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As in Chap. 5 , \((W,S)\) is a Coxeter system, X is a mirrored space over S and \(\mathcal {U}\) ( \(=\mathcal {U} (W,X)\) ) is the result of applying the basic construction to these data. The two main results of this chapter are a formulas for the homology of \(\mathcal {U}\) (Theorem 8.1.2) and for its cohomology with compact supports (Theorem 8.3.1). The two formulas are similar in appearance and in their proofs. (Of course, they give completely different answers whenever W is infinite.) We will give alternative proofs of these formulas in Chap. 15 .

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The Algebraic Topology of \(\mathcal {U}\) and of \(\Sigma \)

  • 章节

  • 发表日期：

    2025年

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  • Michael W. Davis

摘要

As in Chap. 5 , \((W,S)\) is a Coxeter system, X is a mirrored space over S and \(\mathcal {U}\) ( \(=\mathcal {U} (W,X)\) ) is the result of applying the basic construction to these data. The two main results of this chapter are a formulas for the homology of \(\mathcal {U}\) (Theorem 8.1.2) and for its cohomology with compact supports (Theorem 8.3.1). The two formulas are similar in appearance and in their proofs. (Of course, they give completely different answers whenever W is infinite.) We will give alternative proofs of these formulas in Chap. 15 .