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differential cohomology diagram

Context

Differential cohomology

Stable Homotopy theory

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Contents

Idea

In abstract generality, a differential cohomology diagram is the hexagonal diagram in a cohesive (∞,1)-topos formed by the two fracture squares of the unit and counit of, respectively, the shape modality Π\Pi and the flat modality \flat. The exactness properties of this diagram for any stable homotopy type E^\hat E in the corresponding tangent cohesive (∞,1)-topos express E^\hat E as being the coefficients for a differential cohomology refinement by differential form data dRE^\flat_{dR} \hat E of the generalized (Eilenberg-Steenrod) cohomology theory which is represented by the shape spectrum EΠ(E^)E \coloneqq \Pi(\hat E).

Historically, it had been shown in (Simons-Sullivan 07) for ordinary differential cohomology and in (Freed-Lott 10, Simons-Sullivan 08) for differential K-theory that these cohomology theories are characterized as sitting in the middle of a hexagonal diagram of interlocking exact sequences of cohomology groups, which expresses, on the one hand, how every differential cohomology class has underlying it a non-differential cohomology class (“of a principal ∞-bundle”) as well as a curvature differential form datum, and, on the other hand, how the special cases of trivial underlying classes equipped with differential form datum and of flat differential form data sit inside the differential cohomology classes.

At this schematic conceptual level a differential cohomology diagram looks as follows (where all unlabelled arrows are meant to be read as “evident inclusions”);

connectionformsontrivialbundles deRhamdifferential curvatureforms curvature deRhamtheorem flatdifferentialforms geometricbundleswithconnection rationalizedbundle topol.class Cherncharacter geometricbundleswithflatconnection comparison/regulatormap shapeofbundle \array{ && {{connection\;forms}\atop{on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature}\atop{forms}} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ {{flat}\atop{differential\;forms}} && && {{geometric\;bundles}\atop{with\;connection}} && && {{rationalized}\atop{bundle}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles}\atop{with\;flat\;connection}} && \underset{comparison/regulator\;map}{\longrightarrow} && {{shape}\atop{of\;bundle}} }

One characteristic property is that the two outer sequences are exact sequences. This expresses (at this rough schematic level) for instance (for the upper part) that the connections AA on trivial bundles whose curvature vanishes in that dA=0\mathbf{d}A = 0, are exactly the flat connections; as well as (for the lower part) that bundles with flat connections have torsion Chern-clases.

So a differential cohomology theory would be one whose cocycles/cohomology classes have the interpretation of (stable) principal ∞-bundles with connection such as in the middle of this diagram.

The characterization/construction of differential cohomology via homotopy fiber products (of mapping spectra with differential form data) due to (Hopkins-Singer 02) provides an incarnation of this kind of diagram genuinely in stable homotopy theory, so that the outer parts are indeed homotopy fiber sequences and the two squares are homotopy cartesian (are homotopy pullbacks/homotopy pushouts).

In (Bunke-Nikolaus-Völkl 13) it was observed (see Schreiber 13, section 4.1.2 for the generality in which we present this here) that every stable homotopy type E^\hat E in a cohesive (∞,1)-topos H\mathbf{H} canonically sits inside a diagram of this form, being formed from the fracture squares of the units and counits of the shape modality Π\Pi and the flat modality \flat, which in the right part are interpreted as the Maurer-Cartan form θ E\theta_E and the Chern character ch Ech_E:

Π dRE^ d dRE^ θ E^ Π dRE^ E^ Π dRE^ ch E E^ ΠE^. \array{ && \Pi_{dR} {\hat E} && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}{\hat E} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_{\hat E}}} && \searrow \\ \Pi_{dR} \flat {\hat E} && && {\hat E} && && \Pi \flat_{dR} \hat E \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_E}} \\ && \flat {\hat E} && \longrightarrow && \Pi \hat E } \,.

This is theorem 1 below.

For the special case that H=TSmoothGrpd\mathbf{H} = T Smooth\infty Grpd is the tangent (∞,1)-topos of smooth ∞-groupoids (∞-stacks over the site of smooth manifolds) this subsumes the cases mentioned above. But there are many examples of cohomology theories not of the form of (Hopkins-Singer 02) but represented by stable homotopy types in a cohesive (∞,1)-topos (see Schreiber 13, BunkeNikolausVölkl 13) and hence fitting into such a diagram, where the interpretation of the pieces of the diagram is just as it should be.

Therefore it makes sense to define generally that a differential cohomology diagram is the above combined fracture squares with its outer homotopy fiber sequences for shape modality and flat modality in any cohesive (∞,1)-topos.

Definition/Construction

Differential coefficients and Maurer-Cartan forms

Let H\mathbf{H} be a cohesive (∞,1)-topos, write THT\mathbf{H} for its tangent cohesive (∞,1)-topos and write T *HTHT_\ast \mathbf{H} \hookrightarrow T \mathbf{H} for the stable (∞,1)-category of spectrum objects inside it.

As usual, we use the following notation.

Definition (Notation)

Write

restricted to Stab(H)THStab(\mathbf{H}) \hookrightarrow T \mathbf{H}, respectively.

Given EE \in Spectra we say that E^Stab(H)\hat E \in Stab(\mathbf{H}) with an equivalence

Π(E^)E \Pi(\hat E) \simeq E

is a cohesive refinement or differential refinement of EE.

