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Section 9.1 deals with \(\pi _1(\mathcal {U})\) . The universal cover of \(\mathcal {U}\) can be constructed as a space of the form \(\widetilde {\mathcal {U}}=\mathcal {U}(\widetilde {W},\widetilde {X})\) , where \(\widetilde {X}\) is the universal cover of the fundamental chamber X and the Coxeter group \(\widetilde {W}\) has a fundamental generator for each component of the inverse image in \(\widetilde {X}\) of each mirror of X. This gives a short exact sequence computing \(\pi _1(\mathcal {U})\) in terms of \(\pi _1(X)\) and \(\widetilde {W}\) (Theorem 9.1.5).

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The Fundamental Group and the Fundamental Group at Infinity

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  • 发表日期：

    2025年

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  • Michael W. Davis

摘要

Section 9.1 deals with \(\pi _1(\mathcal {U})\) . The universal cover of \(\mathcal {U}\) can be constructed as a space of the form \(\widetilde {\mathcal {U}}=\mathcal {U}(\widetilde {W},\widetilde {X})\) , where \(\widetilde {X}\) is the universal cover of the fundamental chamber X and the Coxeter group \(\widetilde {W}\) has a fundamental generator for each component of the inverse image in \(\widetilde {X}\) of each mirror of X. This gives a short exact sequence computing \(\pi _1(\mathcal {U})\) in terms of \(\pi _1(X)\) and \(\widetilde {W}\) (Theorem 9.1.5).