A parametrized index theorem for the algebraicK-theory Euler class
A parametrized index theorem for the algebraicK-theory Euler class
复制标题
代数K理论欧拉类的参数化指数定理
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
B. Williams
中科院分区：
文献类型：
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作者：
W. Dwyer;M. Weiss;B. Williams
Riemann-Roch定理是断言K理论中某些代数上定义的错误映射与拓扑上定义的错误映射一致或兼容的定理[BFM]。Bismut和Lott [BiLo]证明了光滑纤维束的Riemann-Roch定理，其中拓扑定义的错误映射是Becker-Gottlieb和Dold的同伦转移。我们推广并改进了他们的定理。在此过程中，我们证明了紧拓扑流形纤维束的一个族指数定理，该定理的相关指数方程涉及到代数K理论。我们的方法使我们能够对这样的束做出代数K理论的“普遍”选择。利用这一选择，我们分别得到了具有有限支配纤维的纤维之间紧致拓扑流形束和紧致光滑流形束的指数理论表征。
A Riemann–Roch theorem is a theorem which asserts that some algebraically defined wrong–way map in K –theory agrees or is compatible with a topologically defined one [BFM]. Bismut and Lott [BiLo] proved a Riemann–Roch theorem for smooth fiber bundles in which the topologically defined wrong–way map is the homotopy transfer of Becker–Gottlieb and Dold. We generalize and refine their theorem. In the process, we prove a family index theorem for fiber bundles with compact topological manifold fibers, a theorem in which the relevant index equation involves algebraic K –theory. Our methods enable us to make a “universal” choice of algebraic K –theory for such bundles. With this choice, we obtain index–theoretic characterizations of bundles of compact topological manifolds and bundles of compact smooth manifolds, respectively, among fibrations with finitely dominated fibers.