# 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 cunder 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 of a commutative monoid”) 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

Throughout, let $k$ be either the field of real numbers $\mathbb{R}$ or of complex numbers $\mathbb{C}$ .

We take

1. vector space to mean finite dimensional vector space over $k$.

2. vector bundle to mean topological vector bundle over $k$ of finite rank.

For the most part below we will assume that the base topological space $X$ is a compact Hausdorff space. Because then the following statement holds, which is crucial in some places:

###### Lemma

(over compact Hausdorff space every topological vector bundle is direct summand of a trivial vector bundle)

For every topological vector bundle $E \to X$ over the compact Hausdorff space $X$ there exists a topological vector bundle $\tilde E \to X$ such that the direct sum of vector bundles

$E \oplus_X \tilde E \simeq X \times \mathbb{R}^{n}$

For proof see this prop. at topological vector bundle.

### The K-group

The starting point is the simple observation that the operation of direct sum of vector bundles yields a monoid structure (semi-group with unit) on isomorphism classes of topological vector bundles, which however is lacking inverse elements and hence is not an actual group.

###### Definition

(semi-group of isomorphism classes of topological vector bundles on $X$)

For $X$ a topological space, write $Vect(X)_{/\sim}$ for the set of isomorphism classes of topological vector bundles over $X$. The operation of direct sum of vector bundles

$(-)\oplus_X (-) \;\colon\; Vect(X) \times Vect(X) \longrightarrow Vect(X)$

descends to this quotient by isomorphism

$[E_1] + [E_2] \;\coloneqq\; [E_1 \oplus_X E_2]$

to yield the structure of a monoid (semi-group with unit)

$\left( Vect(X)_{/\sim}, + \right) \,.$
###### Remark

The operation of direct sum of vector bundles on isomorphism classes in def. 1 is indeed not a group:

Let $x \in X$ be a chosen point of $x$ and write

$rk_x \;\colon\; Vect(X)_{/\sim} \longrightarrow \mathbb{N}$

for the function which takes a topological vector bundle to the rank over the connected component of the point $x$.

Then under direct sum of vector bundles the rank is additive

$rk_x(E_1 \oplus_X E_2) \,=\, rk_x(E_1) + rk_x(E_1) \,.$

Now since the natural numbers under addition are just a monoid (semi-group with unit), with no element except zeor having an inverse element under the additive operation, it follows immediately that a necessary condition for the isomorphism class of a topological vector bundle to be invertible under direct sum of vector bundles is that its rank of a vector bundle be zero. But there is only one such class of vector bundles, in fact there is only one such vector bundle, namely the unique rnk-zro bundle $X \times k^0$, necessarily a trivial bundle.

Now for the monoid of natural numbers $(\mathbb{N},+)$ it is a time honored fact that it is interesting and useful to rectify its failure of being a group by universally forcing it to become one. This is a process called group completion and the group completion of the natural numbers is the additive group of integers $(\mathbb{Z},+)$.

The idea is hence to apply group completion also to the monoid $(Vect(X)_{/\sim}, +)$, and so that the rank operation above becomes a homomorphism of abelian groups.

An explicit construction of group completion of a commutative monoid is called the Grothendieck group of a commutative monoid.

###### Definition

(K-group as the Grothendieck group of isomorphism classes of topological vector bundles)

For $X$ a topological space, write

$K(X) \;\coloneqq\; K(Vect(X)_{/\sim}, +)$

for the Grothendieck group of the commutative monoid (abelian semi-group with unit) of isomorphism classes of topological vector bundles on $X$ from def. 1.

This means that $K(X)$ is the group whose elements are equivalence classes of pairs

$([E_+], [E_-]) \; \in Vect(X)_{/\sim} \times Vect(X)_{/\sim}$

of isomorphism classes of topological vector bundles on $X$, with respect to the equivalence relation

$\big( \left( [E_+], [E_-] \right) \;\sim\; \left( [F_+, F_-] \right) \big) \;\Leftrightarrow\; \left( \underset{[G],[H] \in Vect(X)_{/\sim}}{\exists} \left( \left([E_+ \oplus_X G] , [E_- \oplus_X G]\right) \,=\, \left([F_+ \oplus_X H] , [F_- \oplus_X H]\right) \right) \right) \,.$

Here a pair $([E_+], [E_-])$ is also called a virtual vector bundle, and its equivalence class under the above equivalence relation is also denoted

$[E_+] - [E_-] \;\;\in K(X) \,.$

If $X$ is a pointed topological space, hence equipped with a choice of point $x \in X$

then difference of ranks $rk_x(-)$ of the representing vector bundles over the connected component of $x \in X$

$rk_x(E_+) - rk_x(E_-) \in \mathbb{Z}$

is called the virtual rank of the virtual vector bundle.

