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topological K-theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher spin geometry

Contents

Idea

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

of integers.

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

somethinglikeVect. \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 Core(Vect)Core(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 VV are given by their dimension d(V)d(V) \in \mathbb{N}, and direct sum of vector spaces corresponds to addition of these numbers.

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

Vect:DiffGrpd \mathbf{Vect} : Diff \to \infty Grpd
UCore(VectBund(U)) U \mapsto Core(VectBund(U))

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

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

H(X,Vect):=π 0H diff(X,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\mathbf{B} U of the stable unitary group U:=colim nU(n)U := colim_n U(n).

K-theory as a groupoidification of Vect\mathbf{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\mathbf{B} \mathbb{Z} is obtained from B\mathbf{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 Vect\mathbf{Vect}.

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

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

one can similarly look at equivalence classes of pairs (V,W)Vect(U)×Vect(U)(V,W) \in \mathbf{Vect}(U) \times \mathbf{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 2\mathbb{Z}_2-graded vector bundle.

One version of 2\mathbb{Z}_2-graded vector bundles, which lead to a description of twisted KK-theory are vectorial bundles.

Definition

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

Lemma

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

EEI rankE+rankE. E \oplus E' \simeq I^{rank E + rank E'} \,.
Proof

One invokes a partition of unity relative to an open cover on which EE trivializes, constructs EE' 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 EXE \to X over XX by declaring that E 1E 2E_1 \sim E_2 if there exists k,lk,l \in \mathbb{N} such that there is an isomorphism of vector bundles between the (fiberwise) direct sum of E 1E_1 with I kI^k and of E 2E_2 with I lI^l

(E 1E 2):(E 1I kE 2I l). (E_1 \sim E_2) :\Leftrightarrow \exists (E_1 \oplus I^k \simeq E_2 \oplus I^l) \,.

Write

K˜(X):=Vect(X) \tilde K(X) := Vect(X)_\sim

for the quotient set of equivalence classes.

Proposition

With XX compact Hausdorff as in the assumption, we have that fiberwise direct sum of vector bundles equips K˜(X)\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)K(X) is called the topological K-theory ring of XX or just the K-theory group or even just the K-theory of XX, for short.

Proof

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

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

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

for all nn \in \mathbb{N}, because by definition I nI 0I^n \sim I^0. Therefore an inverse of a class [E 1][E_1] is given by a vector bundle E 2E_2 with the property that the direct sum

E 1E 2I n E_1 \oplus E_2 \simeq I^n

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

Proposition

K˜(X)\tilde K(X) is isomorphic to the Grothendieck group of (Vect(X),)(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

We discuss how the classifying space for K 0K^0 is the delooping of the stable unitary group.

Definition

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

Write

[X,BO(n)]:=π 0Top(X,BO(n)) [X, B O(n)] := \pi_0 Top(X, B O(n))

and

[X,BU(n)]:=π 0Top(X,BU(n)) [X, B U(n)] := \pi_0 Top(X, B U(n))

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

Proposition

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

[X,BO(n)]O(n)Bund(X)Vect (X,n), [X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n) \,,
[X,BU(n)]U(n)Bund(X)Vect (X,n). [X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n) \,.
Definition

For each nn let

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

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

[g][1 [0] [0] [g]]. \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)BU(1)BU(2). B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots \,.

Write

BU:=lim nBU(n) B U := {\lim_{\to}}_{n \in \mathbb{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 BUB U is not equivalent to BU()B U(\mathcal{H}) , where U()U(\mathcal{H}) is the unitary group on a separable infinite-dimensional Hilbert space \mathcal{H}. In fact the latter is contractible, hence has a weak homotopy equivalence to the point

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

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

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

Proposition

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

The product spaces

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

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

K˜(X)[X,BU]. \tilde K(X) \simeq [X, B U ] \,.
K(X)[X,BU×]. 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 K˜(X)\tilde K(X):

Since a compact topological space is a compact object in Top (and using that the classifying spaces BU(n)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

Top(X,BU) =Top(X,lim nBU(n)) lim nTop(X,BU(n)). \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)]U(n)Bund(X)[X, B U(n)] \simeq 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,)K˜(X). K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X) \,.

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

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

There is another variant on the classifying space

Definition

Let

U 𝒦={gU()|gid𝒦} U_{\mathcal{K}} = \left\{ g \in U(\mathcal{H}) | g - id \in \mathcal{K} \right\}

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

Palais showed that

Proposition

U 𝒦U_\mathcal{K} is a homotopy equivalent model for BUB U. It is in fact the norm closure? of the evident model of BUB U in U()U(\mathcal{H}).

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

Since U()U(\mathcal{H}) is contractible, it follows that

BU 𝒦U()/U 𝒦 B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{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 KK-theory is modeled in particular by the space FredFred of Fredholm operators.

Chromatic filtration

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

As the shape of the smooth K-theory spectrum

See at differential cohomology diagram.

Relation to algebraic K-theory

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

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory KUKUthird integral SW class W 3W_3spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class w 4w_4
2integral Morava K-theory K˜(2)\tilde K(2)seventh integral SW class W 7W_7Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

cohomology theories of string theory fields on orientifolds

string theoryB-fieldBB-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology H 3H\mathbb{Z}^3
type II superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KR-theory KR KR^\bullet
type IIA superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 1KU^1
type IIB superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 0KU^0
type I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KO-theory KOKO
type I˜\tilde I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KSC-theory KSCKSC

References

Introductions are in

A textbook account of topological K-theory with an eye towards operator K-theory is section 1 of

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)

The comparison map between algebraic and topological K-theory is discussed for instance in

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

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

Revised on April 29, 2014 04:34:37 by Urs Schreiber (89.204.135.30)