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 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
for all $W \in Vect(X)_{/\sim}$. The resulting set of equivalence classes is an abelian group with group operation given on representatives by
and with the inverse of $[V_+,V_-]$ given by
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
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.
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 $n \in \mathbb{N}$.
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 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
H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 9 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Max Karoubi, K-theory. An elementary introduction, lectures given at the Clay Mathematics Academy (arXiv:math/0602082)
Eric Friedlander, An introduction to K-theory (emphasis on algebraic K-theory), 2007 (pdf)
Varvara Karpova, Complex topological K-theory, 2009 (pdf)
Chris Blair, Some K-theory examples, 2009 (pdf)
Klaus Wirthmüller, Vector bundles and K-theory, 2012 (pdf)
Aderemi Kuku, Introduction to K-theory and some applications (pdf)
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
The proof of the Hopf invariant one theorem in terms of topological K-theory is due to