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algebraic K-theory of smooth manifolds

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Context

Higher algebra

Differential cohomology

Contents

Definition

The construction that sends a smooth manifold XX to the algebraic K-theory spectrum K(C (X,))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

KStab(SmoothGrpd)T *SmoothGrpd, \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 K\mathbf{K} is the topological K-theory spectrum kuku (Bunke-Nikolaus-Voelkl 13, lemma 6.3, Bunke 14, (48) with def. 2.21):

ʃKku. ʃ \mathbf{K} \simeq ku \,.

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

Regulator and relation to differential K-theory

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

K reg DD η ʃ η ʃ ku ch DD per \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):

reg conn:Kku conn. \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,K][X,\mathbf{K}] out of a smooth manifold XX, hence to the moduli stacks (here: moduli spectra) of algebraic K-theory cocycles on XX.

The regulator from above induces a map

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

(…)

References

Last revised on November 7, 2015 at 10:07:45. See the history of this page for a list of all contributions to it.