group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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 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
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 reduced 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.
The following discussion of topological K-theory in terms of point-set topology. For more abstract perspectives see for instance Snaith's theorem and other pointers at K-theory.
Assumed background for the following is the content of
Throughout, let $k$ be a topological field, usually the real numbers $\mathbb{R}$ or of complex numbers $\mathbb{C}$.
In the following we take
vector space to mean finite dimensional vector space over $k$.
vector bundle to mean topological vector bundle over $k$ of finite rank.
We say monoid for semigroup with unit.
For the most part below we will invoke the assumption that the base topological space $X$ is a compact Hausdorff space. Because then the following statement holds, which is crucial in some places:
(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
is a trivial vector bundle.
For proof see this prop. at topological vector bundle.
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.
(monoid 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
descends to this quotient by isomorphism
to yield the structure of a monoid (semi-group with unit)
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
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
Now since the natural numbers under addition are just a monoid (semi-group with unit), with no element except zero 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 rank-zero bundle $X \times k^0$, necessarily a trivial vector 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.
(K-group as the Grothendieck group of isomorphism classes of topological vector bundles)
For $X$ a topological space, write
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
of isomorphism classes of topological vector bundles on $X$, with respect to the equivalence relation
Here a pair $([E_+], [E_-])$ is also called a virtual vector bundle, and its equivalence class under the above equivalence relation is also denoted
If $X$ is a pointed topological space, hence equipped with a choice of point $x \in X$ then the difference of ranks $rk_x(-)$ of the representing vector bundles over the connected component of $x \in X$
is called the virtual rank of the virtual vector bundle.
(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
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 monoid (semi-group with unit) of isomorphism classes of vector bundles over the point (def. 1) is the natural numbers:
Accordingly the K-group of the point is the Grothendieck group of the natural numbers, which is the additive group of integers (this example):
and this identification is the assignment of virtual rank (def. 2).
(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) is of the form
(i.e. with negative component represented by a trivial vector bundle).
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
(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 topological vector bundles and makes $(Vect(X)_{\sim}, \oplus, \otimes )$ 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 multiplicative structure passes to the K-group (def. 2) by the formula
Accordingly the ring $(K(X), +,\cdot)$ is also called the K-theory ring of $X$.
(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
is compatible with direct sum of vector bundles as well as with tensor product of vector bundles and hence descends to a homomorphism of commutative rings
between the K-theory rings from remark 2. Moreover, for
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
Finally, of course pullback along an identity function $id_X \colon X \to X$ is the identity group homomorphism.
In summary this says that the assignment of K-groups to topological spaces is a functor
from the opposite category of 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:
(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 vector bundle, both bundles become isomorphic
If $E_1 \sim_{stable} E_2$ we say that $E_1$ and $E_2$ are stably equivalent vector bundles.
(image of plain vector bundles in virtual vector bundles)
Let $X$ be a topological space. There is a homomorphism of semigroups
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. 2), hence this function factors as an epimorphism onto $Vect(X)_{/\sim_{stable}}$ followed by an injection
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
are equivalent precisely if there exist vector bundles $F_1$ and $F_2$ such that
First of all this means that $F_1 \simeq F_2$, hence is equivalent to the existence of a vector bundle $F$ such that
Now, by the assumption that $X$ is compact Hausdorff, lemma 1 implies that there exists a vector bundle $\tilde F$ such that
is the trivial vector bundle of some rank $n \in \mathbb{N}$. This means that the above is equivalent already to the existence of an $n \in \mathbb{N}$ such that
This is the definition of stable equivalence from def. 2.
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 3) this induces a group homomorphism
given by restricting a virtual vector bundle to the basepoint.
The kernel of this map is called the reduced K-theory group of $(X,x)$, denoted
(expressing plain K-groups as reduced K-groups)
Let $X$ be a topological space. Write $X_* \coloneqq X \sqcup \ast$ for its disjoint union space with the point space, and regard this as a pointed topological space with base point the adjoined point.
