cohomology

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# 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

$ℤ$\mathbb{Z}

of integers.

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

$ℤ\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\mathrm{something}\mathrm{like}\mathrm{Vect}\phantom{\rule{thinmathspace}{0ex}}.$\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 $ℕ$, 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 $\mathrm{Core}\left(\mathrm{Vect}\right)$ 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 $d\left(V\right)\in ℕ$, and direct sum of vector spaces corresponds to addition of these numbers.

If we want to use the category $\mathrm{Core}\left(\mathrm{Vect}\right)$ 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

$\mathrm{Vect}:\mathrm{Diff}\to \infty \mathrm{Grpd}$\mathbf{Vect} : Diff \to \infty Grpd
$U↦\mathrm{Core}\left(\mathrm{VectBund}\left(U\right)\right)$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 $\mathrm{Core}\left(\mathrm{Vect}\right)$.

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

$H\left(X,\mathrm{Vect}\right):={\pi }_{0}{H}_{\mathrm{diff}}\left(X,\mathrm{Vect}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $U:={\mathrm{colim}}_{n}U\left(n\right)$.

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

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 $Bℤ$ is obtained from $Bℕ$ 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 $\mathrm{Vect}$.

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

$\left[\left(n,m\right)\right]=\left[\left(n+r,m+r\right)\right]\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\forall r\in ℕ$[(n,m)] = [(n+r, m+r)] \;\; \forall r \in \mathbb{N}

one can similarly look at equivalence classes of pairs $\left(V,W\right)\in \mathrm{Vect}\left(U\right)×\mathrm{Vect}\left(U\right)$ 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 ${I}^{n}={k}^{n}×X$ over $X$ of rank $n\in ℕ$.

###### Lemma

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

$E\oplus E\prime \simeq {I}^{\mathrm{rank}E+\mathrm{rank}E\prime }\phantom{\rule{thinmathspace}{0ex}}.$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 $k,l\in ℕ$ 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 ${I}^{l}$

$\left({E}_{1}\sim {E}_{2}\right):⇔\exists \left({E}_{1}\oplus {I}^{k}\simeq {E}_{2}\oplus {I}^{l}\right)\phantom{\rule{thinmathspace}{0ex}}.$(E_1 \sim E_2) :\Leftrightarrow \exists (E_1 \oplus I^k \simeq E_2 \oplus I^l) \,.

Write

$\stackrel{˜}{K}\left(X\right):=\mathrm{Vect}\left(X{\right)}_{\sim }$\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 $\stackrel{˜}{K}\left(X\right)$ with the structure of an abelian group. Together with the fiberwise tensor product of vector bundles this yields a ring.

Therefore $K\left(X\right)$ 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 $K\left(X\right)$ there is an inverse to the operation of direct sum of vector bundles. Because in $\mathrm{Vect}\left(X\right)$ direct sum acts by addition of the ranks of vector bundles, it clearly has no inverse in $\mathrm{Vect}\left(X\right)$.

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

$\left[{I}^{n}\right]=0$[I^n] = 0

for all $n\in ℕ$, because by definition ${I}^{n}\sim {I}^{0}$. Therefore an inverse of a class $\left[{E}_{1}\right]$ 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

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

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

Write

$\left[X,BO\left(n\right)\right]:={\pi }_{0}\mathrm{Top}\left(X,BO\left(n\right)\right)$[X, B O(n)] := \pi_0 Top(X, B O(n))

and

$\left[X,BU\left(n\right)\right]:={\pi }_{0}\mathrm{Top}\left(X,BU\left(n\right)\right)$[X, B U(n)] := \pi_0 Top(X, B U(n))

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

###### Proposition

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

$\left[X,BO\left(n\right)\right]\simeq O\left(n\right)\mathrm{Bund}\left(X\right)\simeq {\mathrm{Vect}}_{ℝ}\left(X,n\right)\phantom{\rule{thinmathspace}{0ex}},$[X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n) \,,
$\left[X,BU\left(n\right)\right]\simeq U\left(n\right)\mathrm{Bund}\left(X\right)\simeq {\mathrm{Vect}}_{ℂ}\left(X,n\right)\phantom{\rule{thinmathspace}{0ex}}.$[X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n) \,.
###### Definition

