nLab
differential function complex

under construction

Contents

Idea

A differential function complex (HopkinsSinger) is a Kan complex of cocycle s for generalized differential cohomology, hence for differential refinements of generalized (Eilenberg-Steenrod) cohomology theories:

roughly, given a spectrum EE representing a given cohomology theory, its differential function complex over any given smooth manifold UU is the simplicial set whose kk-simplices are triples consisting of

  • a continuous function f:U×Δ kE nf : U \times \Delta^k \to E_{n};

  • a smooth differential form ω\omega on U×Δ kU \times \Delta^k whose corresponding real cohomology class (under the de Rham theorem) is that of the pullback of the real cohomology classes of EE along ff;

  • an explicit coboundary in real cohomology exhibiting this fact.

(More precisely, in order for this construction to yield not just a single simplicial set (which will be a Kan complex) but a suitable spectrum object, there are conditons on the dependency of ω\omega on the tangent vectors to the simplex.)

When applied to the Eilenberg-MacLane spectrum KK\mathbb{Z} this construction reproduces, on cohomology classes, ordinary differential cohomology. Applied to the classifying space BUB U of topological K-theory it gives differential K-theory.

See also at differential cohomology diagram –Hopkins-Singer coefficients.

Definitions

Cocycles with values in graded vector spaces

For the present purposes it will be convenient to collect cocycles of various degrees together to a single cocycle. For that purpose we make the following simple definition.

Definition

For V=V V = V^\bullet a graded vector space over the real numbers set

  • for EE a topological space:

    C (E,V) n:= i+j=nC i(E,V j) C^\bullet(E, V)^n := \oplus_{i + j = n} C^i(E, V^j)
  • and so on

(…)

Differential functions

Definition

For

a differential function on a smooth manifold UU with values in (E,ι)(E,\iota) is a triple (c,h,ω)(c,h,\omega) with

  • c:UEc : U \to E a continuous map;

  • ωΩ n(S)\omega \in \Omega^n(S) a smooth differential form on SS;

  • hC n1(U,)h \in C^{n-1}(U,\mathbb{R}) a cochain in real cohomology on (the topological space underlying) UU;

such that in the abelian group Z n(S,)Z^n(S,\mathbb{R}) of singular cochains the equation

ω=c *ι+δh \omega = c^*\iota + \delta h

holds, where

This is (HopkinsSinger, def.4.1).

In words this is: a continuous map to the topological space together with a smooth refinement of the pullback of the chosen singular cochain.

Differential function complexes

Definition

For

the differential function complex

(E,ι) U (E,\iota)^U

of all differential functions S(X,ι)S \to (X,\iota) is the simplicial set whose kk-simplices are differential functions, def, 2

U×Δ Top k(E,ι). U \times \Delta^k_{Top} \to (E,\iota) \,.

For applications one needs certain sub-complex of this, filtered by the number of legs that ω\omega has along the simplices.

Definition

For ss \in \mathbb{N} write

  • filt sΩ (U×Δ k)filt_s \Omega^\bullet(U \times \Delta^k)

    for the sub-simplicial set of differential forms that vanish when evaluated on more than ss vector fields tangent to the simplex;

  • filt s(X,ι) S(X,ι) Sfilt_s (X,\iota)^S \subset (X,\iota)^S

    for the sub-simplicial set of those differential functions whose differential form component is in filt sΩ (U×Δ k)filt_s \Omega^\bullet(U \times \Delta^k).

This is (HopkinsSinger, def. 4.5).

Proposition

The complex filt s(E,ι) Ufilt_s (E,\iota)^U is (up to equivalence, of course) the homotopy pullback

filt s(E,ι) U filt sΩ cl n(U×Δ ,𝒱) SingE U Z (U×Δ ,𝒱) \array{ filt_s (E,\iota)^U &\to& filt_s \Omega^n_{cl}(U \times \Delta^\bullet, \mathcal{V}) \\ \downarrow && \downarrow \\ Sing E^U &\to& Z^\bullet(U \times \Delta^\bullet, \mathcal{V}) }

in sSet (regarded as equipped with its standard model structure on simplicial sets).

