Olami–Feder–Christensen model

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, C. Then K_k(X) is 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 focussed on complex K-theory, the real case being similar.

As a first example, note that the K-theory of a point are 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 are the integers.

There is also a reduced version of K-theory, $\tilde{K}(X)$, 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 ${ϵ}_1$ and ${ϵ}_2$, so that $E\oplus {ϵ}_1\cong F\oplus {ϵ}_2$. The fact that this equivalence relation results in a group follows from the fact that every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, $\tilde{K}(X)$ can be defined as the kernel of the map $K(X)\to K(x_0)\cong \mathbb{\mathbb{Z}}$ induced by the inclusion of the basepoint 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)

$\tilde{K}(X/A)\to \tilde{K}(X)\to \tilde{K}(A)$

extends to a long exact sequence

$\cdots \to \tilde{K}(SX)\to \tilde{K}(SA)\to \tilde{K}(X/A)\to \tilde{K}(X)\to \tilde{K}(A)$.

Then define ${\tilde{K}}^{-n}(X):=\tilde{K}(S^{n}X)$ for $n\ge 0$ where $S^n$ is the nth reduced suspension of a space. Negative indices are chosen so that the coboundary maps increase dimension. One-point compactification extends this definition to locally compact spaces without basepoints: $K^{-n}(X)={\tilde{K}}^{-n}(X_{+})$. Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

Properties

• $K^n$ respectively ${\tilde{K}}^n$ 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 (Z with the discrete topology), i.e. $K(X)\cong [X_{+},\mathbb{\mathbb{Z}}\times BU]$ where [,] denotes pointed homotopy classes and BU is the colimit of the classifying spaces $B{U}_n\cong G{r}_n({\mathbb{ℂ}}^{\mathrm{\infty }})$ of the unitary groups. Similarly, $\tilde{K}(X)\cong [X,\mathbb{\mathbb{Z}}\times BU]$. For real K-theory use BO.
• There is a natural ring homomorphism $K^{*}(X)\to H^{2*}(X,\mathbf{Q})$, the Chern character, such that $K^{*}(X)\otimes \mathbf{Q}\to H^{2*}(X,\mathbf{Q})$ 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 $K(X)\cong \tilde{K}(T(E))$ where T(E) is the Thom space of the vector bundle E over X.
• 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 = CP¹, i.e. the Riemann sphere.
• ${\tilde{K}}^{n+2}(X)={\tilde{K}}^n(X).$
• 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 J. F. 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.

See also

• KR-theory
• Atiyah-Singer index theorem
• Snaith’s theorem

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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• Max Karoubi (1978), K-theory, an introduction Springer-Verlag
  • Max Karoubi (2006), "K-theory. An elementary introduction", Template:Arxiv
• Allen Hatcher, Vector Bundles & K-Theory, (2003)
• Maxim Stykow, Connections of K-Theory to Geometry and Topology, (2013)