arXiv 1709.074132017-09-21math.GT

双曲$3$ -Orbifolds的体积和同调

Volume and Homology for Hyperbolic $3$-Orbifolds, I

Peter B. Shalen

备注：: 99 pages

摘要：

设${\mathfrak M}$是一个闭的、可定向的、双曲的3-orbifold，其奇异集是一个环，且使得$π_1({\mathfrak M})$不含双曲三角群.我们证明了如果基础流形$|{\mathfrak M}|$是不可约的，并且$|{\mathfrak M}|$对${\mathfrak M}$的每个两层（orbifold）覆盖$\widetilde{\mathfrak M}$都是不可约的，如果${\rm vol} {\mathfrak M}\le1.72$，则$\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$ .此外，如果是${\rm vol} {\mathfrak M}\le1.22$，则是$\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 11$，如果是${\rm vol} {\mathfrak M}\le0.61$，则是$\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 7$。证明是一个应用程序的结果，将用于本文的续集，以获得定性相似的结果，而无需假设不可约的$|{\mathfrak M}|$和$|\widetilde{\mathfrak M}|$。

英文摘要：

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) covering $\widetilde{\mathfrak M}$ of ${\mathfrak M}$, and if ${\rm vol} {\mathfrak M}\le1.72$, then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$. Furthermore, if ${\rm vol} {\mathfrak M}\le1.22$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 11$, and if ${\rm vol} {\mathfrak M}\le0.61$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 7$. The proof is an application of results that will be used in the sequel to this paper to obtain qualitatively similar results without the assumption of irreducibility of $|{\mathfrak M}|$ and $|\widetilde{\mathfrak M}|$.

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