###### Example

(K-group of the point is the integers)

Let $X = \ast$ be the point. Then a topological vector bundle on $X$ is just a vector space

$Vect(\ast) \simeq Vect$

and an isomorphism of vector bundles is just a bijective linear map.

Since finite dimensional vector spaces are isomorphic precisely if they have the same dimension, the semi-group of isomorphism classes of vector bundles over the point (def. 1) is the natural numbers:

$\left( Vect(\ast)_{/\sim}, + \right) \;\simeq\; \left( \mathbb{N}, + \right) \,.$

Accordingly the K-group of the point is the Grothendieck group of the natural numbers, which is the additive group of integers (this example):

$K(\ast) \simeq (\mathbb{Z}, +)$

and this identification is the assignment of virtual rank (def. 2).

###### Remark

(on compact Hausdorff spaces all virtual vector bundles are formal difference by a trivial vector bundle)

If $X$ is a compact Hausdorff space, then every virtual vector bundle on $X$ (def. 2) of the form

$[E] - [X \times k^n]$

(i.e. for the negatice component represented by a trivial vector bundle).

This is because for $X$ compact Hausdorff then lemma 1 implies that for every topological vector bundle $E_-$ there exists a topological vector bundle $\tilde E_-$ with $E_- \oplus_X \tilde E_- \simeq X \times k^n$, and hence

$[E_+] - [E_-] = \underset{[E]}{\underbrace{[E_+] + [\tilde E_-]}} - \underset{[X \times k^n]}{\underbrace{ \left( [E_-] + [\tilde E_-] \right) } } = [E] - [X \times k^n] \,.$
###### Remark

(commutative ring structure on $K(X)$ from tensor product of vector bundles)

Also the operation of tensor product of vector bundles over $X$ descends to isomorphism classes of vector bundles? and makes $(Vect(X)_{\sim}, \oplus, \oplus)$ a semi-ring? (rig).

(This is the shadow under passing to isomorphism classes of the fact that the category $Vect(X)$ is a distributive monoidal category under tensor product of vector bundles).

This mulitlicative structure passes to the K-group (def. 2) by the formula

$[E_+, E_-] \cdot [F_+, F_-] \;\coloneqq\; [ (E_+ \otimes_X F_+) \oplus_X (E_- \otimes_X E_-) \,,\, (E_+ \otimes_X F_-) \oplus_X (E_- \otimes_X F_+) ] \,.$

Accordingly the ring $(K(X), +,\cdot)$ is also called the K-theory ring of $X$.

###### Remark

(functoriality of the K-theory ring assignment)

Let $f \colon X \longrightarrow Y$ be a continuous function between topological spaces. The operation of pullback of vector bundles

$f^\ast \;\colon\; Vect(Y) \longrightarrow Vect(X)$

is compatible with direct sum of vector bundles as well as with tensor prouct of vector bundles? and hence descends to a homomorphism of commutative rings

$f^\ast \;\colon\; K(Y) \longrightarrow K(X)$

between the K-theory rings from remark 3. Moreover, for

$X \overset{f}{\longrightarrow} Y \overset{g}{\longrightarrow} Z$

two consecutive continuous functions, then the consecutive pullback of the vector bundle is isomorphic to the pullback along the composite map, which means that on K-group pullback preserves composition

$(g \circ f)^\ast = f^\ast \circ g^\ast \;\colon\; K(Z) \longrightarrow K(X) \,.$

Finally, of course pullback along an identity function $id_X \colom X \to X$ is the identity group homomorphism.

In summary this says that the assignment of K-groups to topological spaces is a functor

$K(-) \;\colon\; Top \longrightarrow CRing$

from the category Top of topological space to the category CRing of commutative rings.