For each $n$ let

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

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

$\left[g\right]↦\left[\begin{array}{cc}1& \left[0\right]\\ \left[0\right]& \left[g\right]\end{array}\right]\phantom{\rule{thinmathspace}{0ex}}.$\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

$BU\left(0\right)↪BU\left(1\right)↪BU\left(2\right)↪\cdots \phantom{\rule{thinmathspace}{0ex}}.$B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots \,.

Write

$BU:={\underset{\to }{\mathrm{lim}}}_{n\in ℕ}BU\left(n\right)$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 $BU$ is not equivalent to $BU\left(ℋ\right)$, where $U\left(ℋ\right)$ 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

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

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

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

###### Proposition

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

The product spaces

$BO×ℤ\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}BU×ℤ$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

$\stackrel{˜}{K}\left(X\right)\simeq \left[X,BU\right]\phantom{\rule{thinmathspace}{0ex}}.$\tilde K(X) \simeq [X, B U ] \,.
$K\left(X\right)\simeq \left[X,BU×ℤ\right]\phantom{\rule{thinmathspace}{0ex}}.$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 $\stackrel{˜}{K}\left(X\right)$:

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

$\begin{array}{rl}\mathrm{Top}\left(X,BU\right)& =\mathrm{Top}\left(X,{\underset{\to }{\mathrm{lim}}}_{n}BU\left(n\right)\right)\\ & \simeq {\underset{\to }{\mathrm{lim}}}_{n}\mathrm{Top}\left(X,BU\left(n\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left[X,BU\left(n\right)\right]\simeq U\left(n\right)\mathrm{Bund}\left(X\right)$, 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

$K\left(X\right)\simeq {H}^{0}\left(X,ℤ\right)\oplus \stackrel{˜}{K}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X) \,.

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

${H}^{0}\left(X,ℤ\right)\oplus \stackrel{˜}{K}\left(X\right)\simeq \left[X,ℤ\right]×\left[X,BU\right]\simeq \left[X,BU×ℤ\right]\phantom{\rule{thinmathspace}{0ex}}.$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

${U}_{𝒦}=\left\{g\in U\left(ℋ\right)\mid g-\mathrm{id}\in 𝒦\right\}$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 $ℋ$ which differ from the identity by a compact operator.

Palais showed that

###### Proposition

${U}_{𝒦}$ is a homotopy equivalent model for $BU$. It is in fact the norm closure? of the evident model of $BU$ in $U\left(ℋ\right)$.

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

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

$B{U}_{𝒦}≔U\left(ℋ\right)/{U}_{𝒦}$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 $K$-theory is modeled in particular by the space $\mathrm{Fred}$ of Fredholm operators.

### Chromatic filtration

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ring
0ordinary cohomologyEilenberg-MacLane spectrum $Hℤ$
0th Morava K-theory$K\left(0\right)$
1complex K-theorycomplex K-theory spectrum $\mathrm{KU}$
first Morava K-theory$K\left(1\right)$
first Morava E-theory$E\left(1\right)$
2elliptic cohomology${\mathrm{Ell}}_{E}$
tmftmf spectrum
second Morava K-theory$K\left(2\right)$
second Morava E-theory$E\left(2\right)$
algebraic K-theory of KU$K\left(\mathrm{KU}\right)$
$n$$n$th Morava K-theory$K\left(n\right)$
$n$th Morava E-theory$E\left(n\right)$
$n+1$algebraic K-theory applied to chrom. level $n$$K\left({E}_{n}\right)$ (red-shift conjecture)
$\infty$

### 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).

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory $\mathrm{KU}$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 $\stackrel{˜}{K}\left(2\right)$seventh integral SW class ${W}_{7}$Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

## 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)

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 November 12, 2013 12:31:06 by Urs Schreiber (188.200.54.65)