nLab
topological K-theory

Idea

What is called topological K-theory is a collection of generalized (Eilenberg-Steenrod) cohomology theories whose cocycles in degree 0 on a space $X$ can be represented by pairs of vector bundles, real or complex ones, on $X$ modulo a certain equivalence relation.

Notice that “ordinary cohomology” is the generalized (Eilenberg-Steenrod) cohomology that is represented by the Eilenberg-MacLane spectrum which, as a stably abelian infinity-groupoid is just the additive group

$$\mathbb{Z}$$\mathbb{Z}

of integers.

To a large extent K-theory is the cohomology theory obtained by categorifying this once:

$$\mathbb{Z}\mapsto \mathrm{something}\mathrm{like}\mathrm{Vect}.$$\mathbb{Z} \;\; \mapsto something like \mathbf{Vect} \,.

Motivational example: “nonabelian K-cohomology”

To see how this works, first consider the task of generalizing the “nonabelian cohomology” or cohomotopy theory given by the coefficient object $\mathbb{N}$, the additive semi-group of $\mathbb{N}$ of natural numbers.

This does have arbitrarily high deloopings in the context of omega-categories, but not in the context of infinity-groupoids. So for the purposes of cohomology $\mathbb{N}$ is just the monoidal 0-groupoid $\mathbb{N}$ which as a coefficient object induces a very boring cohomology theory: the $\mathbb{N}$-cohomology of anything connected is just the monoidal set $\mathbb{N}$ itself. While we cannot deloop it, we can categorify it and do obtain an interesting nonabelian cohomology theory:

Namely the category $\mathrm{Core}(\mathrm{Vect})$ of finite dimensional vector spaces with invertible linear maps between them would serve as a categorification of $\mathbb{N}$: isomorphism classes of finite dimensional vector spaces $V$ are given by their dimension $d(V)\in \mathbb{N}$, and direct sum of vector spaces corresponds to addition of these numbers.

If we want to use the category $\mathrm{Core}(\mathrm{Vect})$ as the coefficient for a cohomology theory, we should for greater applicability equip it with its natural topological or smooth structure, so that it makes sense to ask what the $\mathrm{Vect}$-cohomology of a topological space or a smooth space would be. The canonical way to do this is to regard $\mathrm{Vect}$ as a generalized smooth space called a smooth infinity-stack and consider it as the assignment

$$\mathrm{Vect}:\mathrm{Diff}\to \mathrm{\infty }\mathrm{Grpd}$$\mathbf{Vect} : Diff \to \infty Grpd

$$U\mapsto \mathrm{Core}(\mathrm{VectBund}(U))$$U \mapsto Core(VectBund(U))

that sends each smooth test space $U$ (a smooth manifold, say), to groupoid of smooth vector bundles over $U$ with bundle isomorphisms betweem them. We regard here a vector bundle $V\to U$ as a smooth $U$-parametrized family of vector spaces (the fibers over each point) and thus as a smooth probe or plot of the category $\mathrm{Core}(\mathrm{Vect})$.

The nonabelian cohomology theory with coefficients in $\mathrm{Vect}$ has no cohomology groups, but at least cohomology monoids

$$H(X,\mathrm{Vect}):={\pi }_0{H}_{\mathrm{diff}}(X,\mathrm{Vect}).$$H(X,\mathbf{Vect}) := \pi_0 \mathbf{H}_{diff}(X, \mathbf{Vect}) \,.

It is equivalent to the nonabelian cohomology with coefficients the delooping $BU$ of the stable unitary group $U:={\mathrm{colim}}_{n}U(n)$.

K-theory as a groupoidification of $\mathrm{Vect}$

The integers $\mathbb{Z}$ are obtained from the natural numbers $\mathbb{N}$ by including “formal inverses” to all elements under the additive operation. Another way to think of this is that the delooped groupoid $B\mathbb{Z}$ is obtained from $B\mathbb{N}$ by groupoidification (under the nerve operation this is fibrant replacement in the model structure on simplicial sets).

The idea of K-cohomology is essentially to apply this groupoidification process to not just to $\mathbb{N}$, but to its categorification $\mathrm{Vect}$.

Just as an integer $k=n-m\in \mathbb{Z}$ may be regarded as an equivalence class of natural numbers $(n,m)\in \mathbb{N}\times \mathbb{N}$ under the relation

$$[(n,m)]=[(n+r,m+r)]\forall r\in \mathbb{N}$$[(n,m)] = [(n+r, m+r)] \;\; \forall r \in \mathbb{N}

one can similarly look at equivalence classes of pairs $(V,W)\in \mathrm{Vect}(U)\times \mathrm{Vect}(U)$ of vector bundles.

This perspective on K-theory was originally realized by Atiyah and Hirzebruch. The resulting cohomology theory is usually called topological K-theory.

As one of several variations, it is useful to regard a pair of vector bundles as a single ${\mathbb{Z}}_2$-graded vector bundle.

One version of ${\mathbb{Z}}_2$-graded vector bundles, which lead to a description of twisted $K$-theory are vectorial bundles.

Spectrum

Being a generalized (Eilenberg-Steenrod) cohomology theory, toplogical K-theory is represented by a spectrum: the K-theory spectrum.

The degree-0 part of this spectrum, i.e. the classifying space for degree 0 topological $K$-theory is modeled in particular by the space $\mathrm{Fred}$ of Fredholm operators.

References

An introductory reference is

• Allen Hatcher, Vector bundles and K-theory (web)

More advanced material is in …

Discussion

The above idea section was originally at K-theory where it triggered the following discussion

Zoran What do you mean by originally ?? K-theory was invented by Grothendieck in algebraic geometry and over there it does NOT correspond to an Eilenberg-Steenrod setup. It seems this entry belongs much more to title “topological K-theory” (as opposed to algebraic). Second even for topological case there are real and complex K-theory etc. so not “a cohomology”. One of the nlab entries should also put emphasis on the K-theory of OPERATOR algebras (and also to the subject of K-homology, which is not known in geometric terms but in terms of operators), what corresponds to a rather central part of K-theory today, including most of the work in nc geometry.

Urs Schreiber: right, I have briefly modified the sentence now and then moved the entire paragraph here to “topological K-theory”. Eventually we need a more exhaustive and more comprehensive discussion here and need something general at K-theory.

I made a blog comment about that here.