Then the reduced K-theory of $X_+$ is the plain K-theory of $X$:
Because every topological vector bundle on $X \sqcup \ast$ is the direct sum of vector bundles of one that has rank zero on $\ast$ and one that has rank zero on $X$ (this example.)
(restriction in K-theory to the point computes virtual rank)
By example 1 we have that
$K(\ast) \simeq \mathbb{Z}$;
under this identification the function $const_x^\ast$ is the assignment of virtual rank
If $X$ is a locally compact Hausdorff space, then a continuous function
is said to vanish at infinity if it extends by zero to the one-point compactification $X^* \coloneqq (X \sqcup \{\infty\}, \tau_{cpt})$
Now the one-point compactification $X^\ast$ is a compact Hausdorff space (by this prop. and this prop.) and canonically a pointed topological space with basepoint the element $\infty \in X^\ast$.
Moreover, every compact Hausdorff space $X$ arises this way as the one-point compactification of the complement subspace of any of its points: $X \simeq (X \setminus \{x\})^\ast$ (by this remark).
Since open subspaces of compact Hausdorff spaces are locally compact, this complement subspace $X \setminus \{x\} \subset X$ is a locally compact Hausdorff space, and every locally compact Hausdorff spaces arises this way (by this prop.).
Therefore one may think of the reduced K-groups $\tilde K(X)$ (def. 3) of compact Hausdorff spaces as the those K-groups of locally compact Hausdorff spaces which “vanish at infinity”.
(functoriality of the reduced K-groups)
By the functoriality of the unreduced K-groups (remark 3) on (the opposite of) the category Top of all topological spaces, the reduced K-groups (def. 3) becomes functorial on the category $Top^{\ast/}$ of pointed topological spaces (whose morphisms are the continuous functions that preserve the base-point):
This follows by the functoriality of the kernel construction (which in turn follows by the universal property of the kernel):
For $(X,x)$ and $(Y,y)$ pointed topological spaces and $f \colon X \longrightarrow Y$ a continuous function which preserves basepoints $f(x) = y$ then
(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 groups (def. 3)
splits and thus yields an isomorphism, which is given by
Here on the left we are using prop. 1 to represent any element of the K-group as a virtual difference of a vector bundle $E$ by a trivial vector bundle, and $rk_x(E) \in \mathbb{N}$ denotes the rank of this vector bundle 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 vector bundle.
By remark 4 the kernel of $const_x^\ast$ is identified with the virtual vector bundles of vanishing virtual rank. By prop. 1 this kernel is identified with the elements of the form
For $S^2$ the Euclidean 2-sphere, write
for the complex topological K-theory class of the basic complex line bundle on the 2-sphere. By prop. 4 its image in reduced K-theory is the virtual vector bundle
This is known as the Bott element, due to its key role in the Bott periodicity of complex topological K-theory, discussed below.
In order to describe $\tilde K(X)$ itself as an equivalence class, we consider the followign refinement of stable equivalence of vector bundles (def. 2):
(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 $X \times \mathbb{R}^{k_2}$
The operation of direct sum of vector bundles descends to these quotients
to yield a commutative semi-group
For $X$ a compact Hausdorff space then the commutative monoid $(Vect(X)_{/\sim_{red}}, +)$ from def. 5 is already an abelian group and is in fact naturally isomorphic to the reduced K-theory group $\tilde K(X)$ (def. 3):
By prop. 4 $\tilde K(X)$ is the subgroup of the Grothendieck group $K(X)$ on the elements of the form $[E]- [X \times k^{rk_x(E)}]$, which are clearly 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
As before, in prop. 1 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
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.
(non-unital commutative ring-structure on $\tilde K(X)$)
In view of the commutative ring structure on the K-group $K(X)$ from remark 2, the reduced K-group $\tilde K(X)$ from def. 3, being the kernel of a ring homomorphism (remark 3) 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 $\tilde K(x)$.)
Let
$X$ be a compact Hausdorff space;
$A \subset X$ a closed subspace.
