nLab differential function complex

under construction

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 $E$ representing a given cohomology theory, its differential function complex over any given smooth manifold $U$ is the simplicial set whose $k$-simplices are triples consisting of

• a continuous function $f : U \times \Delta^k \to E_{n}$;

• a smooth differential form $\omega$ on $U \times \Delta^k$ whose corresponding real cohomology class (under the de Rham theorem) is that of the pullback of the real cohomology classes of $E$ along $f$;

• 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 $\omega$ on the tangent vectors to the simplex.)

When applied to the Eilenberg-MacLane spectrum $K\mathbb{Z}$ this construction reproduces, on cohomology classes, ordinary differential cohomology. Applied to the classifying space $B U$ 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 $V = V^\bullet$ a graded vector space over the real numbers set

• for $E$ a topological space:

  $$C^\bullet(E, V)^n := \oplus_{i + j = n} C^i(E, V^j)$$

• and so on

(…)

Differential functions

Definition

For

• $E$ a topological space;

• $\iota \in Z^n(E,\mathbb{R})$ a cocycle on $E$ for real-valued singular cohomology on $E$,

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

• $c : U \to E$ a continuous map;

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

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

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

$$\omega = c^*\iota + \delta h$$

holds, where

• $\omega$ is here regarded as a singular cochain (that sends a chain to the integral of $\omega$ over it, as discussed at de Rham theorem),

• $\delta$ denotes the coboundary operator,(the Moore complex differential of the singular simplicial complex).

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

• $E$ be a topological space and

• $\iota \in Z^n(E,\mathbb{R})$ a cocycle on $E$ for real-valued singular cohomology on $X$,

• $U$ a smooth manifold,

the differential function complex

$$(E,\iota)^U$$

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

$$U \times \Delta^k_{Top} \to (E,\iota) \,.$$

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

Definition

For $s \in \mathbb{N}$ write

• $filt_s \Omega^\bullet(U \times \Delta^k)$

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

• $filt_s (X,\iota)^S \subset (X,\iota)^S$

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

This is (HopkinsSinger, def. 4.5).

Proposition

The complex $filt_s (E,\iota)^U$ is (up to equivalence, of course) the homotopy pullback

$$\array{ filt_s (E,\iota)^U &\to& filt_s \Omega^n_{cl}(U \times \Delta^\bullet, \mathcal{V}) \\ \downarrow && \downarrow \\ Sing E^U &\to& Z^\bullet(U \times \Delta^\bullet, \mathcal{V}) }$$

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

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

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

Proposition

We have generally

$$\pi_k Z(S \times \Delta^\bullet_{Diff}, \mathcal{V}) = H^{n-m}(S; \mathcal{V})$$

(for instance by the Dold-Kan correspondence).

The simplicial homotopy groups of $filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})$ are

$$\pi_k filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff}) = \left\{ \array{ H_{dR}^{n-k}(S, \mathcal{V}) & | k \lt s \\ \Omega_{cl}(S; \mathcal{V})^{n-s} & | k = s \\ 0 & | k \gt s } \right\} \,.$$

This implies isomorphisms

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

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

Differential $E$-cohomology

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

Proposition

For $S$ a smooth manifold, and $s \in \mathbb{N}$, the sequence of differential function complexes, def. ,

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

forms an Omega-spectrum.

This is the differential function spectrum for $E$, $S$, $s$.

This is ([HopkinsSinger, section 4.6]).

Definition

The differential $E$-cohomology group of the smooth manifold $S$ in degree $n$ is

$$H_{diff}^n(S,E) := \pi_0 filt_0(E_n \iota_n)^S$$

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 $E$ an Omega-spectrum, $S$ a smooth manifold, we have for all $s,n \in \mathbb{N}$, a weak homotopy equivalence

$$\Omega filt_{s+1}(E_{n}; \iota_{n})^S \stackrel{\simeq}{\to} filt_{s}(E_{n-1}; \iota_{n-1})^S \,,$$

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

Relation to differential cohomology in cohesive $(\infty,1)$-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 $E_\bullet$ a spectrum as above,
we have an (∞,1)-pullback square

$$\array{ filt_0 (E_n; \iota_n)^{(-)} &\to& \prod_i \Omega^{n_i}_{cl}(-) \\ \downarrow && \downarrow \\ Disc E_n & \stackrel{}{\to} & \prod_i \mathbf{B}^{n_i} \mathbb{R}_{disc} } \,.$$

Proof

By prop. we have that

• $filt_\infty (E; \iota_n)^S \simeq Sing X^S$;

• $filt_0 \Omega_{cl}(S \times \Delta^\bullet) \simeq \Omega_{cl}(S)$.

The statement then follows with the pasting law for homotopy pullbacks

$$\array{ filt_0 (E_n; \iota_n)^S &\to& \Omega^n_{cl}(S; \mathcal{V}) \\ \downarrow && \downarrow \\ filt_\infty (E_n; \iota_n)^S &\to& filt_\infty \Omega_{cl}(S \times \Delta^\bullet; \mathcal{V}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Sing X^S &\to& Z(S \times \Delta^\bullet; \mathcal{V}) } \,.$$

(…)

Examples

• For $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum, $H_{diff}^n(-,E)$ is ordinary differential cohomology.

• For $E = K U$ the K-theory spectrum, $H_{diff}^n(-,E)$ is differential K-theory.

• For $E = M O, M U$ the Thom spectrum, $H_{diff}^n(-,E)$ is differential cobordism cohomology;

• For $E = tmf$ the tmf spectrum, $H_{diff}^n(-,E)$ is differential cohomology;

Line bundles with connection

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

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

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

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

But since $\hat \omega$ is necessarily closed it follows with $d (g \wedge \tilde \omega) = d t \frac{\partial g}{\partial t} \wedge \tilde \omega + g \wedge d_{S} \tilde \omega$ that $g$ is actually constant.

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

…

Differential K-cocycles

(…)

Higher filtration degree

Example

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

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

Remark

Example for $n = 4$ 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

• Mike Hopkins, I. Singer, Quadratic Functions in Geometry, Topology,and M-Theory

For further references see differential cohomology.

• Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory