基于对应的双变K理论
Bivariant K-theory via correspondences
摘要：
我们使用对应定义了一个纯拓扑等变双变K理论的空间与适当的广群作用。我们关于对应的概念与康纳斯和斯堪的纳维亚人的概念略有不同。我们将光滑K-定向映射替换为一类K-定向法线映射，它们是带有某种因子分解的映射。我们的构造没有使用等变K理论的任何特殊性质。为了突出这一点，我们构造任意等变乘法上同调理论的双变扩展。我们在几何双变K理论中给出了某些对偶同构的充要条件，并在某些情况下验证了这些条件，包括具有李群的光滑余紧作用的光滑流形.其中一个对偶同构将双变K理论简化为具有支持条件的K理论。由于类似的对偶同构存在于卡斯帕罗夫理论中，如果存在这样的对偶同构，则两个双变K-理论都同意。
英文摘要：
We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented maps by a class of K-oriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant K-theories agree if there is such a duality isomorphism.
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