nLab
topological K-theory

Context

Cohomology

Idea
- Motivational example: “nonabelian K-cohomology”
- K-theory as a groupoidification of $Vect$
Definition
Properties
Related concepts
References

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

ℤ

of integers.

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

ℤ \mapsto something like 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 $ℕ$ , the additive semi-group of $ℕ$ 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 $ℕ$ is just the monoidal 0-groupoid $ℕ$ which as a coefficient object induces a very boring cohomology theory: the $ℕ$ -cohomology of anything connected is just the monoidal set $ℕ$ itself. While we cannot deloop it, we can categorify it and do obtain an interesting nonabelian cohomology theory:

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

If we want to use the category $Core (Vect)$ 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 $Vect$ -cohomology of a topological space or a smooth space would be. The canonical way to do this is to regard $Vect$ as a generalized smooth space called a smooth infinity-stack and consider it as the assignment

Vect : Diff \to ∞ Grpd

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 $Core (Vect)$ .

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

H (X, Vect) : = π_{0} H_{diff} (X, Vect) .

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

K-theory as a groupoidification of $Vect$

The integers $ℤ$ are obtained from the natural numbers $ℕ$ by including “formal inverses” to all elements under the additive operation. Another way to think of this is that the delooped groupoid $B ℤ$ is obtained from $B ℕ$ 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 $ℕ$ , but to its categorification $Vect$ .

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

[(n, m)] = [(n + r, m + r)] \forall r \in ℕ

one can similarly look at equivalence classes of pairs $(V, W) \in Vect (U) \times Vect (U)$ 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 $ℤ_{2}$ -graded vector bundle.

One version of $ℤ_{2}$ -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 $n \in ℕ$ .

Lemma

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

E \oplus E' ≃ I^{rank E + rank E'} .

Proof

One invokes a partition of unity relative to an open cover on which $E$ trivializes, constructs $E'$ 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 $k, l \in ℕ$ 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}) : \Leftrightarrow \exists (E_{1} \oplus I^{k} ≃ E_{2} \oplus I^{l}) .

Write

\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 $Vect (X)$ direct sum acts by addition of the ranks of vector bundles, it clearly has no inverse in $Vect (X)$ .

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

for all $n \in ℕ$ , 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} ≃ 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 $(Vect (X), \oplus)$ .

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 $n \in ℕ$ 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 $B U (n), B O (n) \in$ Top $ for the corresponding classifying space.

Write

[X, B O (n)] : = π_{0} Top (X, B O (n))

and

[X, B U (n)] : = π_{0} Top (X, B U (n))

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

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, B O (n)] ≃ O (n) Bund (X) ≃ {Vect}_{ℝ} (X, n),

[X, B U (n)] ≃ U (n) Bund (X) ≃ {Vect}_{ℂ} (X, n) .

Definition

For each $n$ let

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

[g] \mapsto [\begin{matrix} 1 & [0] \\ [0] & [g] \end{matrix}] .

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

B U (0) ↪ B U (1) ↪ B U (2) ↪ \dots .

Write

B U : = {\lim_{\to}}_{n \in ℕ} B U (n)

for the homotopy colimit (the “homotopy direct limit”) over this diagram (see at homotopy colimit the section Sequential homotopy colimits).

Note

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

B U (ℋ) ≃ *

while $B U$ has nontrivial homotopy groups in arbitrary higher degree (by Kuiper's theorem).

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

Proposition

Write $ℤ$ for the set of integers regarded as a discrete topological space.

The product spaces

B O \times ℤ, B U \times 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) ≃ [X, B U] .

K (X) ≃ [X, B U \times 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 $B U (n)$ 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{aligned} Top (X, B U) & = Top (X, {\lim_{\to}}_{n} B U (n)) \\ ≃ {\lim_{\to}}_{n} Top (X, B U (n)) \end{aligned} .

Since $[X, B U (n)] ≃ U (n) Bund (X)$ , 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) ≃ H^{0} (X, ℤ) \oplus \tilde{K} (X) .

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

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

There is another variant on the classifying space

Definition

Let

U_{𝒦} = {g \in U (ℋ) ∣ g - id \in 𝒦}

be the group of unitary operators on a separable Hilbert space $ℋ$ which differ from the identity by a compact operator.

Palais showed that

Proposition

$U_{𝒦}$ is a homotopy equivalent model for $B U$ . It is in fact the norm closure? of the evident model of $B U$ in $U (ℋ)$ .

Moreover $U_{𝒦} \subset U (ℋ)$ is a Banach Lie? normal subgroup.

Since $U (ℋ)$ is contractible, it follows that

B U_{𝒦} ≔ U (ℋ) / U_{𝒦}

is a model for the classifying space of reduced K-theory.

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 $Fred$ 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
- KR-theory
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)

Revised on June 8, 2012 12:32:22 by Urs Schreiber (131.130.239.35)

nLab
topological K-theory

Context

Cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Motivational example: “nonabelian K-cohomology”

K-theory as a groupoidification of $Vect$

Definition

Lemma

Proof

Definition

Proposition

Proof

Proposition

Properties

Classifying space

Definition

Proposition

Definition

Note

Proposition

Proof

Definition

Proposition

Spectrum

Relation to algebraic K-theory

Related concepts

References

nLab topological K-theory

Context

Cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Motivational example: “nonabelian K-cohomology”

K-theory as a groupoidification of Vect

Definition

Lemma

Proof

Definition

Proposition

Proof

Proposition

Properties

Classifying space

Definition

Proposition

Definition

Note

Proposition

Proof

Definition

Proposition

Spectrum

Relation to algebraic K-theory

Related concepts

References

nLab
topological K-theory

K-theory as a groupoidification of $Vect$