group cohomology, nonabelian group cohomology, Lie group cohomology
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For $X$ a compact Hausdorff space, the fundamental product theorem in topological K-theory identifies
the topological K-theory-ring $K(X \times S^2)$ of the product topological space $X \times S^2$ with the 2-sphere $S^2$;
the K-theory ring $K(X)$ of the original space $X$ with a generator $H$ 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 $X \times S^2$ via an $X$-parameterized clutching construction;
showing that all the clutching functions are homotopic to those that are Laurent polynomials as functions on $S^1$, hence products of a polynomial clutching $p$ functions with a monomial $z^{-n}$ of negative power;
observing that the bundle corresponding to a clutching function of the form $f z^n$ is equivalent to the bundle corresponding to $f$ and tensored with the $n$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 $X \times S^2$ yields a virtual sum of external tensor products of vector bundles of bundles on $X$ 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 $S^2 \subset \mathbb{R}^3$ the 2-sphere with its Euclidean subspace topology, write $h$ for the basic line bundle on the 2-sphere. Its image in the topological K-theory ring $K(S^2)$ satisfies the relation
(by this prop.).
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 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 $X$ a topological space 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 tensor product of vector bundles.
(fundamental product theorem in topological K-theory)
For $X$ a compact Hausdorff space, then ring homomorphism $\Phi \colon K(X) \otimes \mathbb{Z}[h]/((h-1)^2) \longrightarrow K(X \times S^2)$ 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. 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 $X$ 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 $X$ be a pointed compact Hausdorff space.
Then there is an isomorphism of reduced K-theory
from that of $X$ to that of its double suspension $\Sigma^2 X$.
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. )
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
is the isomorphism to be established.
(topological K-theory ring of the 2-sphere)
For $X = \ast$ the point space, the fundamental product theorem states that the homomorphism
is an isomorphism.
This means that the relation $(h-1)^2 = 0$ 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 $\mathbb{Z}[h]/((h-1)^2)$ 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
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 09:28:09. See the history of this page for a list of all contributions to it.