Write

Notice that on stable homotopy types we have

dRAcofib(AA)=A/(A). \flat_{dR}A \simeq cofib(\flat A \to A) = A/(\flat A) \,.
Remark

Warning: Elsewhere we often write dRΣ\flat_{dR} \Sigma for dR\flat_{dR} above and Π dRΩ\Pi_{dR}\Omega for what is Π dR\Pi_{dR} here. That other convention has its advanages in the context of unstable cohesion. Here with stable cohesion the present convention is more natural.

Remark

As discussed at structures in a cohesive ∞-topos – de Rham cohomology there (and as is discussed below at de Rham coefficients) for AT *HA \in \mathbf{T}_\ast H a cohesive homotopy type then we may think of Π dRA\Pi_{dR} A and of dRA\flat_{dR} A as being the de Rham complex with coefficients in Π( dRΣA)\Pi(\flat_{dR} \Sigma A), truncated to negative degree and to non-negative degree, respectively; the canonical map

Π dRA d dRA A \array{ \Pi_{dR}A && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}A \\ & \searrow && \nearrow \\ && A }

interpreting as the de Rham differential.

Beware that this is a very general conceptualization of de Rham coeffcients. In standard examples dR\flat_{dR} does come from traditional differential form data (see at Deligne coefficients and at Hopkins-Singer coefficients below), but generally it may have quite different looking models. But in any case dRΣA\flat_{dR} \Sigma A always has the interpretation of the home of the curvature forms of cohomology with coefficients in AA, which makes thinking of dR\flat_{dR} as producing generalized form data useful.

Chern character and the differential fracture squares

Definition

The shape of the Maurer-Cartan form θ\theta, def. 1, we call the Chern character

chΠθ. ch \coloneqq \Pi \theta \,.

Its component on E^T *H\hat E\in T_\ast \mathbf{H} we write ch E^ch_{\hat E} or ch Ech_E, for short.

Proposition

(differential fracture square)

For every E^T *H\hat E \in T_\ast \mathbf{H} the naturality square

E^ dRE^ (pb) Π(E^) ch E Π dRE^ \array{ \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(\hat E) &\stackrel{ch_E}{\longrightarrow}& \Pi \flat_{dR} \hat E }

(of the shape modality applied to the homotopy cofiber of the counit of the flat modality) is an (∞,1)-pullback square (hence also an (∞,1)-pushout).

Dually, also the square

Π dRE^ Π dRE^ E^ E^ \array{ \Pi_{dR} \flat \hat E &\longrightarrow& \Pi_{dR} \hat E \\ \downarrow && \downarrow \\ \flat \hat E & \longrightarrow & \hat E }

is homotopy cartesian.

This fact was observed in (Bunke-Nikolaus-Völkl 13, prop. 3.5). It may be thought of as an incarnation of the concept of a fracture theorem.

Proof

By cohesion and stability we have the diagram

E^ E^ dRE^ Π(E^) Π(E^) Π( dRE^) \array{ \flat \hat E &\longrightarrow & \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E \\ \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow \\ \Pi(\flat \hat E) &\longrightarrow& \Pi(\hat E) &\stackrel{}{\longrightarrow}& \Pi(\flat_{dR} \hat E) }

where both rows are homotopy fiber sequences. By cohesion the left vertical map is an equivalence. The claim now follows with the homotopy fiber characterization of homotopy pullbacks.

The second statement follows dually:

Π dRE^ Π dRE^ Π dR dRE^ E^ E^ dRE^. \array{ \Pi_{dR} \flat \hat E &\longrightarrow& \Pi_{dR} \hat E &\longrightarrow& \Pi_{dR} \flat_{dR} \hat E \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ \flat \hat E & \longrightarrow & \hat E &\longrightarrow&\flat_{dR} \hat E } \,.

Notice that generally dRE^\flat_{dR} \hat E is a flat modalityanti-modal type” in that it is annihilated by \flat:

( dRE^)0. \flat (\flat_{dR} \hat E) \simeq 0 \,.

(\flat-anti-modal types are called “pure” in (Bunke-Nikolaus-Völkl 13, prop. 3.5)).

The following says that not only does, by prop.1, the (Π)(\Pi \dashv \flat)-fracture square exhibit any stable cohesive homotopy type as the pullback of a \flat-anti-modal type along a map of Π\Pi-modal types into its shape, but that conversely all homotopy pullbacks of this form are (Π)(\Pi \dashv \flat)-fracture squares.

Lemma

For

  1. EE a shape-modal type

  2. Ω 0\Omega^{\bullet \geq 0} a flat-anti-modal type;

  3. EfΠΩ >0E \stackrel{f}{\longrightarrow} \Pi \Omega^{\bullet \gt 0} a morphism;

then the homotopy pullback square for the homotopy fiber product

E^E×ΠΩ 0Ω 0 \hat E \coloneqq E \underset{\Pi\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0}

is equivalently the (Π)(\Pi\dashv \flat)-fracture square of E^\hat E according to prop. 1:

E^ dRE^ Ω 0 E ΠE^ fch E^ Π dRE^ ΠΩ 0. \array{ & \hat E &\longrightarrow& \flat_{dR} \hat E & \simeq \Omega^{\bullet \geq 0} \\ & \downarrow && \downarrow \\ E \simeq & \Pi \hat E &\stackrel{f \simeq ch_{\hat E}}{\longrightarrow}& \Pi \flat_{dR} \hat E & \simeq \Pi \Omega^{\bullet \geq 0} } \,.
Proof

First we observe that indeed Ω 0 dRE^\Omega^{\bullet\geq 0}\simeq \flat_{dR} \hat E. For that consider the following morphisms of homotopy pullback diagrams

