group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
spin geometry, string geometry, fivebrane geometry …
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:
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, for greater applicability we should 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
that sends each smooth test space $U$ (a smooth manifold, say) to the 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
It is equivalent to the nonabelian cohomology with coefficients the delooping $\mathbf{B} U$ of the stable unitary group $U := colim_n U(n)$.
To get actual topological K-theory from this, one applies geometric realization (fundamental infinity-groupoid) of the infinity-group completion of $\mathbf{Vect}$ or $\mathbf{B}U$ (Bunke-Nikolaus-Voelkl 13). See at differential cohomology hexagon the section Algebraic K-theory of smooth manifolds.
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.
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
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.
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 $n \in \mathbb{N}$.
For every vector bundle $E \to X$ (with $X$ compact Hausdorff) there exists a vector bundle $E' \to X$ such that
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).
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 $I^l$
Write
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
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 $X$.
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 $\tilde K(X)$
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
is isomorphic to a trivial bundle for some $n$. This is the case by lemma 1.
$\tilde K(X)$ 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.
We discuss how the classifying space for $K^0$ is the delooping of the stable unitary group.
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
and
for the set of homotopy-classes of continuous functions $X \to B U(n)$.
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:
For each $n$ let
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
This induces a corresponding sequence of morphisms of classifying spaces, def. 2, in Top
Write
for the homotopy colimit (the “homotopy direct limit”) over this diagram (see at homotopy colimit the section Sequential homotopy colimits).
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
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.
Write $\mathbb{Z}$ for the set of integers regarded as a discrete topological space.
The product spaces
are classifying spaces for real and complex $K$-theory, respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups
See for instance (Friedlander, prop. 3.2) or (Karoubi, prop. 1.32, theorem 1.33).
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
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
Because $H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]$ it follows that
There is another variant on the classifying space
Let
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
$U_\mathcal{K}$ 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
is a model for the classifying space of reduced K-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))
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 level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 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 cohomology | MU | MR-theory |
See at differential cohomology diagram.
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.
topological K-theory
cohomology theories of string theory fields on orientifolds
string theory | B-field | $B$-field moduli | RR-field |
---|---|---|---|
bosonic string | line 2-bundle | ordinary cohomology $H\mathbb{Z}^3$ | |
type II superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KR-theory $KR^\bullet$ |
type IIA superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^1$ |
type IIB superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^0$ |
type I superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KO-theory $KO$ |
type $\tilde I$ superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KSC-theory $KSC$ |
The “ring of complex vector bundles” $K(X)$ was introduced in
and shown to give a generalized (Eilenberg-Steenrod) cohomology theory in
Early lecture notes in a general context of stable homotopy theory and generalized cohomology theory includes
Introductions include
Allen Hatcher, Vector bundles and K-theory, 2003/2009 (web)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 9 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Eric Friedlander, An introduction to K-theory, 2007 (pdf)
Varvara Karpova, Complex topological K-theory, 2009 (pdf)
Max Karoubi, K-theory: an introduction
H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)
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
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