A vanishing theorem for characteristic classes of odd-dimensional manifold bundles
A vanishing theorem for characteristic classes of odd-dimensional manifold bundles
复制标题
奇维流形丛特征类的消失定理
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发表时间：
2009
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通讯作者：
Johannes Ebert
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作者：
Johannes Ebert
我们展示了如何Atiyah-Singer家庭指数定理，通常和自伴椭圆算子适合自然的框架内的Madsen-Tillmann-韦斯谱。我们的主要定理涉及奇维流形的丛。利用完全泛函分析方法，我们证明了对任意光滑的真定向纤维丛E 对于奇维纤维，奇签名算子在K^1（X）$中的族指数$ind（B）是平凡的。Atiyah-Singer定理使我们可以得出一个拓扑结论：广义Madsen-Tillmann-韦斯映射$alpha：B Diff^+（M^{2 m-1}） o loopinf MTSO（2 m-1）$在有理上同调中杀死了Hirzebruch $cL$-类。如果$m=2$，这意味着$alpha$诱导有理上同调的零映射。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$alpha：B Diff^+（M）的平凡性 o在许多情况下，MTSO（3）$在mod $p$上同调。我们显示了一个适当的版本，这些结果与边界的流形丛。
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 $E o X$ 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 $alpha: B Diff^+ (M^{2m-1}) o loopinf MTSO(2m-1)$ kills the Hirzebruch $cL$-class in rational cohomology. If $m=2$, this means that $alpha$ 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 $alpha: B Diff^+ (M) o MTSO (3)$ in mod $p$ cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.