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arXiv 1303.2986math.GT
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相对群同调中双曲三维流形的不变量

Invariants of hyperbolic 3-manifolds in relative group homology

José Antonio Arciniega-Nevárez, José Luis Cisneros-Molina

备注：
47 pages, 1 figure, PDFLaTeX. The proof of Proposition 3.20 was not correct (and probably also the statement). Results on Sections 2 and 3 were published in a separate article and summarized in Section 2 of version 2. Some parts were rewritten
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摘要：

设$M$是有限体积的完全定向双曲$3$ -流形.利用子群族的分类空间，我们在Adamson相对同调群$H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$中构造了一类$β_P(M)$，其中$\bar{P}$是将$\infty$固定在黎曼球面上的抛物变换的子群.我们还证明了类$F(M)$的高须相对同调群$H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$所构造的Zickert，这是不明确的，并依赖于一个选择的装饰horospheres，都映射到$β_P(M)$通过一个典型的比较同态$H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\to H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$。为此，我们简化了类$F(M)$的建设使用一个更简单的复杂的计算$H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$，得到一个简单的单纯公式为$F(M)$，这反过来又给出了一个更简单，更有效的公式来计算体积和Chern-Simons不变量比Zickert给出的。这些构造可以推广到任何边界抛物$PSL_2(\mathbb{C})$表示。

英文摘要：

Let $M$ be a complete oriented hyperbolic $3$--manifold of finite volume. Using classifying spaces for families of subgroups we construct a class $β_P(M)$ in the Adamson relative homology group $H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$, where $\bar{P}$ is the subgroup of parabolic transformations which fix $\infty$ in the Riemann sphere. We also prove that the classes $F(M)$ in the Takasu relative homology groups $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$ constructed by Zickert, which are not well-defined and depend of a choice of decorations by horospheres, are all mapped to $β_P(M)$ via a canonical comparison homomorphism $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\to H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$. To do this, we simplify the construction of the classes $F(M)$ using a simpler complex which computes $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$, getting a simple simplicial formula for $F(M)$, which in turn gives a simpler and more efficient formula to compute the volume and Chern--Simons invariant than the one given by Zickert. The constructions can be extended for any boundary-parabolic $PSL_2(\mathbb{C})$-representation.

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