Then the relative K-theory group of the pair $(X,A)$, denoted $K(X,A)$ is the reduced K-theory group (def. 3) of the quotient space $X/A$ (this def.):
(expressing plain and reduced K-theory in terms of relative K-theory)
The relative K-theory construction from def. 6 reduces in special cases to the plain K-theory group and to the reduced K-theory group.
Recall that for the case that $A = \emptyset \subset X$ then $X/\emptyset = X_+ = X \sqcup \ast$ (by this example). Therefore:
for $A = \emptyset \subset X$ we have $K(X,\emptyset) = \tilde K(X \sqcup \ast) \simeq K(X)$ (example 2);
for $A = \{x\} \subset X$ we have $K(X, \{x\}) = \tilde K(X/\{x\}) = \tilde K(X)$.
The (reduced) K-theory groups of reduced suspensions of pointed space are called the “K-group in degree 1”:
(graded K-groups)
For $X$ a pointed topological space write
for the reduced K-theory of the reduced suspension of $X$.
For $X$ a compact Hausdorff space and $A \subset X$ a closed subspace, write
for the reduced K-theory of the reduced suspension of the quotient space.
We say these are the K(-cohomology)-groups in degree 1. For emphasis one says that the original K-groups are in degree zero and writes
The groups are collected to the graded K-groups, which are the direct sums
and
regarded as $\mathbb{Z}/2$-graded groups.
Under tensor product of vector bundles this becomes a non-unital $\mathbb{Z}/2$- graded-commutative ring (discussed below).
Recall from example 2 and from example 7 the identifications of plain, reduced and relative K-groups, which with the degree-zero notation from def. 8 read:
The analogue is true for the K-groups in degree 1 from def. 8, though this is no longer completely trivial:
By prop. 1 the topological K-theory groups of compact topological spaces are represended by homotopy classes of continuous functions into the classifying spaces $B O \times \mathbb{Z}$ and $B U \times \mathbb{Z}$, respectively (def. 14).
There are various ways of generalizing this situation to non-compact spaces:
(Grothendieck group topological K-theory)
The Grothendieck group construction on the monoid $(Vect(X)/_\sim, \oplus)$ of isomorphism classes of topological vector bundles makes sense for every topological space $X$. For non-compact $X$ this is usually just called that “the Groothencieck group of vector bundles on $X$”, sometimes denoted
(representable topological K-theory)
The group of homotopy classes of continuous functions into a (classifying) space is of course well defined for any domain space, hence for any topological space $X$ one may set
This is called representable K-theory.
Representable K-theory over paracompact topological spaces was discussed in (Karoubi 70).
(inverse limit topological K-theory)
Let $X$ be a topological space with the structure of a CW-complex, hence a colimit (“direct limit”) $X \simeq \underset{\longrightarrow}{\lim}_n X_n$ such that each $X_n$ is a finite cell complex, hence in particular a compact topological space. Then the limit (inverse limit) of the corresponding K-group
is called the inverse limit K-theory of $X$.
$\,$
No two of these definitions are equivalent to each other on all of their domain of defintion (e.g. Anderson-Hodgkin 68, Jackowski-Oliver).
Representable and direct limit K-theory of spaces that are sequential colimits of compact spaces differ in general by a lim^1-term (Segal-Atiyah 69, prop. 4.1).
(topological K-theory ring of the point space)
We have already seen in example 1 that
(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
(complex topological K-theory ring of the 2-sphere)
For $X = \ast$ the point space, the fundamental product theorem in topological K-theory 12 states that the homomorphism
is an isomorphism.
This means that the relation $(h-1)^2 = 0$ satisfied by the basic complex 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,
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
(complex topological K-theory of the torus)
Consider the torus $S^1 \times S^1$, the product topological space of the circle with itself (with the Euclidean subspace topology).
By example 9 we have
Since the smash product of the circle with itself is the 2-sphere, and since, the complex K-theory of the circle vanishes by example 5, this shows that the topological K-theory of the torus coincides with that of the 2-sphere:
(homotopy invariance of K-groups)
Let $X$ and $Y$ be paracompact Hausdorff spaces, and let
be a continuous function which is a homotopy equivalence. Then the pullback operation on (reduced) K-groups along $f$ (remark 3, remark 5) is an isomorphism:
and
This is an immediate consequence of the fact that over paraompact Hausdorff spaces isomorphism classes of topological vector bundles are homotopy invariant (this example.)