0 Ω 0 Ω 0 Ω 0 ΠΩ 0 ΠΩ 0 0 f E E 0 lim() E^ E^ dRE^ \array{ 0 \simeq & \flat \Omega^{\bullet \geq 0} &\longrightarrow& \Omega^{\bullet \geq 0} &\longrightarrow& \Omega^{\bullet \geq 0} \\ & \downarrow && \downarrow && \downarrow \\ & \flat \Pi \Omega^{\bullet \geq 0} &\stackrel{\simeq}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} &\longrightarrow& 0 \\ & \uparrow && \uparrow^{\mathrlap{f}} && \uparrow \\ & \flat E &\stackrel{\simeq}{\longrightarrow}& E &\longrightarrow& 0 \\ \\ \underset{\leftarrow}{\lim}(-) \simeq& \flat \hat E &\longrightarrow& \hat E &\longrightarrow& \flat_{dR}\hat E }

Here in the middle column we are showing the homotopy fiber product defining E^\hat E. On the left we have its image under \flat, with the \flat-counits running horizontally and using that \flat preserves (∞,1)-limits and that EE and ΠΩ 0\Pi \Omega^{\bullet \geq 0} are \flat-modal types by assumption and by cohesion, while Ω 0\Omega^{\bullet \geq 0} is \flat-anti modal by assumption. This and using that by stability all finite (∞,1)-limits and finite (∞,1)-colimits commute produces the homotopy cofiber morphisms going to the right.

This shows that Ω 0 dRE^\Omega^{\bullet \geq 0} \simeq \flat_{dR} \hat E and that the projection E×Π dRΩ 0Ω 0Ω 0E \underset{\Pi_{dR}\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} \to \Omega^{\bullet \geq 0} is equivalently the dR\flat_{dR}-unit on E^\hat E.

Applying the shape modality to the right half of this diagram, using that by stability it preserves the homotopy pullbacks gives

ΠΩ 0 ΠΩ 0 ΠΩ 0 0 f E 0 lim() E ch E^ Π dRE^. \array{ & \Pi \Omega^{\bullet \geq 0} &\stackrel{\simeq}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} \\ & \downarrow^{\mathrlap{\simeq}} && \downarrow \\ & \Pi \Omega^{\bullet \geq 0} &\longrightarrow& 0 \\ & \uparrow^{\mathrlap{f}} && \uparrow \\ & E &\longrightarrow& 0 \\ \\ \underset{\leftarrow}{\lim}(-) \simeq & E &\stackrel{ch_{\hat E}}{\longrightarrow}& \Pi \flat_{dR}\hat E } \,.

This shows that ch E^fch_{\hat E} \simeq f.

Finally, the dual argument

Ω 0 ΠΩ 0 ΠΩ 0 ΠΩ 0 E ΠE E lim() E^ Π(E^) E \array{ & \Omega^{\bullet \geq 0} &\stackrel{}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} \\ & \downarrow && \downarrow^{\mathrlap{\simeq}} \\ & \Pi \Omega^{\bullet \geq 0} &\stackrel{}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} \\ & \uparrow && \uparrow \\ & E &\stackrel{}{\longrightarrow}& \Pi E & \simeq E \\ \\ \underset{\leftarrow}{\lim}(-) \simeq & \hat E &\longrightarrow& \Pi(\hat E) & \simeq E }

shows that the other projection E×Π dRΩ 0Ω 0EE \underset{\Pi_{dR}\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} \to E is equivalently the shape-unit of E^\hat E.

The following proposition says that the construction in lemma 1 extends to a decomposition of the (∞,1)-category of cohesive stable types as a homotopy fiber product of the (∞,1)-category of moprhisms of stable shape-modal types with that of flat-anti-modal types:

Proposition

There is an (∞,1)-pullback of (∞,1)-categories of the form

Stab(H) dRΣ ker() ch Π Spectra I cod Spectra, \array{ Stab(\mathbf{H}) &\stackrel{\flat_{dR} \Sigma}{\longrightarrow}& ker(\flat) \\ \downarrow^{\mathrlap{ch}} && \downarrow^{\mathrlap{\Pi}} \\ Spectra^I &\stackrel{cod}{\longrightarrow}& Spectra } \,,

where the bottom map is the codomain fibration of the (∞,1)-category of spectra, Π\Pi denotes the shape modality restricted to stable \flat-anti-modal types, chch assigns the Chern character map of def. 2 and dRΣ\flat_{dR} \Sigma assigns de Rham coefficients as in def. 1.

(Bunke-Nikolaus-Völkl 13, prop. 3.5)

Dually:

Proposition

For every AT *HA \in T_\ast \mathbf{H} the naturality square

(Π dRΣ 1A) Π dR(Σ 1A) A A \array{ \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Sigma^{-1} A) \\ \downarrow && \downarrow \\ \flat A &\stackrel{}{\longrightarrow}& A }

(of the flat modality applied to the homotopy fiber of the unit of the shape modality) is an (∞,1)-pullback square.

Proof

As before but dually, the diagram extends to a morphism of homotopy cofiber diagrams of the form

(Π dRA) Π dR(A) A A Π(A) Π(A), \array{ \flat(\Pi_{dR} A) &\longrightarrow & \Pi_{dR}(A) \\ \downarrow && \downarrow \\ \flat A &\stackrel{}{\longrightarrow}& A \\ \downarrow && \downarrow \\ \flat \Pi(A) &\stackrel{\simeq}{\longrightarrow}& \Pi(A) } \,,

and by cohesion the bottom horizontal morphism is an equivalence.

