cohomology

spin geometry

string 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 topological space $X$ may be represented by pairs of vector bundles, real or complex ones, on $X$ modulo a certain equivalence relation.

The following is the quick idea. For a detailed introduction see Introduction to Topological K-Theory.

First, recall that for $k$ a field then a $k$-vector bundle over a topological space $X$ is a map $V \to X$ whose fibers are vector spaces which vary over $X$ in a controlled way. Explicitly this means that there exits an open cover $\{U_i \to X\}$ of $X$, a natural number $n \in \mathbb{N}$ (the rank of the vector bundle) and a homeomorphism $U_i \times k^n \to V|_{U_i}$ over $U_i$ which is fiberwise a $k$-linear map.

Vector bundles are of central interest in large parts of mathematics and physics, for instance in Chern-Weil theory and cobordism theory. But the collection $Vect(X)_{/\sim}$ of isomorphism classes of vector bundles over a given space is in general hard to analyze. One reason for this is that these are classified in degree-1 nonabelian cohomology with coefficients in the (nonabelian) general linear group $GL(n,k)$. K-theory may roughly be thought of as the result of forcing vector bundles to be classified by an abelian cohomology theory.

To that end, observe that all natural operations on vector spaces generalize to vector bundles by applying them fiber-wise. Notably there is the fiberwise direct sum of vector bundles, also called the Whitney sum operation. This operation gives the set $Vect(X)_{/\sim}$ of isomorphism classes of vector bundles the structure of an semi-group (monoid) $(Vect(X)_{/\sim},\oplus)$.

Now as under direct sum, the dimension of vector spaces adds, similarly under direct sum of vector bundles their rank adds. Hence in analogy to how one passes from the additive semi-group (monoid) of natural numbers to the addtitive group of integers by adjoining formal additive inverses, so one may adjoin formal additive inverses to $(Vect(X)_{/\sim},\oplus)$. By a general prescription (“Grothendieck group”) this is achieved by first passing to the larger class of pairs $(V_+,V_-)$ of vector bundles (“virtual vector bundles”), and then quotienting out the equivalence relation given by

$(V_+, V_-) \sim (V_+ \oplus W , V_- \oplus W)$

for all $W \in Vect(X)_{/\sim}$. The resulting set of equivalence classes is an abelian group with group operation given on representatives by

$[V_+, V_-] \oplus [V'_+, V'_-] \coloneqq (V_+ \oplus V'_+, V_- \oplus V'_-)$

and with the inverse of $[V_+,V_-]$ given by

$-[V_+, V_-] = [V_-, V_+] \,.$

This abelian group obtained from $(Vect(X)_{/\sim}, \oplus)$ is denoted $K(X)$ and often called the K-theory of the space $X$. Here the letter “K” (due to Alexander Grothendieck) originates as a shorthand for the German word Klasse, referring to the above process of forming equivalence classes of (isomorphism classes) of vector bundles.

This simple construction turns out to yield remarkably useful groups of homotopy invariants. A variety of deep facts in algebraic topology have fairly elementary proofs in terms of topolgical K-theory, for instance the Hopf invariant one problem (Adams-Atiyah 66).

One defines the “higher” K-groups of a topological space to be those of its higher suspensions

$K^{-n}(X) = K(\Sigma^n X) \,.$

The assignment $X \mapsto K^\bullet(X)$ turns out to share many properties of the assignment of ordinary cohomology groups $X \mapsto H^n(X,\mathbb{Z})$. One says that topological K-theory is a generalized (Eilenberg-Steenrod) cohomology theory. As such it is represented by a spectrum. For $k = \mathbb{C}$ this is called KU, for $k = \mathbb{R}$ this is called KO. (There is also the unification of both in KR-theory.)

One of the basic facts about topological K-theory, rather unexpected from the definition, is that these higher K-groups repeat periodically in the degree $n$. For $k = \mathbb{R}$ the periodicity is 8, for $k = \mathbb{C}$ it is 2. This is called Bott periodicity.

It turns out that an important source of virtual vector bundles representing classes in K-theory are index bundles?: Given a Riemannian spin manifold $B$, then there is a vector bundle $S \to B$ called the spin bundle of $B$, which carries a differential operator, called the Dirac operator $D$. The index of a Dirac operator is the formal difference of its kernel by its cokernel $[ker D, coker D]$. Now given a continuous family $D_x$ of Dirac operators/Fredholm operators, parameterized by some topological space $X$, then these indices combine to a class in $K(X)$.

It is via this construction that topological K-theory connects to spin geometry (see e.g. Karoubi K-theory) and index theory.

As the terminology indicates, both spin geometry and Dirac operator originate in physics. Accordingly, K-theory plays a central role in various areas of mathematical physics, for instance in the theory of geometric quantization (“spin^c quantization”) in the theory of D-branes (where it models D-brane charge and RR-fields) and in the theory of Kaluza-Klein compactification via spectral triples (see below).

