trivialized and $n$-di …

If $P \to X$ is a $G$-principal bundle, then the space $map_G (P;EG)$ is contractible. Let $map_P(X;BG)$ be the space of all maps $f$ with $f^{\ast} EG \cong P$. There is an obvious map $map_G (P;EG) …

Let $G$ be the group of all invertible operators on a Hilbert space $H$ that are of the form $1+K$, $K$ compact. Then the space of all invertible operators $GL(H)$ is a model for $EG$ and $BG$ is the …

I can think of several versions, besides those that have been mentioned in the earlier answers: You can in fact construct $B GL_n (\mathbb{C})$ as a manifold, but of course an infinite-dimensional o …