The hexagon diagram

Combining these two statements yields the following (Bunke-Nikolaus-Völkl 13).

Theorem

For H\mathbf{H} a cohesive (∞,1)-topos with shape modality Π\Pi and flat modality \flat, then for every stable homotopy type AStab(H)THA \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H} the canonical hexagon diagram

Π dRA d dRA θ A Π dRA a A b Π dRA ch A A ΠA, \array{ && \Pi_{dR} A && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR} A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \Pi_{dR} \flat A && \Downarrow^{\mathrlap{a}} && A && \Downarrow^{\mathrlap{b}} && \Pi \flat_{dR} A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_A}} \\ && \flat A && \longrightarrow && \Pi A } \,,

formed from the Π\Pi-unit and \flat-counit – the “differential cohomology hexagon” – is homotopy exact in that

  1. the two squares are homotopy pullback squares (“fracture squares”);

  2. the two diagonals are the homotopy fiber sequences of the Maurer-Cartan form θ A\theta_A and its dual;

  3. the bottom morphism is the canonical points-to-pieces transform;

  4. the top and bottom outer sequences are long homotopy fiber sequences.

Proof

Only the last statement remains to be shown, for that use the pasting law: this gives the following diagram in which every square and every pasting rectangle is a homotopy pullback

Π dRA A 0 f 1 a 1 Π dRA f 2 A θ A dRA b f 3 0 ΠA Π dRA. \array{ \Pi_{dR} \flat A &\to & \flat A &\to& 0 \\ \downarrow^{\mathrlap{f_1}} &\swArrow_{\mathrlap{a^{-1}}}& \downarrow && \downarrow \\ \Pi_{dR} A &\stackrel{f_{2}}{\to}& A &\stackrel{\theta_A}{\to}& \flat_{dR} A \\ \downarrow && \downarrow & \swArrow_{\mathrlap{b}} & \downarrow^{\mathrlap{f_3}} \\ 0 &\to& \Pi A &\to& \Pi \flat_{dR} A } \,.

This exhibits the sequence f 1θ Af 2f 3\stackrel{f_1}{\to}\stackrel{\theta_A \circ f_{2}}{\longrightarrow} \stackrel{f_3}{\to}, which is the top part of the hexagon, as a homotopy fiber sequence.

The dual argument shows that the bottom part of the hexagon is a homotopy cofiber sequence.

Remark

Theorem 1 in particular implies that for stable cohesive homotopy types AA there are natural equivalences

  • Π dRAΠ dR dRA\Pi_{dR} A \simeq \Pi_{dR} \flat_{dR} A

  • Π dRAΠ dRA\flat \Pi_{dR} A\simeq \Pi \flat_{dR} A

Given that conceptually, as made explicit in the Idea-section above, we may think of cocycles in Π dRA\Pi \flat_{dR} A as the rationalized characteristic classes and of coycles in Π dRA\flat \Pi_{dR} A as flat differential forms, this expresses the conceptual content of the de Rham theorem.

Examples

Ordinary differential cohomology

We discuss differential refinements in H=\mathbf{H} = Smooth∞Grpd of ordinary cohomology, hence of cohomology with coefficients in chain complexes or equivalently, via the stable Dold-Kan correspondence, cohomology represented by spectra underlying Eilenberg-MacLane spectrum-module spectra.

De Rham coefficients

We briefly recall the smooth spectra given by the de Rham complex (tensored with any chain complex) from smooth spectrum – Examples – De Rham spectra.

Write Ch Ch_\bullet for the (∞,1)-category of chain complexes (of abelian groups, hence over the ring \mathbb{Z} of integers). It is convenient to choose for A Ch A_\bullet \in Ch_\bullet the grading convention

A 1 A 0 A 1 \array{ \vdots \\ \downarrow \\ A_{-1} \\ \downarrow \\ A_0 \\ \downarrow \\ A_1 \\ \downarrow \\ \vdots }

such that under the stable Dold-Kan correspondence

DK:Ch Spectra DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra

the homotopy groups of spectra relate to the homology groups by

π n(DK(A ))H n(A ). \pi_n(DK(A_\bullet)) \simeq H_{-n}(A_\bullet) \,.

In particular for AA \in Ab an abelian group then A[n]A[n] denotes the chain complex concentrated on AA in degree n-n in this counting.

The grading is such as to harmonize well with the central example of a sheaf of chain complexes over the site of smooth manifolds, which is the de Rham complex, regarded as a smooth spectrum via the discussion at smooth spectrum – from chain complexes of smooth modules

Ω Sh (SmthMfd,Ch )Sh (SmthMfd,Spectra)TH \Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
Ω :X(00Ω 0(X)dΩ 1(X)d) \Omega^{\bullet} \;\colon\; X\mapsto (\cdots \to 0 \to 0 \to \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X)\stackrel{\mathbf{d}}{\to} \cdots)

with Ω 0(X)=C (X,)\Omega^0(X) = C^\infty(X, \mathbb{R}) in degree 0.

Throughout we keep the inclusion

CCh DKSpectraDiscStab(H) C \in Ch_\bullet \stackrel{DK}{\longrightarrow} Spectra \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})

notationally implicit, regarding bare chain complexes as geometrically discrete spectrum objects in smooth differential geometry.

