Torsion homology and regulators of isospectral manifolds
Torsion homology and regulators of isospectral manifolds
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DOI：
10.1112/jtopol/jtw023
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发表时间：
2016-01
影响因子：
1.1
通讯作者：
Alex Bartel;Aurel Page
中科院分区：
文献类型：
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作者：
Alex Bartel;Aurel Page
给定一个有限群G，一个封闭黎曼流形的G‐覆盖，以及一个所谓的G‐关系，Sunada的构造产生了一对强等谱流形M1和M2。这些流形具有相同的维数和相同的体积，它们的有理同构群是同构的。本文研究了它们之间的积分同调关系。cheeger - m<s:1> ller定理表明，对于M1，有一定的扭转同调阶和调节阶的乘积与对于M2，有一定的乘积是一致的。通过对商Regi(M1)/Regi（M2）表示的理论解释，我们证明了M1和M2的积分同调中的扭转与覆盖流形的积分同调的G‐模结构之间的联系。进一步证明了对于所有素数p∤#G， M1同调中的p∞-扭转与M2同构。对于p < 71，我们给出了p -扭转同调不同的强等谱双曲3 -流形对的例子，并推测对所有素数p都存在这样的例子。
Given a finite group G , a G ‐covering of closed Riemannian manifolds, and a so‐called G ‐relation, a construction of Sunada produces a pair of manifolds M1 and M2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. Here, we investigate the relationship between their integral homology. The Cheeger–Müller Theorem implies that a certain product of orders of torsion homology and of regulators for M1 agrees with that for M2 . We exhibit a connection between the torsion in the integral homology of M1 and M2 on the one hand, and the G ‐module structure of integral homology of the covering manifold on the other, by interpreting the quotients Regi(M1)/Regi(M2) representation theoretically. Further, we prove that the p∞ ‐torsion in the homology of M1 is isomorphic to that of M2 for all primes p∤#G . For p⩽71 , we give examples of pairs of strongly isospectral hyperbolic 3‐manifolds for which the p ‐torsion homology differs, and we conjecture such examples to exist for all primes p .