奇维流形丛特征类的一个消灭定理
A vanishing theorem for characteristic classes of odd-dimensional manifold bundles
摘要：
我们展示了如何Atiyah-Singer家庭指数定理，通常和自伴椭圆算子自然适合的框架的Madsen-Tillmann-Weiss谱。我们的主要定理涉及奇维流形的丛。利用完全泛函分析的方法，证明了对于任意具有奇维纤维的光滑真定向纤维丛$E \to X$，奇签名算子的族指数$\ind (B) \in K^1 (X)$是平凡的. Atiyah-Singer定理允许我们得出一个拓扑结论：广义Madsen-Tillmann-Weiss映射$α: B \Diff^+ (M^{2m-1}) \to \loopinf \MTSO(2m-1)$杀死了Hirzebruch $\cL$ -类的有理上同调。如果是$m=2$，这意味着$α$在有理上同调中诱导零映射。特别是，马德森-韦斯定理的三维模拟是错误的。对于3-流形$M$，我们还证明了在许多情况下$α: B \Diff^+ (M) \to \MTSO (3)$在模$p$上同调中的平凡性.我们显示了一个适当的版本，这些结果与边界的流形丛。
英文摘要：
We show how the Atiyah-Singer family index theorem for both, usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds. Using completely functional-analytic methods, we show that for any smooth proper oriented fibre bundle $E \to X$ with odd-dimensional fibres, the family index $\ind (B) \in K^1 (X)$ of the odd signature operator is trivial. The Atiyah-Singer theorem allows us to draw a topological conclusion: the generalized Madsen-Tillmann-Weiss map $α: B \Diff^+ (M^{2m-1}) \to \loopinf \MTSO(2m-1)$ kills the Hirzebruch $\cL$-class in rational cohomology. If $m=2$, this means that $α$ induces the zero map in rational cohomology. In particular, the three-dimensional analogue of the Madsen-Weiss theorem is wrong. For 3-manifolds $M$, we also prove the triviality of $α: B \Diff^+ (M) \to \MTSO (3)$ in mod $p$ cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.
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