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arXiv · 1903.03749

On the moduli spaces of commuting elements in the projective unitary groups

Alejandro Adem·Man Chuen Cheng
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Abstract

We provide descriptions for the moduli spaces $\text{Rep}(\Gamma, PU(m))$, where $\Gamma$ is any finitely generated abelian group and $PU(m)$ is the group of $m\times m$ projective unitary matrices. As an application we show that for any connected CW-complex $X$ with $\pi_1(X)\cong \mathbf{Z}^n$, the natural map $\pi_0(\text{Rep}(\pi_1(X), PU(m)))\to [X, BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over $X$.

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Alejandro Adem, Man Chuen Cheng. 2019-03-09. On the moduli spaces of commuting elements in the projective unitary groups. https://doi.org/10.1063/1.5097922

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