group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For a compact Hausdorff space, the fundamental product theorem in topological K-theory identifies
the topological K-theory-ring of the product topological space with the 2-sphere ;
the K-theory ring of the original space with a generator for the basic line bundle on the 2-sphere adjoined:
This theorem in particular serves as a substantial step in a proof of Bott periodicity for topological K-theory (cor. below).
The usual proof proceeds by
realizing all vector bundles on via an -parameterized clutching construction;
showing that all the clutching functions are homotopic to those that are Laurent polynomials as functions on , hence products of a polynomial clutching functions with a monomial of negative power;
observing that the bundle corresponding to a clutching function of the form is equivalent to the bundle corresponding to and tensored with the th tensor product of vector bundles-power of the basic complex line bundle on the 2-sphere;
showing that some direct sum of vector bundles of the vector bundle corresponding to a polynomial clutching function with one coming from a trivial clutching function is given by a linear clutching function;
showing that bundles coming from linear clutching functions are direct sums of one coming from a trivial clutching function with the one coming from the homogeneously linear part;
Applying these steps to a vector bundle on yields a virtual sum of external tensor products of vector bundles of bundles on with powers of the basic complex line bundle on the 2-sphere. This means that the function in the fundamental product theorem is surjective. By similar means one shows that it is also injective.
For the 2-sphere with its Euclidean subspace topology, write for the basic line bundle on the 2-sphere. Its image in the topological K-theory ring satisfies the relation
(by this prop.).
Notice that is the image of in the reduced K-theory of under the splitting (by this prop.). This element
is called the Bott element of complex topological K-theory.
It follows 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 a topological space this induces the composite ring homomorphism
to the topological K-theory ring of the product topological space , where the second map is the external tensor product of vector bundles.
(fundamental product theorem in topological K-theory)
For a compact Hausdorff space, then ring homomorphism is an isomorphism.
(e.g. Hatcher, theorem 2.2)
More generally, for a complex line bundle with class and with denoting its projective bundle then
(e.g. Wirthmuller 12, p. 17)
As a special case this implies the first statement above:
For the product theorem prop. says in particular that the first of the two morphisms in the composite is an isomorphism (example below) and hence by the two-out-of-three-property for isomorphisms it follows that
(external product theorem)
For a compact Hausdorff space we have that the external tensor product of vector bundles 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. ) yields the statement of Bott periodicity of topological K-theory:
Let be a pointed compact Hausdorff space.
Then there is an isomorphism of reduced K-theory
from that of to that of its double suspension .
By this example there is for any two pointed compact Hausdorff spaces and 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 we have (this prop.) the isomorphism of the external product theorem (cor. )
becomes
Multiplying out and chasing through the constructions to see that this reduces to an isomorphism on the common summand , 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
is the isomorphism to be established.
(topological K-theory ring of the 2-sphere)
For the point space, the fundamental product theorem states that the homomorphism
is an isomorphism.
This means that the relation 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 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 has infinitely many copies of , one for each , for ).
It follows (by this prop.) that the reduced K-theory group of the 2-sphere is
Review:
Klaus Wirthmüller, section 6 (from p. 19 on) in: Vector bundles and K-theory, 2012 (pdf)
Allen Hatcher, section 2.1 (from p. 45 on) in: Vector bundles and K-theory (web)
Varvara Karpova, Section 5.2 in: Complex Topological K-Theory, 2009 (pdf, pdf)
Last revised on October 22, 2021 at 13:28:09. See the history of this page for a list of all contributions to it.