叶状流形、代数K理论和一个次不变量
Foliated manifolds, algebraic K-theory, and a secondary invariant
摘要：
引入了具有稳定框架和部分平坦向量丛的叶状流形的$\mathbb{C}/\mathbb{Z}$-值不变量。这个不变量可以用微分$K$-理论中的积分表示，或者用狄拉克算子和局部修正项的$η$-不变量表示。最初，在$\mathbb{C}/\mathbb{Z}$中构造元素涉及其他选择。但是如果叶理的余维足够小，那么这个元素与这些选择无关，因此是上面列出的数据的不变量。我们表明，不变量包括各种经典的不变量，如亚当斯的$e$不变，$ρ$不变的扭曲狄拉克运营商，或Godbillon-Vey不变量从叶理理论。利用微分上同调理论的方法，我们构建了一个调节映射的代数$K$-理论的光滑函数的流形，其连接$K$-理论与$\mathbb{C}/\mathbb{Z}$系数。我们的主要结果是一个公式的不变量在这个调节器和集成代数和拓扑$K$理论。
英文摘要：
We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $η$-invariants of Dirac operators and local correction terms. Initially, the construction of the element in $\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' $e$-invariant, the $ρ$-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic $K$-theory of smooth functions on a manifold to its connective $K$-theory with $\mathbb{C}/\mathbb{Z}$ coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological $K$-theory.
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