We consider for nn \in \mathbb{N} the truncated sheaves of de Rham complexes with coefficients in a given chain complex:

Definition

Write

Ω nSh (SmthMfd,Ch )Sh (SmthMfd,Spectra)TH \Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}

for the smooth spectrum given under the stable Dold-Kan correspondence by the sheaf of truncated de Rham complexes

Ω n:X(00Ω n(X)dΩ n+1(X)d) \Omega^{\bullet \geq n} \;\colon\; X\mapsto (\cdots \to 0 \to 0 \to \Omega^n(X) \stackrel{\mathbf{d}}{\to} \Omega^{n+1}(X)\stackrel{\mathbf{d}}{\to} \cdots)

with Ω n(X)\Omega^n(X) in degree nn (the ppth stage in the Hodge filtration).

More genereally:

Definition

For CCh C \in Ch_\bullet a chain complex,

write

(ΩC) n(\Omega \otimes C)^{\bullet \geq n} for the smooth spectrum given over each manifold XX by the tensor product of chain complexes followed by truncation as indicated:

= ( 0 0 kΩ k(X)C nk d±d C kΩ k(X)C nk+1 d±d C ) deg n1 n n+1 . \begin{array} (\Omega \otimes C)^{\bullet \geq n} &=& (\cdots &\to& 0 &\to& 0 &\to& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k} &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k+1}&\stackrel{\mathbf{d}\pm d_{C}}{\to}& \cdots) \\ \\ deg && && && n-1 && n && n+1 && \end{array} \,.

where the first non-trivial term displayed is in taken to be in degree nn;

and write (ΩC) <n(\Omega \otimes C)^{\bullet \lt n} for

= ( d±d C kΩ k(X)C n2k d±d C kΩ k(X)C n1k0 0 ) deg n1 n n+1 \begin{array} (\Omega \otimes C)^{\bullet \lt n} &=& ( \cdots &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-2-k} &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-1-k} 0 &\to& 0 &\to& \cdots ) \\ \\ deg && && n-1 && n && n+1 && \end{array}

where again the first non-trivial term displayed is taken to be in degree nn.

Remark

Beware of the degree conventions in def. 4: in the first clause the tensor product of complexes is truncated without any shifting, while in the second case the truncation is shifted by one. This notation turns out to well reflect the way that the hexagon, theorem 1, decomposes these smooth spectra by prop. 4 and remark 5 below.

Proposition

For CCh ()C \in Ch_\bullet(\mathbb{R}) a chain complex of real vector spaces and for all nn \in \mathbb{Z}

  1. Π((Ω C) n)C \Pi ((\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n})\simeq C^\bullet

  2. ((Ω C) n)C n\flat ((\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n}) \simeq C^{\bullet \geq n}

  3. dR(Ω C) n(Ω C n1) n\flat_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq (\Omega \otimes_{\mathbb{R}} C^{\bullet \leq n-1})^{\bullet \geq n}

  4. Π dR(Ω C) n(Ω C) n1[1]\Pi_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq (\Omega \otimes_{\mathbb{R}} C )^{\bullet \leq n-1}[-1]

  5. Π dR(Ω C) nC n1\Pi \flat_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq C^{\bullet \leq n-1}

  6. ch:CC n1ch \;\colon\; C \to C_{\bullet\leq n-1} is the canonical projection.

(Bunke-Nikolaus-Völkl 13, lemma 4.4)

Remark

Prop. 4 means first of all that the de Rham complex (Ω C) n(\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} is a differential refinement, def. 1, of the chain complex CC (for all nn)

Moreover it says that dRΣ()\flat_{dR} \Sigma (-) and Π dRΩ()\Pi_{dR} \Omega(-) are similarly the high degree and low degree truncation, respectively, of the de Rham complex with coefficients in Π dRΣ()\Pi \flat_{dR} \Sigma (-).

Specifically for n=0n = 0, only the connected part

DC 1 D C_{\bullet \leq -1}

appears in the de Rham coefficients, and we have

Π dR() d dR() Ω(,D) <0 d dR±d C Ω(,D) 0. \array{ \Pi_{dR}(\cdots) && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}(\cdots) \\ \simeq && \simeq && \simeq \\ \Omega(-,D)^{\bullet \lt 0} && \stackrel{\mathbf{d}_{dR} \pm d_{C}}{\longrightarrow} && \Omega(-,D)^{\bullet \geq 0} } \,.

Deligne coefficients

We discuss the differential cohomology hexagon for smooth Deligne cohomology and hence smooth circle n-bundles with connection realizedin H=\mathbf{H}= Smooth∞Grpd.

Definition

For CCh ()C \in Ch_\bullet(\mathbb{Z}) be a chain complex of abelian groups. For nn \in \mathbb{N} an integer, write C conn,nC_{conn,-n} for the homotopy fiber product in

C conn,n (Ω C) n C C Π((Ω C) n), \array{ C_{conn,-n} &\longrightarrow& (\Omega \otimes_{\mathbb{Z}} C)^{\bullet \geq n} \\ \downarrow && \downarrow \\ C &\longrightarrow& C \otimes \mathbb{R} & \simeq \Pi((\Omega \otimes_{\mathbb{Z}} C)^{\bullet \geq n}) } \,,

where the right map is the unit of the shape modality according to prop. 4.

