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.

Definition

Let $X$ be a compact Hausdorff topological space. Write $k$ for either the field of real numbers $ℝ$ or of complex numbers $C$ . By a vector space we here mean a vector space over $k$ of finite dimension. By a vector bundle we mean a topological $k$-vector bundle. We write $I^n\to X$ for the trivial vector bundle $I^n=k^n\times X$ over $X$ of rank $n\in \mathbb{N}$.

Lemma

For every vector bundle $E\to X$ (with $X$ compact Hausdorff) there exists a vector bundle $E\prime \to X$ such that

$$E\oplus E\prime \simeq I^{\mathrm{rank}E+\mathrm{rank}E\prime }.$$E \oplus E' \simeq I^{rank E + rank E'} \,.

Proof

One invokes a partition of unity relative to an open cover on which $E$ trivializes, constructs $E\prime$ locally and glues.

For details see for instance (Hatcher, prop. 1.4) or (Friedlander, prop. 3.1).

Definition

Define an equivalence relation on the set of finite-rank vector bundles $E\to X$ over $X$ by declaring that $E_1\sim E_2$ if there exists $k,l\in \mathbb{N}$ such that there is an isomorphism of vector bundles between the (fiberwise) direct sum of $E_1$ with $I^k$ and of $E_2$ with $I^l$

$$(E_1\sim E_2):\iff \exists (E_1\oplus I^k\simeq E_2\oplus I^l).$$(E_1 \sim E_2) :\Leftrightarrow \exists (E_1 \oplus I^k \simeq E_2 \oplus I^l) \,.

Write

$$\tilde{K}(X):=\mathrm{Vect}(X{)}_{\sim }$$\tilde K(X) := Vect(X)_\sim

for the quotient set of equivalence classes.

Proposition

With $X$ compact Hausdorff as in the assumption, we have that fiberwise direct sum of vector bundles equips $\tilde{K}(X)$ with the structure of an abelian group. Together with the fiberwise tensor product of vector bundles this yields a ring.

Therefore $K(X)$ is called the topological K-theory ring of $X$ or just the K-theory group or even just the K-theory of $X$, for short.

Proof

The non-trivial part of the statement is that in $K(X)$ there is an inverse to the operation of direct sum of vector bundles. Because in $\mathrm{Vect}(X)$ direct sum acts by addition of the ranks of vector bundles, it clearly has no inverse in $\mathrm{Vect}(X)$.

On the other hand, clearly the K-class $[I^n]$ of any trivial bundle $I^n$ is the neutral element in $K(X)$

$$[I^n]=0$$[I^n] = 0

for all $n\in \mathbb{N}$, because by definition $I^n\sim I^0$. Therefore an inverse of a class $[E_1]$ is given by a vector bundle $E_2$ with the property that the direct sum

$$E_1\oplus E_2\simeq I^n$$E_1 \oplus E_2 \simeq I^n

is isomorphic to a trivial bundle for some $n$. This is the case by lemma 1.

Proposition

$\tilde{K}(X)$ is isomorphic to the Grothendieck group of $(\mathrm{Vect}(X),\oplus )$.

However, def. 1 is more directly related to the definition of K-theory by a classifying space, hence by a spectrum. This we discuss below.

Properties

Classifying space

Definition

For $n\in \mathbb{N}$ write $U(n)$ for the unitary group in dimension $n$ and $O(n)$ for the orthogonal group in dimension $n$, both regarded as topological groups in the standard way. Write $BU(n),BO(n)\in$ Top $ for the corresponding classifying space.

Write

$$[X,BO(n)]:={\pi }_0\mathrm{Top}(X,BO(n))$$[X, B O(n)] := \pi_0 Top(X, B O(n))

and

$$[X,BU(n)]:={\pi }_0\mathrm{Top}(X,BU(n))$$[X, B U(n)] := \pi_0 Top(X, B U(n))

for the set of homotopy-classes of continuous functions $X\to BU(n)$.

Proposition

This is equivalently the set of isomorphism classes of $O(n)$- or $U(n)$-principal bundles on $X$ as well as of rank-$n$ real or complex vector bundles on $X$, respectively:

$$[X,BO(n)]\simeq O(n)\mathrm{Bund}(X)\simeq {\mathrm{Vect}}_{ℝ}(X,n),$$[X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n) \,,

$$[X,BU(n)]\simeq U(n)\mathrm{Bund}(X)\simeq {\mathrm{Vect}}_{ℂ}(X,n).$$[X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n) \,.

