nLab algebraic K-theory of smooth manifolds

Definition

The construction that sends a smooth manifold $X$ to the algebraic K-theory spectrum $K(C^\infty(X,\mathbb{C}))$ of its ring of smooth functions (with values in the complex numbers) presents (after infinity-stackification) a sheaf of spectra on the site of smooth manifolds, hence a smooth spectrum

$$\mathbf{K} \in Stab(Smooth\infty Grpd) \simeq T_\ast Smooth \infty Grpd \,,$$

i.e. an object of the tangent cohesive (∞,1)-topos of Smooth∞Grpd. (See also this definition at differential cohomology hexagon.)

Properties

Shape and relation to topological K-theory

The shape of this $\mathbf{K}$ is the topological K-theory spectrum $ku$ (Bunke-Nikolaus-Voelkl 13, lemma 6.3, Bunke 14, (48) with def. 2.21):

$$ʃ \mathbf{K} \simeq ku \,.$$

Hence $\mathbf{K}$ is a differential cohomology refinement of $ku$, a form of differential K-theory.

Regulator and relation to differential K-theory

There is also the more standard differential K-theory refinement $\mathbf{ku}_{conn}$ of $ku$ (Hopkins-Singer 05, Bunke-Nikolaus-Voelkl 13) which is obtained by pulling back suitable sheaves of ($\mathbb{C}$-valued) differential forms $\mathbf{DD}^-$ along the usual Chern character map $ch \colon ku \longrightarrow DD^{per}$. This Chern character lifts through the shape modality to a regulator map (Bunke 14, (50))

$$\array{ \mathbf{K} &\stackrel{\mathbf{reg}}{\longrightarrow}& \mathbf{DD}^- \\ \downarrow^{\mathrlap{\eta^{ʃ}}} && \downarrow^{\mathrlap{\eta^{ʃ}}} \\ ku &\stackrel{ch}{\longrightarrow}& DD^{per} }$$

Moreover, this induces a differential regulator (BNV 13, p.40 and example 6.9, Bunke 14, def. 2.29):

$$\mathbf{reg}_{conn} \;\colon\; \mathbf{K} \longrightarrow \mathbf{ku}_{conn} \,.$$

See also this proposition at differential cohomology diagram.

Moduli stacks

In (Bunke 14) all this is generalized to the mapping spaces $[X,\mathbf{K}]$ out of a smooth manifold $X$, hence to the moduli stacks (here: moduli spectra) of algebraic K-theory cocycles on $X$.

The regulator from above induces a map

$$[X,\mathbf{reg}_{conn}] \;\colon\; [X,\mathbf{K}] \longrightarrow [X,\mathbf{ku}_{conn}] \,.$$

(…)

References

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, section 6 of Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures October 2014 (arXiv:1311.3188)

• Ulrich Bunke, A regulator for smooth manifolds and an index theorem (arXiv:1407.1379)

• Ulrich Bunke, Smooth aspects of algebraic K-theory (pdf)