(Bunke-Nikolaus-Völkl 13, 4.3)

Proposition

For all nn \in \mathbb{Z}

  1. Π(C conn,n)C\Pi(C_{conn,-n}) \simeq C

  2. dRC conn,n(ΩC n1) n\flat_{dR} C_{conn,-n} \simeq (\Omega \otimes C_{\leq n-1})^{\bullet \geq n}

(Bunke-Nikolaus-Völkl 13, lemma 4.5)

Example

(ordinary Deligne cohomology)

For C=[n+1]C = \mathbb{Z}[n+1] we have

(Ω([n+1])) 0(Ω[n+1]) 0Ω cl n+1() (\Omega \otimes (\mathbb{Z}[n+1]\otimes \mathbb{R}))^{\bullet \geq 0} \simeq (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq 0} \simeq \Omega^{n+1}_{cl}(-)

(with the last term being in degree 0) and more generally for nn \in \mathbb{N}

(Ω([n+1])) n(Ω[n+1]) n(Ω 1()dΩ 2()ddΩ cl n+1()) (\Omega \otimes (\mathbb{Z}[n+1]\otimes \mathbb{R}))^{\bullet \geq -n} \simeq (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -n} \simeq (\Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^{n+1}_{cl}(-) )

(with the last term being in chain degrees n-n to 0 (hence homotopy group degrees nn to 0)).

We have a pasting diagram of homotopy pullbacks

B nU(1) conn Ω cl n+1 (Ω[n+1]) 1 B nU(1) (Ω[n+1]) n B n+1 B n+1 \array{ B^n U(1)_{conn} &\to& \Omega^{n+1}_{cl} \\ \downarrow && \downarrow \\ &\stackrel{}{\longrightarrow}& (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -1} \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{}{\longrightarrow}& (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -n} \\ \downarrow && \downarrow \\ B^{n+1}\mathbb{Z} &\to& B^{n+1} \mathbb{R} }

where on the right runs the Hodge filtration, with the notation B nU(1)=[n+1] conn,n\mathbf{B}^n U(1) = \mathbb{Z}[n+1]_{conn,n} as at circle n-group and B nU(1) conn=[n+1] conn,0\mathbf{B}^n U(1)_{conn} = \mathbb{Z}[n+1]_{conn,0} as at circle n-bundle with connection.

Hence for all nn \in \mathbb{N}

[n+1] conn,nB nU(1)(C (,U(1))00) \mathbb{Z}[n+1]_{conn,n} \simeq \mathbf{B}^n U(1) \simeq (C^\infty(-,U(1)) \to 0 \to \cdots\to 0)

and

[n+1] conn,0B nU(1) conn(C (,U(1))dlogΩ 1()ddΩ n). \mathbb{Z}[n+1]_{conn,0} \simeq \mathbf{B}^n U(1)_{conn} \simeq (C^\infty(-,U(1))\stackrel{\mathbf{d}log}{\to}\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to}\Omega^n) \,.

is represented by the Deligne complex.

The intermediate cases in between these two as in (Schreiber 13, def. 3.9.46, FRS 13, remark 2.3.15) (discussed also for instance at Courant algebroid – Relation to Atiyah Lie 2-algebroid).

Notice that by prop. 5 the above homotopy pullback diagrams are all indeed the right part of the differential cohomology hexagon, theorem 1, e.g.

B nU(1) conn θ B nU(1) conn Ω cl n+1 B nU(1) θ B nU(1) (Ω 1Ω cl n+1) B n+1 Π(θ B nU(1)) B n+1. \array{ \mathbf{B}^n U(1)_{conn} &\stackrel{\theta_{\mathbf{B}^n U(1)_{conn}}}{\longrightarrow}& \Omega^{n+1}_{cl} \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{\theta_{\mathbf{B}^n U(1)}}{\longrightarrow}& (\Omega^1 \to \cdots\to \Omega^{n+1}_{cl}) \\ \downarrow && \downarrow \\ \mathbf{B}^{n+1}\mathbb{Z} &\stackrel{\Pi(\theta_{\mathbf{B}^n U(1)})}{\longrightarrow}&\mathbf{B}^{n+1}\mathbb{R} } \,.

The differential cohomology hexagon obtained from this is on cohomology classes for each smooth manifold XX and each nn \in \mathbb{N}

Ω n(X)/im(d) d Ω cl n+1(X) a H n(X,) H 0(X,B nU(1) conn) H n+1(X,) H n(X,U(1)) H n+1(X,) \array{ && \Omega^{n}(X)/im(\mathbf{d}) && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && H^0(X,\mathbf{B}^n U(1)_{conn}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && H^{n}(X,U(1)) && \underset{}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) }

It is common (e.g. Simons-Sullivan 07) to display this after quotienting out the kernel of aa (which is the group Ω n(X) \Omega^n(X)_{\mathbb{Z}} of closed differential forms with integral periods), which is such as to make the top-left to bottom-right diagonal sequence be not just exact at the middle term, but be a short exact sequence:

0 0 Ω n(X)/Ω n(X) d Ω cl n+1(X) a H n(X,) H 0(X,B nU(1) conn) H n+1(X,) H n(X,U(1)) H n+1(X,) 0 0. \array{ 0 & && && && & 0 \\ & \searrow && && && \nearrow \\ && \Omega^{n}(X)/\Omega^n(X)_{\mathbb{Z}} && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && H^0(X,\mathbf{B}^n U(1)_{conn}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && H^{n}(X,U(1)) && \underset{}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) \\ & \nearrow && && && \searrow \\ 0 & && && && & 0 } \,.

Here the diagonals are now the “curvature exact sequence” and the “characteristic class exact sequence” as discuss at ordinary differential cohomology – Properties – curvature and characteristic class.