Definition

For each $n$ let

$$U(n)\to U(n+1)$$U(n) \to U(n+1)

be the inclusion of topological groups given by inclusion of $n\times n$ matrices into $(n+1)\times (n+1)$-matrices given by the block-diagonal form

$$\left[g\right]\mapsto \left[\begin{array}{cc}1& [0]\\ [0]& [g]\end{array}\right].$$\left[g\right] \mapsto \left[ \array{ 1 & [0] \\ [0] & [g] } \right] \,.

This induces a corresponding sequence of morphisms of classifying spaces, def. 2, in Top

$$BU(0)\hookrightarrow BU(1)\hookrightarrow BU(2)\hookrightarrow \cdots .$$B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots \,.

Write

$$BU:={\underrightarrow{\lim }}_{n\in \mathbb{N}}BU(n)$$B U := {\lim_{\to}}_{n \in \mathbb{N}} B U(n)

for the homotopy colimit (the “homotopy direct limit”) over this diagram.

Note

The topological space $BU$ is not equivalent to $BU(\mathscr{H})$, where $U(\mathscr{H})$ is the unitary group on a separable infinite-dimensional Hilbert space $\mathscr{H}$. In fact the latter is contractible, hence has a weak homotopy equivalence to the point

$$BU(\mathscr{H})\simeq *$$B U(\mathcal{H}) \simeq *

while $BU$ has nontrivial homotopy groups in arbitrary higher degree. (See below).

But there is the group $U(\mathscr{H}{)}_{𝒦}\subset U(\mathscr{H})$ of unitary operators that differ from the identity by a compact operator. This is essentially $U=\Omega BU$ (…). See below (…)

Proposition

Write $\mathbb{Z}$ for the set of integers regarded as a discrete topological space.

The product spaces

$$BO\times \mathbb{Z},BU\times \mathbb{Z}$$B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z}

are classifying spaces for real and complex $K$-theory, respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups

$$\tilde{K}(X)\simeq [X,BU].$$\tilde K(X) \simeq [X, B U ] \,.

$$K(X)\simeq [X,BU\times \mathbb{Z}].$$K(X) \simeq [X, B U \times \mathbb{Z}] \,.

See for instance (Friedlander, prop. 3.2) or (Karoubi, prop. 1.32, theorem 1.33).

Proof

First consider the statement for reduced cohomology $\tilde{K}(X)$:

Since a compact topological space is a compact object in Top (and using that the classifying spaces $BU(n)$ are (see there) paracompact topological spaces, hence normal, and since the inclusion morphisms are closed inclusions (…)) the hom-functor out of it commutes with the filtered colimit

$$\begin{array}{rl}\mathrm{Top}(X,BU)& =\mathrm{Top}(X,{\underrightarrow{\lim }}_{n}BU(n))\\ & \simeq {\underrightarrow{\lim }}_n\mathrm{Top}(X,BU(n))\end{array}.$$\begin{aligned} Top(X, B U) &= Top(X, {\lim_\to}_n B U(n)) \\ & \simeq {\lim_\to}_n Top(X, B U (n)) \end{aligned} \,.

Since $[X,BU(n)]\simeq U(n)\mathrm{Bund}(X)$, in the last line the colimit is over vector bundles of all ranks and identifies two if they become isomorphic after adding a trivial bundle of some finite rank.

For the full statement use that by prop. \ref{missing} we have

$$K(X)\simeq H^0(X,\mathbb{Z})\oplus \tilde{K}(X).$$K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X) \,.

Because $H^0(X,\mathbb{Z})\simeq [X,\mathbb{Z}]$ it follows that

$$H^0(X,\mathbb{Z})\oplus \tilde{K}(X)\simeq [X,\mathbb{Z}]\times [X,BU]\simeq [X,BU\times \mathbb{Z}].$$H^0(X, \mathbb{Z}) \oplus \tilde K(X) \simeq [X, \mathbb{Z}] \times [X, B U] \simeq [X, B U \times \mathbb{Z}] \,.

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.

Relation to algebraic K-theory

The topological K-theory over a space $X$ is not identical with the algebraic K-theory of the ring of functions on $X$, but the two are closely related. See for instance (Paluch, Rosenberg).

Related concepts

• K-theory

• topological K-theory

  • topological index

• twisted K-theory

• differential K-theory

• twisted differential K-theory

References

Introductions are in

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

• Eric Friedlander, An introduction to K-theory (pdf)

• Max Karoubi, K-theory: an introduction

A discussion of the topological K-theory of classifying spaces of Lie groups is in

• Stefan Jackowski and Bob Oliver, Vector bundles over classifying spaces of compact Lie groups (pdf)

Relations to algebraic K-theory are discussed in

• Michael Paluch, Algebraic $K$-theory and topological spaces K-theory 0471 (web)

• Jonathan Rosenberg, Comparison Between Algebraic and Topological K-Theory for Banach Algebras and $C^{*}$-Algebras, (pdf)