nLab fundamental product theorem in topological K-theory

Idea

For $X$ a compact Hausdorff space, the fundamental product theorem in topological K-theory identifies

1. the topological K-theory-ring $K(X \times S^2)$ of the product topological space $X \times S^2$ with the 2-sphere $S^2$;

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:

$$K(X) \otimes_{\mathbb{Z}} \mathbb{Z}[H]/(H-1)^2 \overset{\simeq}{\longrightarrow} K(X \times S^2) \,.$$

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

1. realizing all vector bundles on $X \times S^2$ via an $X$-parameterized clutching construction;

2. 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;

3. 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;

4. 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;

5. 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.

Statement

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

$$2 h = h^2 + 1 \;\;\Leftrightarrow\;\; (h-1)^2 = 0$$

(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

$$h - 1 \in \tilde K_{\mathbb{C}}(S^2)$$

is called the Bott element of complex topological K-theory.

It follows that there is a ring homomorphism of the form

$$\array{ \mathbb{Z}[h]/\left( (h-1)^2 \right) &\overset{}{\longrightarrow}& K(S^2) \\ h &\overset{\phantom{AAA}}{\mapsto}& h }$$

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

$$\array{ \Phi \colon & K(X) \otimes \mathbb{Z}[h]/((h-1)^2) & \longrightarrow & K(X) \otimes K(S^2) & \overset{\boxtimes}{\longrightarrow} & K(X \times S^2) \\ & (E, h) &\overset{\phantom{AAA} }{\mapsto}& (E,H) &\overset{\phantom{AAA}}{\mapsto}& (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H) }$$

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.

Proposition

(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)

Remark

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

$$K(X)[h]/((h-1)(l \cdot h -1)) \simeq K(P(1 \oplus L))$$

(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

Corollary

(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

$$\boxtimes \;\colon\; K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2)$$

is an isomorphism in topological K-theory.

Bott periodicity

When restricted to reduced K-theory then the external product theorem (cor. ) yields the statement of Bott periodicity of topological K-theory:

Corollary

(Bott periodicity)

Let $X$ be a pointed compact Hausdorff space.

Then there is an isomorphism of reduced K-theory

$$(h-1) \widetilde \boxtimes (-) \;\colon\; \tilde K(X) \overset{\simeq}{\longrightarrow} \tilde K(\Sigma^2 X)$$

from that of $X$ to that of its double suspension $\Sigma^2 X$.

Proof

By this example there is for any two pointed compact Hausdorff spaces $X$ and $Y$ an isomorphism

$$\tilde K(Y \times X) \simeq \tilde K(Y \wedge X) \oplus \tilde K(Y) \oplus \tilde K(X)$$

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. )

$$K(S^2) \otimes K(X) \underoverset{\simeq}{\boxtimes}{\longrightarrow} K(S^2 \times X)$$

becomes

$$\left( \tilde K(S^2) \oplus \mathbb{Z} \right) \otimes \left( \tilde K(X) \oplus \mathbb{Z} \right) \;\simeq\; \left( \tilde K(S^2 \times X) \oplus \mathbb{Z} \right) \simeq \left( \tilde K(S^2 \wedge X) \oplus \tilde K(S^2) \oplus \tilde K(X) \oplus \mathbb{Z} \right) \,.$$

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

$$\tilde K(S^2) \otimes \tilde K(X) \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow} \tilde K(S^2 \wedge X) = \tilde K(\Sigma^2 X) \,,$$

where on the right we used that smash product with the 2-sphere is the same as double suspension.

Finally there is an isomorphism

$$\array{ \mathbb{Z} &\underoverset{\simeq}{ \beta }{\longrightarrow}& \tilde K_{\mathbb{C}}(S^2) \\ 1 &\overset{\phantom{AAA}}{\mapsto}& (h-1) }$$

(example ). The composite

$$\array{ \tilde K_{\mathbb{C}}(X) & \simeq \mathbb{Z} \otimes \tilde K_{\mathbb{C}}(X) \overset{ \beta \otimes id }{\longrightarrow} \tilde K_{\mathbb{C}}(S^2) \otimes \tilde K_{\mathbb{C}}(X) \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow} & \tilde K_{\mathbb{C}}(S^2 \wedge X) = \tilde K_{\mathbb{C}}(\Sigma^2 X) \\ E - rk_x(E) &\overset{\phantom{AAAA}}{\mapsto}& (h-1) \widetilde \boxtimes (E - rk_x(E)) }$$

is the isomorphism to be established.

Examples

Example

(topological K-theory ring of the 2-sphere)

For $X = \ast$ the point space, the fundamental product theorem states that the homomorphism

$$\array{ \mathbb{Z}[h]/((h-1)^2) &\longrightarrow& K(S^2) \\ h &\mapsto& h }$$

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,

$$K(S^2) \simeq \mathbb{Z} \oplus \mathbb{Z} = \langle 1, h\rangle$$

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

$$\tilde K(S^2) \simeq \mathbb{Z} \,.$$

Related concepts

• equivariant K-theory of projective G-space

References

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)