Hopkins-Singer coefficients

We discuss how the differential function complexes of (Hopkins-Singer 05), providing differential refinements of geometrically discrete spectra, and how they fit into their differential cohomology hexagon (following Bunke-Nikolaus-Völkl 13, section 4.4.).

Let again H=\mathbf{H} = Smooth∞Grpd.

Throughout, we leave notationally implicit

  1. the stable Dold-Kan correspondence DK:Ch SpectraDK \;\colon\;Ch_\bullet \longrightarrow Spectra;

  2. the inclusion SpectraStabStab(H)Spectra \stackrel{Stab}{\hookrightarrow} Stab(\mathbf{H});

hence always regard a chain complex as the corresponding spectrum and regard any bare spectrum always as a geometrically discrete smooth spectrum.

Definition

For

  1. ESpectraDiscStab(H)E \in Spectra \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H}) a spectrum,

  2. CCh ()C \in Ch_\bullet(\mathbb{R}) a chain complex of real vector spaces;

  3. c:ECc \colon E \longrightarrow C a homomorphism of spectra from EE to the image of CC under the stable Dold-Kan correspondence;

  4. nn \in \mathbb{N}

write E conn c,nE_{conn_c,-n} for the homotopy fiber product in

E conn c,n (Ω C) n E C Π((ΩC) n), \array{ E_{conn_c,-n} &\longrightarrow& (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq -n} \\ \downarrow && \downarrow \\ E &\longrightarrow& C & \simeq \Pi((\Omega \otimes C)^{\bullet \geq -n}) } \,,

where the map on the right is the unit of the shape modality according to prop. 4.

We abbreviate

E conn cE conn c,0. E_{conn_c} \coloneqq E_{conn_c,0} \,.

This appears in (Bunke-Nikolaus-Völkl 13, 4.4), with an earlier version in (Bunke-Gepner 13, def. 2.1).

Remark

Specifically when CEDK()C \simeq E \wedge DK(\mathbb{R}) is a model for the rationalization (realification) of EE and c:ECc \colon E \to C is the canonical map, then it is a model for the Chern character and def. 6 is essentially the definition of (Hopkins-Singer 05).

Proposition

For all nn \in \mathbb{Z}

  1. dRE conn c,n(Ω C n1) n\flat_{dR} E_{conn_c,-n} \simeq (\Omega \otimes_{\mathbb{R}} C_{\bullet \leq -n -1})^{\bullet\geq n}

  2. Π(E conn c,n)E\Pi(E_{conn_c,-n}) \simeq E.

(Bunke-Nikolaus-Völkl 13, lemma 4.7)

Remark

Beware that in prop. 6 the intrinsic de Rham coefficients dRΣE conn c,n\flat_{dR} \Sigma E_{conn_c,-n} in general differ by a truncation from the de Rham complex (Ω C) n(\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq -n} which is being pulled back in definition 6.

But by prop. 6 and theorem 1 composition with the truncation map preserves the homotopy pullback, hence in the differential cohomology hexagon the two differ only in the exactness property in the top right entry.

Corollary

Given E conn cE conn c,0E_{conn_c} \coloneqq E_{conn_c,0} as in def. 6, then for each smooth manifold XX (indeed for each smooth ∞-groupoid XX) the differential cohomology diagram on cohomology classes is of the following form

(Ω(X) C) 1/im(d) d (Ω(X) C) cl 0 H 1(X,C) E conn c 0(X) H 0(X,C) E 0(X) E 0(X). \array{ && (\Omega(X) \otimes_{\mathbb{R}} C)^{-1}/{im(\mathbf{d})} && \stackrel{\mathbf{d}}{\longrightarrow} && (\Omega(X)\otimes_{\mathbb{R}} C)^0_{cl} \\ & \nearrow && \searrow && \nearrow && \searrow \\ H^{-1}(X,C) && && E_{conn_c}^0(X) && && H^0(X,C) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && \flat E^0(X) && \longrightarrow && E^0(X) } \,.

(Bunke-Nikolaus-Völkl 13, (25)-(26))

Differential K-theory

We discuss several differential refinements, def. 1, to smooth spectra, of the complex topological K-theory spectrum KU or of its connective cover kuku, hence versions of differential K-theory.

Hopkins-Singer differential kuku

Let [b]\mathbb{C}[b] be the polynomial ring with complex numbers coefficients on a single generator bb. With bb regarded as in degree 2, regard this as a chain complex with vanishing differentials.

Then the complex ordinary Chern character on K-theory is a map

ch:ku[b]. ch\;\colon\; ku \longrightarrow \mathbb{C}[b] \,.
Definition

Write

ku conn chStab(SmoothGrpd) ku_{conn_ch} \in Stab(Smooth\infty Grpd)

for the smooth spectrum induced by def. 6) from the standard Chern character cchc \coloneqq ch on connective ku.

(Hopkins-Singer 02, section 4.4, Bunke-Nikolaus-Völkl 13, section 6.1)

Algebraic K-theory of smooth manifolds and the ee-invariant

Write

  1. Vect Sh(SmthMfd)H=Sh (SmthMfd)\mathbf{Vect}^{\oplus} \in Sh(SmthMfd) \hookrightarrow \mathbf{H} = Sh_\infty(SmthMfd) for the stack of (complex) vector bundles equipped with the direct sum;

  2. Vect conn Sh(SmthMfd)\mathbf{Vect}_{conn}^{\oplus} \in Sh(SmthMfd) for the stack of (complex) vector bundles with connection, equipped with direct sum;

Write

𝒦:CMon (Cat)Spectra \mathcal{K} \;\colon\; CMon_\infty(\infty Cat) \longrightarrow Spectra

for the (∞,1)-functor from symmetric monoidal (∞,1)-categories into spectra which produces the algebraic K-theory of symmetric monoidal (∞,1)-categories.

