topological K-theory

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

As a first example, note that the {{mvar|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 {{mvar|K}}-theory, $\backslash widetilde\{K\}(X)$, defined for {{mvar|X}} a compact pointed space (cf. reduced homology). This reduced theory is intuitively {{math|K(X)}} modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles {{mvar|E}} and {{mvar|F}} are said to be stably isomorphic if there are trivial bundles $\backslash varepsilon\_1$ and $\backslash varepsilon\_2$, so that $E\; \backslash oplus\; \backslash varepsilon\_1\; \backslash cong\; F\backslash oplus\; \backslash varepsilon\_2$. 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, $\backslash widetilde\{K\}(X)$ can be defined as the kernel of the map $K(X)\backslash to\; K(x\_0)\; \backslash cong\; \backslash Z$ induced by the inclusion of the base point {{math|x₀}} into {{mvar|X}}.

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

:$\backslash widetilde\{K\}(X/A)\; \backslash to\; \backslash widetilde\{K\}(X)\; \backslash to\; \backslash widetilde\{K\}(A)$

extends to a long exact sequence

:$\backslash cdots\; \backslash to\; \backslash widetilde\{K\}(SX)\; \backslash to\; \backslash widetilde\{K\}(SA)\; \backslash to\; \backslash widetilde\{K\}(X/A)\; \backslash to\; \backslash widetilde\{K\}(X)\; \backslash to\; \backslash widetilde\{K\}(A).$

Let {{math|Sⁿ}} be the {{mvar|n}}-th reduced suspension of a space and then define

:$\backslash widetilde\{K\}^\{-n\}(X):=\backslash widetilde\{K\}(S^nX),\; \backslash qquad\; n\backslash geq\; 0.$

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:

:$K^\{-n\}(X)=\backslash widetilde\{K\}^\{-n\}(X\_+).$

Here $X\_+$ is $X$ with a disjoint basepoint labeled '+' adjoined.{{cite book|last1=Hatcher|title=Vector Bundles and K-theory|pages=57|url=https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf|accessdate=27 July 2017}}

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

• $K^n$ (respectively, $\backslash widetilde\{K\}^n$) is a contravariant functor from the homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the {{mvar|K}}-theory over contractible spaces is always $\backslash Z.$
• The spectrum of {{mvar|K}}-theory is $BU\backslash times\backslash Z$ (with the discrete topology on $\backslash Z$), i.e. $K(X)\; \backslash cong\; \backslash left\; [\; X\_+,\; \backslash Z\; \backslash times\; BU\; \backslash right\; ],$ where {{math|[ , ]}} denotes pointed homotopy classes and {{math|BU}} is the colimit of the classifying spaces of the unitary groups: $BU(n)\; \backslash cong\; \backslash operatorname\{Gr\}\; \backslash left\; (n,\; \backslash Complex^\{\backslash infty\}\; \backslash right\; ).$ Similarly, $$\backslash widetilde\{K\}(X)\; \backslash cong\; [X,\; \backslash Z\; \backslash times\; BU].$$ For real {{mvar|K}}-theory use {{math|BO}}.
• There is a natural ring homomorphism $K^0(X)\; \backslash to\; H^\{2*\}(X,\; \backslash Q),$ the Chern character, such that $K^0(X)\; \backslash otimes\; \backslash Q\; \backslash to\; H^\{2*\}(X,\; \backslash Q)$ is an isomorphism.
• The equivalent of the Steenrod operations in {{mvar|K}}-theory are the Adams operations. They can be used to define characteristic classes in topological {{mvar|K}}-theory.
• The Splitting principle of topological {{mvar|K}}-theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
• The Thom isomorphism theorem in topological {{mvar|K}}-theory is $$K(X)\backslash cong\backslash widetilde\{K\}(T(E)),$$ where {{math|T(E)}} is the Thom space of the vector bundle {{mvar|E}} over {{mvar|X}}. This holds whenever {{mvar|E}} is a spin-bundle.
• The Atiyah-Hirzebruch spectral sequence allows computation of {{mvar|K}}-groups from ordinary cohomology groups.
• Topological {{mvar|K}}-theory can be generalized vastly to a functor on C*-algebras, see operator K-theory and KK-theory.

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

• $K(X\; \backslash times\; \backslash mathbb\{S\}^2)\; =\; K(X)\; \backslash otimes\; K(\backslash mathbb\{S\}^2),$ and $K(\backslash mathbb\{S\}^2)\; =\; \backslash Z[H]/(H-1)^2$ where H is the class of the tautological bundle on $\backslash mathbb\{S\}^2\; =\; \backslash mathbb\{P\}^1(\backslash Complex),$ i.e. the Riemann sphere.
• $\backslash widetilde\{K\}^\{n+2\}(X)=\backslash widetilde\{K\}^n(X).$
• $\backslash Omega^2\; BU\; \backslash cong\; BU\; \backslash times\; \backslash Z.$

In real {{mvar|K}}-theory there is a similar periodicity, but modulo 8.

Topological {{mvar|K}}-theory has been applied in John Frank Adams’ proof of the “Hopf invariant one” problem via Adams operations.{{Cite book |last=Adams |first=John |title=On the non-existence of elements of Hopf invariant one |date=1960 |publisher=Ann. Math. 72 1}} Adams also proved an upper bound for the number of linearly-independent vector fields on spheres.{{Cite journal |last=Adams |first=John |date=1962 |title=Vector Fields on Spheres |journal=Annals of Mathematics |volume=75 |issue=3 |pages=603-632}}

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

:$ch\; :\; K^*\_\{\backslash text\{top\}\}(X)\backslash otimes\backslash Q\; \backslash to\; H^*(X;\backslash Q)$

such that

:$\backslash begin\{align\}$

K^0_{\text{top}}(X)\otimes \Q & \cong \bigoplus_k H^{2k}(X;\Q) \\

K^1_{\text{top}}(X)\otimes \Q & \cong \bigoplus_k H^{2k+1}(X;\Q)

\end{align}

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

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

{{Reflist}}

• {{cite book |last1=Atiyah |first1=Michael Francis |author1-link=Michael Atiyah |year=1989 |title=K-theory |series=Advanced Book Classics |edition=2nd |publisher=Addison-Wesley |isbn=978-0-201-09394-0 |mr=1043170}}
• {{cite book |editor1-last=Friedlander |editor1-first=Eric |editor2-last=Grayson |editor2-first=Daniel |year=2005 |title=Handbook of K-Theory |location=Berlin, New York |publisher=Springer-Verlag |isbn=978-3-540-30436-4 |mr=2182598 |doi=10.1007/978-3-540-27855-9}}
• {{cite book |last1=Karoubi |first1=Max |author1-link=Max Karoubi |year=1978 |title=K-theory: an introduction |series=Classics in Mathematics |publisher=Springer-Verlag |isbn=0-387-08090-2 |doi=10.1007/978-3-540-79890-3}}
• {{cite arXiv |last1=Karoubi |first1=Max |author1-link=Max Karoubi |year=2006 |title=K-theory. An elementary introduction | eprint =math/0602082}}
• {{cite web |last1=Hatcher |first1=Allen |authorlink1=Allen Hatcher |year=2003 |title=Vector Bundles & K-Theory |url=https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html}}
• {{cite web |last1=Stykow |first1=Maxim |authorlink1=Maxim Stykow |year=2013 |title=Connections of K-Theory to Geometry and Topology |url=https://www.researchgate.net/publication/330505308}}

Category:K-theory