We consider next the image of plain vector bundles in virtual vector bundles:

###### Definition

(stable equivalence of vector bundles)

Let $X$ be a topological space. Define an equivalence relation $\sim_{stable}$ on topological vector bundles over $X$ by declaring two vector bundles $E_1 E_2 \in Vect(X)$ to be equivalent if there exists a trivial vector bundle $X \times k^n$ of some rank $n$ such that after tensor product of vector bundles with this trivial bundle, both bundles become isomorphic

$\left( E_1 \sim_{stable} E_2 \right) \;\Leftrightarrow\; \underset{n \in \mathbb{N}}{\exists} \left( E_1 \otimes_X (X \times k^n) \;\simeq\; E_2 \otimes_X (X \times k^n) \right) \,.$

If $E_1 \sim_{stable} E_2$ we say that $E_1$ and $E_2$ are stably equivalent vector bundles.

###### Proposition

(image of plain vector bundles in virtual vector bundles)

Let $X$ be a topological space. There is a homomorphism of semigroups

$\array{ Vect(X)_{/\sim} & \longrightarrow & K(X) \\ [E_1] &\overset{\phantom{AAA}}{\mapsto}& \left( [E_1], [X \times k^0] \right) }$

from the isomorphism classes of topological vector bundles (def. 1) to the K-group of $X$ (def. 2 ).

If $X$ is a compact Hausdorff space, then the image of this function is the stable equivalence classes of vector bundles (def. 1), hence this function factors as an epimorphism onto $Vect(X)_{/\sim_{stable}}$ followed by an injection

$Vect(X)_{/\sim} \longrightarrow Vect(X)_{/\sim_{stable}} \hookrightarrow K(X) \,.$
###### Proof

The homomorphism of commutative monoids $Vect(X)_{/\sim} \to K(X)$ is the one given by the universal property of the Grothendieck group construction (this prop.).

By definition of the Grothendieck group (this def.), two elements of the form

$\left( [E_1], [X \times k^0] \right) \phantom{AA} \text{and} \phantom{AA} \left( [E_2], [X \times k^0] \right)$

are equivalent precisely if there exist vector bundles $F_1$ and $F_2$ such that

$\left( [ E_1 \oplus_X F_1 ], [ F_1] \right) \;=\; \left( [E_2 \oplus_X F_2], [F_2] \right) \,.$

First of all this means that $F_1 \simeq F_2$, hence is equivalent to the existence of a vector bundle $F$ such that

$[E_1 \oplus F] \;=\; [E_2 \oplus F] \,.$

Now, by the assumption that $X$ is compact Hausdorff, lemma 1 implies that there exists a vector bundle $\tilde F$ such that

$F \oplus_X \tilde F \simeq X \times k^n$

is the trivial vector bundle of some rank n \in \mathbbN}. This means that the above is equivalent already to the existence of an $n \in \mathbbN}n \in \mathbb{N}$ such that

$[E_1 \oplus (X \times k^n)] \;=\; [E_2 \oplus (X \times k^n)] \,.$

This is the definition of stable equivalence from def. 1.

### The reduced K-group

###### Definition

(reduced K-theory)

Let $X$ be a pointed topological space, hence a topological space equipped with a choice of point $x \in X$, hence with a continuous function $const_x \colon \ast \to X$ from the point space.

By the functoriality of the K-groups (remark 4) this induces a group homomorphism

$const_x^\ast \;\colon\; K(X) \longrightarrow K(\ast) \,.$

The kernel of this map is called the reduced K-theory group of $(X,x)$, denoted

$\tilde K(X) \;\coloneqq\; ker(const_x^\ast) \,.$
###### Remark

(restriction in K-theory to the point computes virtual rank)

By example 1 we have that

1. $K(\ast) \simeq \mathbb{Z}$;

2. under this identification the function $const_x^\ast$ is the assignment of virtual rank

$\array{ K(X) &\overset{const_x^\ast}{\longrightarrow}& K(\ast) &\overset{\simeq}{\to}& \mathbb{Z} \\ [E]- [F] &\overset{\phantom{AAA}}{\mapsto}& [E_x] - [F_x] &\mapsto& rk_x(E) - rk_x(F) }$
###### Proposition

(over compact Hausdorff spaces $\tilde K(X)$ is a direct summand of $K(X)$)

If $(X,x)$ is a pointed compact Hausdorff space then the defining short exact sequence of reduced K-theory group (def. 3)

$0 \to \tilde K(X) \hookrightarrow K(X) \overset{const_x^\ast}{\longrightarrow} K(\ast) \simeq \mathbb{Z}$

splits and thus yields an isomorphism, which is given by

$\array{ K(X) &\overset{\phantom{A}\simeq \phantom{A}}{\longrightarrow}& \tilde K(X) \oplus \mathbb{Z} \\ [E] - [X \times k^n] &\overset{\phantom{AAA}}{\mapsto}& ([E], rk_x(E) - n) } \,.$

Here on the left we are using remark 2 to represent any element of the K-group as a virtual difference of a vector bundle $E$ by a trivial bundle, and $rk_x(E) \in \mathbb{N}$ denotes the rank of this vector bundl over the connected component of $x \in X$.