Definition

Forming objectwise the algebraic K-theory of symmetric monoidal (∞,1)-categories and then ∞-stackifying (which we leave notationally implicit) produces smooth spectra

𝒦(Vect conn ),𝒦(Vect )Stab(SmoothGrpd). \mathcal{K}(\mathbf{Vect}_{conn}^\oplus),\;\mathcal{K}(\mathbf{Vect}^\oplus) \in Stab(Smooth\infty Grpd) \,.

This may be called the algebraic K-theory of smooth manifolds.

Proposition

The smooth spectra 𝒦(Vect)\mathcal{K}(\mathbf{Vect}) and 𝒦(Vect conn)\mathcal{K}(\mathbf{Vect}_{conn}) are both differential refinements, def. 1, of connective ku, both whose underlying geometrically discrete spectrum is the algebraic K-theory K𝒞K \mathcal{C} of the complex numbers:

  1. Π(𝒦(Vect ))Π(𝒦(Vect conn ))ku\Pi(\mathcal{K}(Vect^\oplus)) \simeq \Pi(\mathcal{K}(Vect_{conn}^\oplus))\simeq ku;

  2. (𝒦(Vect ))(𝒦(Vect conn ))K\flat(\mathcal{K}(Vect^\oplus)) \simeq \flat(\mathcal{K}(Vect_{conn}^\oplus))\simeq K \mathbb{C}.

  3. in both cases the points-to-pieces transform Π\flat \to \Pi is the comparison map between algebraic and topological K-theory KkuK \mathbb{C}\to ku.

(Bunke-Nikolaus-Völkl 13, lemma 6.3, corollary 6.5)

Proposition

The traditonal Chern character induces, via the universal property of the homotopy pullback in def. 6, a morphism

𝒦(Vect conn )ku conn ch \mathcal{K}(\mathbf{Vect}_{conn}^\oplus) \longrightarrow ku_{conn_{ch}}

from the differential refinement of kuku given by def. 8, to that given by def. 7.

Its secondary characteristic class, hence its image under the flat modality, is in homotopy groups the regulator

K tor / K π 2k1 / \array{ K\mathbb{C}_{tor} &\stackrel{\simeq}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} \\ \downarrow && \downarrow \\ K\mathbb{C} &\stackrel{\pi_{2k-1}}{\longrightarrow}& \mathbb{C}/\mathbb{Z} }

related to the Adams e-invariant via (Bunke 11, section 5.3).

(Bunke-Nikolaus-Völkl 13, example 6.9)

Via smooth Snaith’s theorem

The ordinary Snaith theorem realizes KU as the localization of the ∞-group ∞-ring of the circle 2-group BU(1)B U(1) at the Bott element bπ 2(KU)b\in \pi_2(KU).

KU(Σ + BU(1))[b 1]. KU \simeq (\Sigma_+^\infty B U(1))[b^{-1}] \,.

With due care to define smooth geometric looping and taking the Hopf fibration as an actual smooth bundle over the smooth 2-sphere, this lifts to the smooth circle 2-group BU(1)\mathbf{B}U(1) and with more care to BU(1) conn\mathbf{B}U(1)_{conn} to produce smooth spectra

(Σ +BU(1))[b 1] (\Sigma_+ \mathbf{B}U(1))[b^{-1}]

and

(Σ +BU(1) conn)[b 1] (\Sigma_+ \mathbf{B}U(1)_{conn})[b^{-1}]

which both are differential refinements, def. 1, of KU.

(Bunke-Nikolaus-Völkl 13, section 6.3)

Complex-analytic differential generalized cohomology

The above examples all take place in the TTSmooth∞Grpd, modelling higher differential geometry. Another fundamental example of a cohesive (∞,1)-topos is ComplexAnalytic∞Grpd, modelling complex analytic higher geometry.

Differential cohomology in the complex-analytic context is discussed in (Hopkins-Quick 12), see at complex analytic ∞-groupoid – Properties – Differential cohomology for more.

E E_\infty-Arithmetic geometry

cohesion in E-∞ arithmetic geometry:

cohesion modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃ&#643;torsion approximation
dR-shape modalityʃ dR&#643;_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,

References

The differential cohomology hexagon was maybe first highlighted in the context of ordinary differential cohomology

and in the context of differential K-theory in

proven also in

An artistic impression of the differential cohomology hexagon as a dance choreography inspired by a talk by James Simons is at

That the differential cohomology hexagon exists not just at the level of exact sequences of cohomology classes but already at the level of homotopy fiber sequences of cocycle spaces was observed for ordinary differential cohomology and for differential K-theory in

  • Man-Ho Ho, Refined hexagons for differential cohomology (arXiv:1310.0582)

That generally the homotopy fiber product-construction of differential refinements of generalized (Eilenberg-Steenrod) cohomology theories due to

constitutes the right one of the two squares in the homotopy-theoretic version of the diagram is discussed explicitly for instance in prop 4.57 of

That every stable homotopy type in a cohesive (∞,1)-topos naturally sits in a differential cohomology diagram was observed (focusing on Smooth∞Grpd) in

following

where in section 4.1.2 a fully general abstract account is given. This in turn follows

Surveys/expositions of this include

See also

Revised on October 12, 2016 11:15:31 by Urs Schreiber (195.37.209.183)