Wiki. “拓扑 K-理论” [拓扑K-理论]
Wiki. “拓扑 K-理论” [拓扑K-理论]
拓扑 K-理论是由拓扑空间上的向量丛构造的一种广义上同调理论. K-理论这个名字最早由 Grothendieck 使用, 来自德文 Klasse.
粗略地说, K-理论试图用 Abel 上同调去表达向量丛的分类, (向量丛的分类本身是一种非 Abel 上同调, 即 $GL_n$ 系数的上同调).
It turns out that an important source of virtual vector bundles representing classes in $K$-theory are index bundles: Given a Riemannian spin manifold $B$, then there is a vector bundle $S\to B$ called the spin bundle of $B$, which carries a differential operator, called the Dirac 算子 $D$. The index of a Dirac operator is the formal difference of its kernel by its cokernel. Now given a continuous family $D_x$ of Dirac operators/Fredholm operators, parameterized by some topological space $X$, then these indices combine to a class in $K(X)$. It is via this construction that topological K-theory connects to spin geometry (see e.g. Karoubi K-theory) and index theory.
As the terminology indicates, both spin geometry and Dirac operator originate in physics. Accordingly, K-theory plays a central role in various areas of mathematical physics, for instance in the theory of geometric quantization (“$\mathrm{spin}^\mathrm{c}$ quantization”) in the theory of D-branes (where it models D-brane charge and RR-fields) and in the theory of Kaluza-Klein compactification via spectral triples (see below).
All these geometric constructions have an operator algebraic incarnation: by the topological Serre-Swan theorem then vector bundles of finite rank are equivalently modules over the $C^*$-algebra of continuous functions on the base space. Using this relation one may express K-theory classes entirely operator algebraically, this is called operator K-theory. Now Dirac operators are generalized to Fredholm operators.
There are more $C^*$-algebras than arising as algebras of functions of topological space, namely non-commutative C-algebras. One may think of these as defining non-commutative geometry, but the definition of operator K-theory immediately generalizes to this situation (see also at KK-theory).
While the $C^*$-algebra of a Riemannian spin manifold remembers only the underlying topological space, one may algebraically encode the smooth structure and Riemannian structure by passing from Fredholm modules to “spectral triples”. This may for instance be used to algebraically encode the spin physics underlying the standard model of particle physics and operator K-theory plays a crucial role in this.