Equivalently this means that every element of $K(X)$ decomposes as follows into a piece that has vanishing virtual rank over the connected component of $x$ and a virtual trivial bundle.

$[E]- [X \times k^n] = \underset{\in \tilde K(X) \subset K(X)}{\underbrace{\left( [E] - [X \times k^{rk_x(E)}] \right)}} - \underset{\in \mathbb{Z} \subset K(X) }{\underbrace{[X \times k^{n-rk_x(E)}]}} \,.$
###### Proof

By remark 5 the kernel of $const_x^\ast$ is identified with the virtual vector bundles of vanishing virtual rank. By remark 2 this kernel is identified with the elements of the form

$[E] - [X \times k^{rk_x(E)}] \,.$

In order to describe $\tilde K(X)$ itself as an equivalence class, we consider the followign refinement of stable equivalence of vector bundles (def. 1):

###### Definition

(equivalence relation for reduced K-theory on compact Hausdorff spaces)

For $X$ a topological space, define an equivalence relation on the set of topological vector bundles $E \to X$ over $X$ by declaring that $E_1 \sim E_2$ if there exists $k_1, k_2 \in \mathbb{N}$ such that there is an isomorphism of topological vector bundles between the direct sum of vector bundles of $E_1$ with the trivial vector bundle $X \times \mathbb{R}^{k_1}$ and of $E_2$ with $X \times \mathbb{R}^{k_2}$

$(E_1 \sim_{red} E_2) \Leftrightarrow \left( \underset{k_1,k_2 \in \mathbb{N}}{\exists} \left( (E_1 \oplus_X (X \times \mathbb{R}^{k_1}) \;\simeq\; (E_2 \oplus_X (X \times \mathbb{R}^{k_2}) \right) \right) \,.$

The operation of direct sum of vector bundles descends to these quotients

$[E_1] + [E_2] \;\coloneqq\; [ E_1 \oplus_X E_2 ]$

to yield a commutative semi-group

$\left(Vect(X)_{/\sim_{red}}, +\right) \,.$
###### Proposition

For $X$ a compact Hausdorff space then the commutative semi-group $(Vect(X)_{/\sim_{red}}, +)$ from def. 4 and an actual abelian group and is in fact naturally isomorphic to the reduced K-theory group $\tilde K(X)$ (def. 3):

$\tilde K(X) \simeq (Vect(X)_{/\sim_{red}}, +) \,.$
###### Proof

By prop. 3 $\tilde K(X)$ is the subgroup of the Grothendieck group $K(X)$ on the elements of the form $[E]- [X \times k^{rk_x(E)}]$, which are cleraly entirely determined by $[E] \in Vect(X)_{/\sim}$. Hence we need to check if the equivalence relation of the Gorthendieck goup coincides with $\sim_{red}$ on these representatives.

The relation in the Grothendieck group is given by

$\left( [E_1] \sim [E_2] \right) \Leftrightarrow \left( \underset{[G], [H] \in Vect(X)_{/\sim}}{\exists} \left( ( [E_1]+ [G], [X \times k^{rk_x(E_1)}] + [G] ) \;=\; ( [E_2] + [H], [X \times k^{rk_x(E_2)}] + [H] ) \right) \right)$

As before, in remark 2 we may assume without restriction that $G = X \times k^{n_1}$ and $H = X \times k^{n_2}$ are trivial vector bundles. Then the above equality on the first component

$[E_1] + [X \times k^{n_1}] = [E_2] + [X \times k^{n_2}]$

is the one that defines $\sim_{red}$, and since isomorphic vector bundles necessarily have the same rank, it implies the equality of the second component.

###### Remark

(non-unital commutative ring-structure on $\tilde K(X)$)

In view of the commutative ring structure on the K-group $K(X)$ from remark 3, the reduced K-group $\tilde K(X)$ from def. 3, being the kernel of a ring homomorphism (remark 4) is an ideal in $K(X)$, hence itself a non-unital commutative ring.

(The ring unit of $K(X)$ is the class $[X \times k^1, X \times k^0]$ of the trivial line bundle on $X$, which has virtual rank 1, and hence is not in $\tilde K(x)$.)