We discuss the long exact sequences in cohomology for topological K-theory.
What makes these work in prop. 7 below, turns out to be the following property of topological vector bundles:
(topological vector bundle trivial over closed subspace of compact Hausdorff space is pullback of bundle on quotient space)
Let $X$ be a compact Hausdorff space and let $A \subset X$ be a closed subspace.
If a topological vector bundle $E \overset{p}{\to} X$ is such that its restriction $E\vert_A$ is trivializable, then $E$ is isomorphic to the pullback bundle $q^\ast E'$ of a topological vector bundle $E' \to X/A$ over the quotient space..
The proof of lemma 2 is given at topological vector bundle here. What makes that proof work, in turn, is the Tietze extension theorem, via this lemma.
(exact sequence in reduced topological K-theory)
Let
$X$ be a pointed compact Hausdorff space;
Write $X/A$ for the corresponding quotient space (this def.). Denote the continuous functions of subspace inclusion and of quotient space co-projection by
respectively. Then the induced sequence of reduced K-theory groups (def. 3, remark 5)
is exact, meaning that they induce an isomorphism
between the image of $g^\ast$ and the kernel of $i^\ast$.
Similarly the sequence of unreduced and relative K-groups (def. 6) is exact:
(e.g. Wirthmuller 12, p. 32 (34 of 67), Hatcher, prop. 2.9)
First observe that both statements are equivalent to each other: By naturality of the construction of kernels the following diagram commutes:
Here the top vertical morphisms, the kernel inclusions, are monomorphisms, and hence the top horizonal row is exact precisely if the middle horizontal row is.
Hence it is sufficient to consider the top row. First of all the composite function $A \overset{i}{\to} X \overset{q}{\to} X/A$ is a constant function, constant on the basepoint, and hence $i^\ast \circ q^\ast$ is a constant function, constant on zero. This says that we have an inclusion
Hence it only remains to see for $x \in \tilde K(X)$ a class with $i^\ast(x) = 0$ that $x = q^\ast(y)$ comes from a class on the quotient $X/A$. But by compactness, the class $x$ is given by a virtual vector bundle of the form $E - rk(E)$ (prop. 1, prop. 4). and by prop. 5 the triviality of $i^\ast(E- rk(E))$ means that there is $n \in \mathbb{N}$ such that $i \ast(E) \oplus_A (A \times \mathbb{C}^n)$ is a trivializable vector bundle. Therefore lemma 2 gives that $E \oplus_X (X \times \mathbb{C}^n)$ is isomorphic to the pullback bundle of a vector bundle $E'$ on $X/A$. This proves the claim.
(long exact sequence in reduced topological K-theory)
For $X$ a compact Hausdorff space and for $A \subset X$ a closed subspace inclusion, there is a long exact sequence of reduced K-theory groups of the form
where $\Sigma(-)$ denotes reduced suspension.
The sequence is induced by functoriality (remark 5) from the long cofiber sequence
obtained by consecutively forming mapping cones. By the discussion at topological cofiber sequence this may be rearranged as
(this prop. and this).
The claim hence follows by the homotopy invariance of the K-groups (prop. 6).
$\,$
We discuss some useful consequences of the long exact sequences in cohomology.
(direct sum decomposition of K-theory groups over retractions)
Let $X$ be a (pointed) compact topological space and $A \subset X$ a (pointed) closed subspace, such that the subspace inclusion $A \overset{i}{\to}$ X as a retraction, i.e. a continuous function $r \colon X \to A$ such that the composite
is the identity function.
Then there is a splitting of the K-theory group of $X$ as a direct sum of the K-theory of $A$ and the relative K-theory of the quotient space $X/A$:
and in the pointed case a splitting of the reduced K-theory groups
The long exact sequence from cor 1 together with the retraction yields
The splitting makes the morphisms $i^\ast$ and its suspension be surjections, so that the long exact sequence decomposes into short exact sequences which are split exact:
(finite wedge axiom)
Let $(X,x)$ and $(Y,y)$ be two pointed compact Hausdorff spaces with wedge sum
(i.e. the quotient of their disjoint union space by re-identifying the base points).
