nLab
differential function complex

Contents

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,

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. ,

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. .

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. 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. 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 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.

Last revised on April 27, 2014 at 08:31:40. See the history of this page for a list of all contributions to it.