## Examples

###### Example

(topological K-theory ring of the point space)

We have already see in example 1 that

$K(\ast) \simeq \mathbb{Z} \,.$
###### Example

(complex topological K-theory ring of the circle)

Since the complex general linear group $GL(n,\mathbb{C})$ is path-connected (this prop.) and hence the classifying space $B GL(n,\mathbb{C})$ is simply-connected, hence its fundamental group is trivial $\pi_1(B GL(n,\mathbb{C})) \simeq [S^1, B GL(n,\mathbb{C})] = 1$. Accordingly, all complex vector bundles on $S^1$ are isomorphic toa trivial vector bundle.

It follows that

$K_{\mathbb{C}}(S^1) \simeq \mathbb{Z} \phantom{AA} \text{and} \phantom{AA} \tilde K_{ \mathbb{C} } \simeq 0 \,.$
###### Example

(complex topological K-theory ring of the 2-sphere)

For $X = \ast$ the point space, the fundamental product theorem in topological K-theory 5 states that the homomorphism

$\array{ \mathbb{Z}[h]/((h-1)^2) &\longrightarrow& K_{\mathbb{C}}(S^2) \\ h &\mapsto& h }$

is an isomorphism.

This means that the relation $(h-1)^2 = 0$ satisfied by the basic line bundle on the 2-sphere (this prop.) is the only relation is satisfies in topological K-theory.

Notice that the underlying abelian group of $\mathbb{Z}[h]/((h-1)^2)$ is two direct sum copies of the integers,

$K_{\mathbb{C}}(S^2) \simeq \mathbb{Z} \oplus \mathbb{Z} = \langle 1, h\rangle$

one copy spanned by the trivial complex line bundle on the 2-sphere, the other spanned by the basic complex line bundle on the 2-sphere. (In contrast, the underlying abelian group of the polynomial ring $\mathbb{R}[h]$ has infinitely many copies of $\mathbb{Z}$, one for each $h^n$, for $n \in \mathbb{N}$).

It follows (by this prop.) that the reduced K-theory group of the 2-sphere is

$\tilde K_{\mathbb{C}}(S^2) \simeq \mathbb{Z} \,.$

## Properties

### Fundamental product theorem

For $S^2 \subset \mathbb{R}^3$ the 2-sphere with its Euclidean subspace topology, write $h$ for the basic line bundle on the 2-sphere. Its image in the topological K-theory ring $K(S^2)$ satisfies the relation

$2 h = h^2 + 1 \;\;\Leftrightarrow\;\; (h-1)^2 = 0$

(by this prop.).

(Notice that $h-1$ is the image of $h$ in the reduced K-theory $\tilde K(X)$ of $S^2$ under the splitting $K(X) \simeq \tilde K(X) \oplus \mathbb{Z}$ (by prop. 3.)

It follows that there is a ring homomorphism of the form

$\array{ \mathbb{Z}[h]/\left( (h-1)^2 \right) &\overset{}{\longrightarrow}& K(S^2) \\ h &\overset{\phantom{AAA}}{\mapsto}& H }$

from the polynomial ring in one abstract generator, quotiented by this relation, to the topological K-theory ring.

More generally, for $X$ a topological space, then this induces the composite ring homomorphism

$\array{ K(X) \otimes \mathbb{Z}[h]/((h-1)^2) & \longrightarrow & K(X) \times K(S^2) & \longrightarrow & K(X \times S^2) \\ (E, h) &\overset{\phantom{AAA} }{\mapsto}& (E,H) &\overset{\phantom{AAA}}{\mapsto}& (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H) }$

to the topological K-theory ring of the product topological space X \times S^^2, where the second map is the external tensor product of vector bundles.

###### Proposition

(fundamental product theorem in topological K-theory)

For $X \times S^^X$ a compact Hausdorff space, then this ring homomorphism is an isomorphism.

(e.g. Hatcher, theorem 2.2)

For $X = \ast$ the product theorem prop. 5 says in particular that the first of the two morphisms in the composite is an isomorphism (example 4 below) and hence by the two-out-of-three-property for isomorphisms it follows that

###### Corollary

(external product theorem)

For $X$ a compact Hausdorff space we have that the external tensor product of vector bundles

$K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2)$

is an isomorphism in topological K-theory.

### 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. 5, 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 28, 2017 10:21:03 by Urs Schreiber (92.218.150.85)