Here E UE^U is the internal hom in Top and Sing()Sing(-) denotes the singular simplicial complex.

The following proposition gives the simplicial homotopy groups of these differential function complexes in dependence of the parameter ss.

Proposition

We have generally

π kZ(S×Δ Diff ,𝒱)=H nm(S;𝒱) \pi_k Z(S \times \Delta^\bullet_{Diff}, \mathcal{V}) = H^{n-m}(S; \mathcal{V})

(for instance by the Dold-Kan correspondence).

The simplicial homotopy groups of filt sΩ cl n(S×Δ Diff )filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff}) are

π kfilt sΩ cl n(S×Δ Diff )={H dR nk(S,𝒱) |k<s Ω cl(S;𝒱) ns |k=s 0 |k>s}. \pi_k filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff}) = \left\{ \array{ H_{dR}^{n-k}(S, \mathcal{V}) & | k \lt s \\ \Omega_{cl}(S; \mathcal{V})^{n-s} & | k = s \\ 0 & | k \gt s } \right\} \,.

This implies isomorphisms

π kfilt s(X;ι) S{π kX S |k<s H nk1(S;𝒱)/π k+1X S|k>s. \pi_k filt_s(X; \iota)^S \stackrel{\simeq}{\to} \left\{ \array{ \pi_k X^S & | k \lt s \\ H^{n-k-1}(S; \mathcal{V})/ \pi_{k+1} X^S | k \gt s } \right. \,.

This appears as HopkinsSinger, p. 36 and corollary D15.

Differential EE-cohomology

Let E E_\bullet be an Omega-spectrum. Let ι \iota_\bullet be the canonical Chern character class (…).

Proposition

For SS a smooth manifold, and ss \in \mathbb{N}, the sequence of differential function complexes, def. 3,

filt s+n(E n;ι n) SΩfilt s+(n+1)(E n+1;ι n+1) S filt_{s + n}(E_n; \iota_n)^S \stackrel{\simeq}{\to} \Omega filt_{s + (n + 1)}(E_{n+1}; \iota_{n+1})^S

forms an Omega-spectrum.

This is the differential function spectrum for EE, SS, ss.

This is ([HopkinsSinger, section 4.6]).

Definition

The differential EE-cohomology group of the smooth manifold SS in degree nn is

H diff n(S,E):=π 0filt 0(E nι n) S H_{diff}^n(S,E) := \pi_0 filt_0(E_n \iota_n)^S

This is (HopkinsSinger, def. 4.34).

Properties

Homotopy groups

For reference, we repeat from above the central statements about the homotopy types of the differential function complexes, def. 3.

Proposition

For EE an Omega-spectrum, SS a smooth manifold, we have for all s,ns,n \in \mathbb{N}, a weak homotopy equivalence

Ωfilt s+1(E n;ι n) Sfilt s(E n1;ι n1) S, \Omega filt_{s+1}(E_{n}; \iota_{n})^S \stackrel{\simeq}{\to} filt_{s}(E_{n-1}; \iota_{n-1})^S \,,

identifying the loop space object (at the canonical base point) of the differential function complex of E nE_{n} at filtration level s+1s+1 with that differential function complex of E n1E_{n-1} at filtration level ss.

Relation to differential cohomology in cohesive (,1)(\infty,1)-toposes

The following is a simple corollary or slight rephrasing of some of the above constructions, which may serve to show how differential function complexes present differential cohomology in the cohesive (∞,1)-topos of smooth ∞-groupoids.

Proposition

For E E_\bullet a spectrum as above,
we have an (∞,1)-pullback square

filt 0(E n;ι n) () iΩ cl n i() DiscE n iB n i disc. \array{ filt_0 (E_n; \iota_n)^{(-)} &\to& \prod_i \Omega^{n_i}_{cl}(-) \\ \downarrow && \downarrow \\ Disc E_n & \stackrel{}{\to} & \prod_i \mathbf{B}^{n_i} \mathbb{R}_{disc} } \,.
Proof

By prop. 2 we have that

  • filt (E;ι n) SSingX Sfilt_\infty (E; \iota_n)^S \simeq Sing X^S;

  • filt 0Ω cl(S×Δ )Ω cl(S)filt_0 \Omega_{cl}(S \times \Delta^\bullet) \simeq \Omega_{cl}(S).