All these geometric constructions have an operator algebraic incarnation: by the topological Serre-Swan theorem then vector bundles of finite rank are equivalently modules over the C*-algebra of continuous functions on the base space. Using this relation one may express K-theory classes entirely operator algebraically, this is called operator K-theory. Now Dirac operators are generalized to Fredholm operators.

There are more C*-algebras than arising as algebras of functions of topological space, namely non-commutative C-algebras. One may think of these as defining non-commutative geometry, but the definition of operator K-theory immediately generalizes to this situation (see also at KK-theory).

While the C*-algebra of a Riemannian spin manifold remembers only the underlying topological space, one may algebraically encode the smooth structure and Riemannian structure by passing from Fredholm modules to “spectral triples”. This may for instance be used to algebraically encode the spin physics underlying the standard model of particle physics and operator K-theory plays a crucial role in this.

## Definition

under construction

Let $X$ be a compact Hausdorff topological space. Write $k$ for either the field of real numbers $\mathbb{R}$ 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 of finite rank of a 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 $E' \to X$ such that

$E \oplus E' \simeq I^{rank E + rank E'} \,.$
###### Proof

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

See at direct sum of vector bundles this prop for more

###### 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 $I^l$

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

Write

$\tilde K(X) := Vect(X)/\sim$

for the quotient set of equivalence classes. We also define the slightly coarser equivalence relation $E_1 \sim_s E_2$ where in the above definition of $\sim$ we force $k=l$. The set of equivalence classes for this equivalence relation is denoted

$K(X) := Vect(X)/{\sim_s}.$
###### Proposition

With $X$ compact Hausdorff as in the assumption, we have that fiberwise direct sum of vector bundles equips $\tilde K(X)$ and $K(X)$ with the structure of an abelian group. Together with the fiberwise tensor product of vector bundles this makes $K(X)$ a ring and $\tilde K(X)$ a nonunital 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. The smaller ring $\tilde K(X)$ is called the reduced K-theory of $X$.

###### Proof

The non-trivial part of the statement is that in $\tilde K(X)$ and $K(X)$ 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 $\tilde K(X)$

$[I^n] = 0$

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

$E_1 \oplus E_2 \simeq I^n$

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

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

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

Write

$[X, B O(n)] := \pi_0 Top(X, B O(n))$

and

$[X, B U(n)] := \pi_0 Top(X, B U(n))$

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:

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

For each $n$ let

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

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

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

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

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

Write

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

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

###### Remark

The topological space $B U$ is not equivalent to $B U(\mathcal{H})$, where $U(\mathcal{H})$ 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

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

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

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

###### Proposition

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

The product spaces

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

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

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

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 $B U(n)$ 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)) \\ & \simeq {\lim_\to}_n Top(X, B U (n)) \end{aligned} \,.

Since $[X, B U(n)] \simeq U(n) Bund(X)$, 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

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

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

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

There is another variant on the classifying space

###### Definition

Let

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

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

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

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

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

$B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}}$

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

### As a generalized cohomology theory

That topological K-theory satisfies the axioms of a generalized (Eilenberg-Steenrod) cohomology theory was shown (at least) in (Atiyah-Hirzebruch 61, 1.8](#AtiyahHirzebruch61))

### Spectrum

Being a generalized (Eilenberg-Steenrod) cohomology theory, topological 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.

### Chromatic filtration

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

### 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). 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 $KU$third integral SW class $W_3$spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class $w_4$
2integral Morava K-theory $\tilde K(2)$seventh integral SW class $W_7$Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

cohomology theories of string theory fields on orientifolds

string theoryB-field$B$-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology $H\mathbb{Z}^3$
type II superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KR-theory $KR^\bullet$
type IIA superstringsuper line 2-bundle$B GL_1(KU)$KU-theory $KU^1$
type IIB superstringsuper line 2-bundle$B GL_1(KU)$KU-theory $KU^0$
type I superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KO-theory $KO$
type $\tilde I$ superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KSC-theory $KSC$

## References

The “ring of complex vector bundles” $K(X)$ was introduced in

• M. F. Atiyah, F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math Soc. vol. 65 (1959) pp. 276-281.

and shown to give a generalized (Eilenberg-Steenrod) cohomology theory in

Early lecture notes on topological K-theory in a general context of stable homotopy theory and generalized cohomology theory includes

Textbook accounts on topological K-theory include

• M. F. Atiyah, K-theory, Benjamin New-York (1967)

• Max Karoubi, K-theory: an introduction, Grundlehren der Math. Wissen. 226 Springer 1978, Reprinted in Classics in Mathematics (2008)

• Allen Hatcher, Vector bundles and K-theory, 2003/2009 (web)

Further introductions include

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 $K$-theory and topological spaces K-theory 0471 (web)

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

Discussion from the point of view of smooth stacks and differential K-theory is in

The proof of the Hopf invariant one theorem in terms of topological K-theory is due to

Revised on May 25, 2017 16:51:14 by Urs Schreiber (92.218.150.85)