HEAT KERNELS ON COVERING SPACES AND TOPOLOGICAL INVARIANTS
HEAT KERNELS ON COVERING SPACES AND TOPOLOGICAL INVARIANTS
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DOI：
10.4310/jdg/1214448084
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发表时间：
1992
影响因子：
2.5
通讯作者：
J. Lott
中科院分区：
文献类型：
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作者：
J. Lott
众所周知，作用在闭定向流形M上的微分形式上的热流与M的拓扑之间存在关系。根据霍奇理论，可以从热流恢复M的贝蒂数。Ray和Singer [42]定义了M上无圈平坦丛的一个光滑不变量&解析挠率，并证明了它等于经典的Reidemeister挠率。Cheeger [7]和Muller [40]分别证明了这个猜想是正确的。解析挠率仅在奇维流形上是非零的，并且在某些方面表现为欧拉特征线的奇维对应物[20]。如果M不是单连通的，则存在Betti数的覆盖空间类似物。利用万有覆盖M上的热流，可以定义M [1]的L2-Betti数，方法是不取通常意义上的热核的迹，而是取某个II型von Neumann代数的元素。更具体地说，这相当于在M中的基本域上积分热核的局部迹。在第11节中，我们总结了这个理论。我们考虑解析挠率的覆盖空间模拟。这个iΛ解析挠率与iΛ上同调的关系与普通解析挠率与de Rham上同调的关系相同。在§111中，我们在一个技术性假设下定义了iΛ解析挠率<9^（M），并证明了它的基本性质。证明了&^（M）是L2-Betti数为零的流形的光滑不变量.这个证明类似于普通解析挠率的类似陈述，但需要一些注意，因为热核可能在很长时间内缓慢衰减。为了知道是否有有趣的例子&^（M），我们在M允许双曲度量的情况下计算它。很明显，它与双曲度规的体积成比例，
It is well known that there are relationships between the heat flow, acting on differential forms on a closed oriented manifold M, and the topology of M. From Hodge theory, one can recover the Betti numbers of M from the heat flow. Furthermore, Ray and Singer [42] defined an analytic torsion, a smooth invariant of acyclic flat bundles on M, and conjectured that it equals the classical Reidemeister torsion. This conjecture was proved to be true independently by Cheeger [7] and Muller [40]. The analytic torsion is nonzero only on odd-dimensional manifolds, and behaves in some ways as an odd-dimensional counterpart of the Euler characteristic [20]. If M is not simply-connected, then there is a covering space analog of the Betti numbers. Using the heat flow on the universal cover M, one can define the L2-Betti numbers of M [1] by taking the trace of the heat kernel not in the ordinary sense, but as an element of a certain type II von Neumann algebra. More concretely, this amounts to integrating the local trace of the heat kernel over a fundamental domain in M. In §11, we summarize this theory. We consider the covering space analog of the analytic torsion. This iΛanalytic torsion has the same relation to the iΛcohomology as the ordinary analytic torsion bears to de Rham cohomology. In §111 we define the iΛanalytic torsion <9^(M), under a technical assumption which we discuss later, and prove its basic properties. We show that &^(M) is a smooth invariant of manifolds whose L2-Betti numbers vanish. The proof is similar to that of the analogous statement for the ordinary analytic torsion, but requires some care because of the possible slow decay of the heat kernels for large time. In order to know if there are interesting examples of &^(M), we compute it in the case where M admits a hyperbolic metric. It is clear that is proportionate to the volume of the hyperbolic metric, which is