Topological K-theory

Short description: Branch of algebraic topology

In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.

Definitions

Let X be a compact Hausdorff space and k=ℝ or ℂ. Then Kk(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional k-vector bundles over X under Whitney sum. Tensor product of bundles gives K-theory a commutative ring structure. Without subscripts, K(X) usually denotes complex K-theory whereas real K-theory is sometimes written as KO(X). The remaining discussion is focused on complex K-theory.

As a first example, note that the K-theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers is the integers.

There is also a reduced version of K-theory, K~(X), defined for X a compact pointed space (cf. reduced homology). This reduced theory is intuitively K(X) modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles E and F are said to be stably isomorphic if there are trivial bundles ε1 and ε2, so that E⊕ε1≅F⊕ε2. This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, K~(X) can be defined as the kernel of the map K(X)→K(x0)≅ℤ induced by the inclusion of the base point x₀ into X.

K-theory forms a multiplicative (generalized) cohomology theory as follows. The short exact sequence of a pair of pointed spaces (X, A)

K~(X/A)→K~(X)→K~(A)

extends to a long exact sequence

⋯→K~(SX)→K~(SA)→K~(X/A)→K~(X)→K~(A).

Let Sⁿ be the n-th reduced suspension of a space and then define

K~−n(X):=K~(SnX),n≥0.

Negative indices are chosen so that the coboundary maps increase dimension.

It is often useful to have an unreduced version of these groups, simply by defining:

K−n(X)=K~−n(X+).

Here X+ is X with a disjoint basepoint labeled '+' adjoined.

Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

Properties

• Kn (respectively, K~n) is a contravariant functor from the homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the K-theory over contractible spaces is always ℤ.
• The spectrum of K-theory is BU×ℤ (with the discrete topology on ℤ), i.e. K(X)≅[X+,ℤ×BU], where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: BU(n)≅Gr⁡(n,ℂ∞). Similarly, K~(X)≅[X,ℤ×BU]. For real K-theory use BO.
• There is a natural ring homomorphism K0(X)→H2*(X,ℚ), the Chern character, such that K0(X)⊗ℚ→H2*(X,ℚ) is an isomorphism.
• The equivalent of the Steenrod operations in K-theory are the Adams operations. They can be used to define characteristic classes in topological K-theory.
• The Splitting principle of topological K-theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
• The Thom isomorphism theorem in topological K-theory is K(X)≅K~(T(E)), where T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle.
• The Atiyah-Hirzebruch spectral sequence allows computation of K-groups from ordinary cohomology groups.
• Topological K-theory can be generalized vastly to a functor on C*-algebras, see operator K-theory and KK-theory.

Bott periodicity

The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way:

• K(X×𝕊2)=K(X)⊗K(𝕊2), and K(𝕊2)=ℤ[H]/(H−1)2 where H is the class of the tautological bundle on 𝕊2=ℙ1(ℂ), i.e. the Riemann sphere.
• K~n+2(X)=K~n(X).
• Ω2BU≅BU×ℤ.

In real K-theory there is a similar periodicity, but modulo 8.

Applications

The two most famous applications of topological K-theory are both due to Frank Adams. First he solved the Hopf invariant one problem by doing a computation with his Adams operations. Then he proved an upper bound for the number of linearly independent vector fields on spheres.

Chern character

Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex X with its rational cohomology. In particular, they showed that there exists a homomorphism

ch:Ktop*(X)⊗ℚ→H*(X;ℚ)

such that

Ktop0(X)⊗ℚ≅⨁kH2k(X;ℚ)Ktop1(X)⊗ℚ≅⨁kH2k+1(X;ℚ)

There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety X.

See also

• Atiyah–Hirzebruch spectral sequence (computational tool for finding K-theory groups)
• KR-theory
• Atiyah–Singer index theorem
• Snaith's theorem
• Algebraic K-theory

References

• Atiyah, Michael Francis (1989). K-theory. Advanced Book Classics (2nd ed.). Addison-Wesley. ISBN 978-0-201-09394-0. 
• Friedlander, Eric; Grayson, Daniel, eds (2005). Handbook of K-Theory. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-27855-9. ISBN 978-3-540-30436-4. 
• Karoubi, Max (1978). K-theory: an introduction. Classics in Mathematics. Springer-Verlag. doi:10.1007/978-3-540-79890-3. ISBN 0-387-08090-2. 
• Karoubi, Max (2006). "K-theory. An elementary introduction". arXiv:math/0602082.
• Hatcher, Allen (2003). "Vector Bundles & K-Theory". http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html. 
• Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology". https://www.researchgate.net/publication/330505308.