Reference for $E_{\infty}$-ness of the Chern Character

Question

I would like a reference/proof for the fact that the Chern character map: $$KU_{\mathbb{Q}} \rightarrow H\mathbb{Q}[u, u^{-1}]$$

is an $E_{\infty}$-ring map. Thank you in advance!

Answer 1

The answer really depends on one's desired choice of definitions for KU, HQ, and the Chern character itself; some definitions allow one to produce a very short definition of the Chern character as an E_∞-ring map.

For example, start with the Chern-Weil morphism (Vect^∇,⊕,⊗)→(Ω[u],+,∧), which gives a morphism of stacks of E_∞-monoids in symmetric monoidal groupoids. Using Segal's functor, we promote this to a stack of E_∞-rings. (Here u denotes the formal variable in degree 2 used in the Chern-Weil map.)

Using the universal property of the homotopy group completion, extend this to a morphism (VirtVect^∇,⊕,⊗)→(Ω[u],+,∧), where VirtVect^∇ stands for virtual vector bundles with connection, which are defined as the homotopy group completion of Vect^∇.

Now apply the concordification a.k.a. homotopification functor F↦hocolim_k F(Δ^k). (Here F(Δ^k) denotes the E_∞-ring given by evaluating the stack F on the smooth manifold Δ^k=R^k, the extended k-simplex.) The source becomes ku, the connective K-theory, and the target becomes HR[u], the periodic real Eilenberg-MacLane spectrum. We have KU=ku[β^{−1}], so to extend this morphism from ku to KU it suffices to observe that the image of the Bott class β is invertible in the target.

The advantage of the above approach with smooth stacks is that one can instantly extract variants of the E_∞-Chern character for differential cohomology, equivariant cohomology, and twisted cohomology (and any combination thereof, such as twisted equivariant differential cohomology) by replacing the last step (concordification) with the appropriate construction (the Bunke-Nikolaus-Völkl construction for differential cohomology or evaluation on orbits for equivariant cohomology) and adjusting the first step (using modules over E_∞-rings) for twisted cohomology.