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 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 $\mathbb{Z}$ of integers.

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

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

### 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 $Core(Vect)$ 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 $Core(Vect)$ 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

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

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 $\mathbf{Vect}$ has no cohomology groups, but at least cohomology monoids

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

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

Note: Topological complex K-theory is defined on pairs of spaces $K(X,U)$, such that the section of the complex bundle over $U$ is trivial (we might choose a trivialization). If no second space is listed, we assumed that K-theory of our manifold $X$ is taken with respect to the empty set – $K(X) \equiv K(X, \emptyset)$ – in this case, the bundle can be nowhere trivial.

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

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 $\mathbf{B} \mathbb{Z}$ is obtained from $\mathbf{B} \mathbb{N}$ 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 $\mathbf{Vect}$.

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

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

one can similarly look at equivalence classes of pairs $(V,W) \in \mathbf{Vect}(U) \times \mathbf{Vect}(U)$ 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 $\mathbb{Z}_2$-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 $\mathbb{R}$ 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. 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.

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 $I^l$

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

Write

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

for the quotient set of equivalence classes.

###### 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 $n \in \mathbb{N}$, 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).

###### Note

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.

### 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.

### 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

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 $K$-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 January 27, 2015 21:30:23 by Catherine Ray (192.31.105.153)