Then there is an isomorphism
We have retracts
and
Applying prop. 8 to each of these consecutively yields an isomorphism that establishes the claim:
This proves the claim.
Alternatively, we may again argue directly from the long exact sequence:
Consider the subspace inclusion
This is a closed subspace because its complement is $(X \vee Y) \setminus X = Y \setminus \{y\}$ which is open because all points in a Hausdorff space (which is in particular $T_1$-separated) are closed. Moreover, by definition of wedge sum the corresponding quotient space is $Y$:
Similary for the inclusion of $Y$. Hence in particular these inclusions and quotients are retractions in that they factor the identity maps as
By functoriality (remark 5) this implies that similarly
In particular these maps are injections and surjections, respectively.
Therefore by prop. 7 there are short exact sequences of the form
which are split exact. This implies the claim.
(external product in K-theory)
Let $X$ and $Y$ be topological spaces. Then the external tensor product of topological vector bundles over $X$ and $Y$
induces on K-groups an external product
We want to see that this restricts to an operation on reduced K-theory. To this end we need the following proposition:
(reduced K-theory of product space)
Let $(X,x_0)$ $(Y,y_0)$ be two pointed compact Hausdorff spaces with $X \times Y$ their product topological space and $X \wedge Y$ their smash product. Then there is an isomorphism of reduced K-theory groups
Be definition, the smash product is the quotient topological space of the product topological space by the wedge sum:
for the inclusion
This quotient is still a compact topological space because continuous images of compact spaces are compact and and it is still Hausdorff topological space because compact subspaces in Hausdorff spaces are separated by neighbourhoods from points, so that the point $(X \vee Y)/ (X \vee Y) \in (X \times Y)/(X \vee Y)$ is separated by open neighbourhoods from points in $(X \times Y) \setminus (X \vee Y)$.
Hence corollary 1 yields a long exact sequence of the form
By example 8 the two terms involving reduced topological K-theory of a wedge sum are direct sums of the reduced K-theory of the wedge summands:
Now observe that, via example 8, the morphisms $i^\ast$ and $\Sigma i^\ast$ are split epimorphisms, with section given by “external direct sum”
This means that the long exact sequence decomposes into short exact sequences
which moreover are split exact. This yields the claim.
It follows that:
(external product on reduced K-groups)
Let $X$ and $Y$ be pointed compact Hausdorff spaces. Then the external product on K-groups (def. 12) restricts to reduced K-groups under the inclusion $\tilde K(-) \hookrightarrow K(-)$ from prop. 4 and the inclusion $\tilde K(-\wedge -) \hookrightarrow K(-\times -)$ from prop. 9, in that there is a morphism $\tilde \boxtimes$ that makes the following diagram commute:
By prop. 4 the elements in $\tilde K(X)$ and $\tilde K(Y)$ are represented by virtual vector bundles which vanish when restricted to the base points $x \in X$ and $y \in Y$, respectively. But this implies that their external tensor product of vector bundles vanishes over $X \times \{y\}$ and $\{x\} \times Y$. From the proof of prop. 9 it is the restriction of the product to to these subspaces that gives the map
and hence on these element this component vanishes.
In order to compute K-classes, one needs the computation of some basic cases, such as that of the K-theory groups of n-spheres and of product spaces with $n$-spheres. The fundamental product theorem in K-theory determines these K-theory grous. Its result is most succinctly summarized by the statement of Bott periodicity, to which we turn below.
