Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what not give you this? Is it a generalization of some nice finite-dimensional concept? Does it have a deep connection to (finite-dimensional) vector bundles?



Fredholm operators in $K$-theory?
A classical connection is the Atiyah-Jänich Theorem, see 


Klaus Jänich: Vektorraumbündel und der Raum der Fredholm-Operatoren. Math. Ann. 161 (1965) 129–142. 


Let ${\mathcal{F}}$ be the space of Fredholm operators with the operator norm, and $X$ any compact space. To a map $F\colon X\to {\mathcal{F}}$ one can associate the virtual vector bundle $(ker(F(x)))\_x-(coker(F(x)))\_x$ from Qiaochu Yuan's comment. (In general, there might be points where dimension of kernel and cokernel jump, but upon homotopy of $F$ one can assume that kernel and cokernel are indeed vector bundles.) 


Jänich proves that this yields a bijection $$\left[X,{\mathcal{F}}\right]\cong K(X),$$
where $\left[.,.\right]$ means homotopy classes.