The statement then follows with the pasting law for homotopy pullbacks

filt 0(E n;ι n) S Ω cl n(S;𝒱) filt (E n;ι n) S filt Ω cl(S×Δ ;𝒱) SingX S Z(S×Δ ;𝒱). \array{ filt_0 (E_n; \iota_n)^S &\to& \Omega^n_{cl}(S; \mathcal{V}) \\ \downarrow && \downarrow \\ filt_\infty (E_n; \iota_n)^S &\to& filt_\infty \Omega_{cl}(S \times \Delta^\bullet; \mathcal{V}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Sing X^S &\to& Z(S \times \Delta^\bullet; \mathcal{V}) } \,.

(…)

Examples

Line bundles with connection

Let X=U(1)K(,2)X = \mathcal{B} U(1) \simeq K(\mathbb{Z},2) be the Eilenberg-MacLane space that is the classifying space for U(1)U(1)-principal bundles. It carries the canonical cocycle ι:=Id:U(1)U(1)K(,2)\iota := Id : \mathcal{B}U(1) \to \mathcal{B}U(1) \simeq K(\mathbb{Z},2) representing in H 2(X,)H^2(X,\mathbb{Z}) the class of the universal complex line bundle LXL \to X on XX.

Accordingly, for c:SU(1)c : S\to \mathcal{B}U(1) a continuous map, we have the corresponding line bundle c *Lc^* L on SS.

One checks (…details…Example 2.7 in HopSin) that a refinement of cc to a differential function (c,ω,h)(c,\omega,h) corresponds to equipping c *Lc^* L with a smooth connection.

Now consider ((c,ω,h)(c,ω,h))filt 0(U(1),Id) S((c,\omega,h) \to (c',\omega', h')) \in filt_0 (\mathcal{B}U(1),Id)^S a morphism between two such (U(1),Id)(\mathcal{B}U(1),Id)-differential functions. By definition this is now a U(1)U(1)-principal bundle L^\hat L with connection on S×Δ Diff iS \times \Delta^i_{Diff}, whose curvature form ω^Ω 2(S×Δ Diff 1)\hat \omega \in \Omega^2(S \times \Delta^1_{Diff}) is of the form gω˜g \cdot \tilde \omega, where ω˜\tilde \omega is a 2-form on SS and gg is a smooth function on Δ Diff 1\Delta^1_{Diff}, both pulled back to S×Δ Diff 1S \times \Delta^1_{Diff} and multiplied there.

But since ω^\hat \omega is necessarily closed it follows with d(gω˜)=dtgtω˜+gd Sω˜d (g \wedge \tilde \omega) = d t \frac{\partial g}{\partial t} \wedge \tilde \omega + g \wedge d_{S} \tilde \omega that gg is actually constant.

This means that that the parallel transoport of the connection ^\hat \nabla on S×Δ Diff 1S \times \Delta^1_{Diff} induces a insomorphism between the two line bundles on SS over the endpoints of S×Δ Diff 1S \times \Delta^1_{Diff} that respects the connections.

Differential K-cocycles

(…)

Higher filtration degree

Example

For E=HE = H \mathbb{Z} the Eilenberg-MacLane spectrum, prop. 4 states that filt 1(H n+1;ι n) Sfilt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S is an n-groupoid such that the automorphisms of the 0-object form ordinary differential cohomology in degree nn.

Ωfilt 1(H n+1;ι n) SH diff n(S). \Omega filt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S \simeq \mathbf{H}_{diff}^n(S) \,.
Remark

Example 1 for n=4n = 4 plays a central role in the description of T-duality by twisted differential K-theory in (KahleValentino).

References

Differential function complexes were introduced and studied in

For further references see differential cohomology.

Revised on April 27, 2014 08:31:40 by Urs Schreiber (82.113.98.143)