Before discussing the product theorem, it is useful to recall the analogous situation in ordinary cohomology $H^\bullet(-) \coloneqq H^\bullet(\mathbb{Z})$. Here it is immediate to determine the cohomology groups of the n-spheres, in particular one finds that for the 2-sphere is $H^\bullet(S^2) = \mathbb{Z}\langle e\rangle \oplus \mathbb{Z}\langle h\rangle$, for $h \in \tilde H^2(S^2)$ the first Chern class of the basic complex line bundle on the 2-sphere. As a ring this has the trivial product $h^2 = 0$, since by degree-reasons the cup product goes $H^2(S^2) \otimes H^2(S^2) \to H^4(S^2) = 0$.
Therefore me may write the ordinary cohomology ring of the 2-sphere as the following quotient ring of the polynomial ring in the generator $h$:
Notice that in ordinary cohomology $h$ is also the generator of the reduced cohomology group $\tilde H^\bullet(S^2) \simeq \mathbb{Z}\langle h\rangle$. Now as an element of $K_{\athbb{C}}(S^2)$ the basic complex line bundle on the 2-sphere is not reduced, but its image in reduced K-theory is the Bott element virtual vector bundle $\beta = h-1$ (def. 3). The fundamental product theorem in topological K-theory says, in particular, that the complex topological K-theory of the 2-sphere behaves in just the same way as the ordinary cohomology, if only one replaces the generator $h$ by $\beta = h-1$.
First of all, the Bott element also squares to zero:
(nilpotency of the Bott element)
For $S^2 \subset \mathbb{R}^3$ the 2-sphere with its Euclidean subspace topology, write $h \in Vect_{\mathbb{C}}(S^2)_{/\sim}$ for the basic complex line bundle on the 2-sphere. Its image in the topological K-theory ring $K(S^2)$ satisfies the relation
A proof of this may be obtained by analysis of the relevant clutching function, see here.
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 this prop.). This element
is the Bott element of complex topological K-theory (def. 3).
It follows from prop. 11 that there is a ring homomorphism of the form
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
to the topological K-theory ring of the product topological space $X \times S^2$, where the second map $\boxtimes$ is the external product from def. 12.
(fundamental product theorem in topological K-theory)
For $X$ a compact Hausdorff space, then this ring homomorphism is an isomorphism.
(e.g. Hatcher, theorem 2.2)
More generally, for $L\to X$ a complex line bundle with class $l \in K(X)$ and with $P(1 \oplus L)$ denoting its projective bundle then
(e.g. Wirthmuller 12, p. 17)
As a special case this implies the first statement above:
For $X = \ast$ the product theorem prop. 12 says in particular that the first of the two morphisms in the composite is an isomorphism (example 6 below) and hence by the two-out-of-three-property for isomorphisms it follows that:
(external product theorem in topological K-theory)
For $X$ a compact Hausdorff space we have that the external product in K-theory $\boxtimes$ (def. 12) with vector bundles on the 2-sphere
is an isomorphism in topological K-theory.
When restricted to reduced K-theory then the external product theorem (cor. 2) yields the statement of Bott periodicity of topological K-theory:
Let $X$ be a pointed compact Hausdorff space.
Then the external product $\tilde X$ in reduced K-theory (prop. 12) with the image of the basic complex line bundle on the 2-sphere in reduced K-theory yields an isomorphism of reduced K-groups
from that of $X$ to that of its double suspension $\Sigma^2 X$.
e.g. Wirthmuller 12, p. 34 (36 of 67)
By this example there is for any two pointed compact Hausdorff spaces $X$ and $Y$ an isomorphism
relating the reduced K-theory of the product topological space with that of the smash product.
Using this and the fact that for any pointed compact Hausdorff space $Z$ we have $K(Z) \simeq \tilde K(Z) \oplus \mathbb{Z}$ (this prop.) the isomorphism of the external product theorem (cor. 2)
becomes
Multiplying out and chasing through the constructions to see that this reduces to an isomorphism on the common summand $\tilde K(S^2) \oplus \tilde K(X) \oplus \mathbb{Z}$, this yields an isomorphism of the form
where on the right we used that smash product with the 2-sphere is the same as double suspension.
Finally there is an isomorphism
(example 6). The composite
is the isomorphism to be established.
The external product on reduced K-groups from prop. 10 allows to extend the commutative ring structure from the plain K-groups (remark 2) to a ring structure on the graded K-groups from def. 8. This is def. 13 below.
