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Yes, this is possible. The construction below is fairly standard and I'm not sure where I learned it. The aim is to show that for each compact Lie group $G$, there is a closed $n$-connected manifold $ …

This is a substantially more interesting question than the votes indicate; while the initial phrasing is more naive than the real situation permits, a slight generalization is indeed true. The answer …

Theorem. Let $G$ be a group. There exists a finite 3-complex $X_G$ with $\pi_1 X_G = G$ and $H_i X_G = 0$ for $i > 1$ if, and only if, $G$ is finitely presentable and has second group homology $H_2(G) …

Let $X$ be any paracompact space. Then Hilbert vector bundles over $X$ are classified by homotopy classes of maps $[X, BU(\mathcal H)]$. But when $\mathcal H$ is infinite-dimensional, the group $U(\ma …

Define the Wu manifold $W(n) = SU(n)/SO(n)$, the inclusion $SO \to SU$ given by thinking of $\Bbb C^n = \Bbb R^n \otimes \Bbb C$ (that is, including real matrices into complex matrices). Note that $W( …