To state this definition, recall that
for $X$ a pointed topological space then the diagonal map to its product topological space $X \times X$ induced a diagonal to the smash product $X \wedge Y = (X \times X)/(X \vee Y)$
since reduced suspension is equivalently smash product with the circle $\Sigma X \simeq S^1 \wedge X$, there are induced “partial diagonal maps” of the form
etc.
(product on graded K-groups)
For $X$ a pointed compact Hausdorff space, the product on graded K-groups
is the linear map which on the direct summands $\tilde K^0(X) \coloneqq \tilde K(X)$ and $\tilde K^1(X) \coloneqq \tilde K(\Sigma X)$ is given by the following morphisms, which are composites of the external product $\tilde \boxtimes$ on reduced K-groups from prop. 10 with pullbacks along the above suspended diagonal maps:
where the last isomorphism on the right is Bott periodicity isomorphism (prop. 13).
We discuss how the classifying space for $\tilde K^0$ is the delooping of the stable unitary group.
(classifying space 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. 14, 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.
For $G$ a compact Lie group with classifying space $B G$ (in general non-compact) then the map from the Grothendieck group $\mathbb{K}(B G) \coloneqq Grp(Vect(B G)/_\sim, \oplus)$ (def. 9) to the representable K-theory $K(B G)_{rep} \coloneqq [X, B U \times\mathbb{Z}]$ (def. 10) is injective
Topological K-theory satisfies the axioms of a generalized (Eilenberg-Steenrod) cohomology theory (Atiyah-Hirzebruch 61, 1.8).
This is essentially the statement of the long exact sequences above.
A multiplicative generalized (Eilenberg-Steenrod) cohomology theory $E$ is called complex orientable if the element $1 \in E(\ast) \simeq \tilde E(S^0)$ is in the image of the pullback morphism
If so, then a choice of pre-image $c^E_1 \in E^2(B U(1))$ is a choice of complex orientation (this def.).
Now for $E = K_{\mathbb{C}}$ being complex topological K-theory regarded as a generalized cohomology theory as above, then by Bott periodicity (prop. 13) and by $\tilde K_{\mathbb{Z}}(S^2) \simeq \mathbb{Z} \cdot (h-1)$ (example 5) this reduces to the statement that there is an element $c^K_1 \in \tilde K_{\mathbb{C}}(B U(1))$ such that its image under
is the Bott element $h-1$, the virtual vector bundle difference between the basic complex line bundle on the 2-sphere and the trivial complex line bundle.
By the very nature of the basic complex line bundle on the 2-sphere $h$, it is the restriction of the universal complex line bundle $\mathcal{O}(1)$ on $B U(1) \simeq \mathbb{C}P^\infty$ along the defining cell inclusion $i \colon S^2 \hookrightarrow \mathbb{C}P^\infty \simeq B U(1)$. Hence if we set
then this is a complex orientation for complex topological K-theory.
From this we obtain the formal group law associated with topological K-theory (from this prop.):
By the nature of the classifying space $B U(1)$ we have that for
the group product operation, which classifies the tensor product of line bundles, that
where
are the two projections out of the Cartesian product. Hence
This shows that the formal group law associated with the complex orientation of complex topological K-theory is that of the formal multiplicative group given by
Being a generalized (Eilenberg-Steenrod) cohomology theory, topological K-theory is represented by a spectrum: the K-theory spectrum.
e.g. Switzer 75, p. 216
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.
This K-theory spectrum has the structure of a ring spectrum
(e.g. Switzer 75, section 13.90, around p. 300,
see also p. 205 (213 of 251) in A Concise Course in Algebraic Topology)
(…)
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
Representable K-theory over non-compact spaces was considered in
and (over classifying spaces in the context of equivariant K-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, (pdf)
Robert Switzer, sections 11 and 13.90 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
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
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
Topological K-theory of Eilenberg-MacLane spaces is discussed in
Topological topological K-theory of classifying spaces of Lie groups is in
For references on D-branes in terms of K-theory see there.