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Lie integration

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Integration theory

Contents

Idea

Lie integration is a process that assigns to a Lie algebra 𝔤\mathfrak{g} – or more generally to an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by 𝔤\mathfrak{g}. It is essentially the reverse operation to Lie differentiation, except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties.

Classically, Lie integration of Lie algebras is part of Lie's three theorems, which in particular finds an unique (up to isomorphism) simply connected Lie group integrating a given finite-dimensional Lie algebra.

One may observe that the simply connected Lie group integrating a (finite-dimensional) Lie algebra is equivalently realized as the collection of equivalence classes of Lie algebra valued 1-forms on the interval where two such are identified if they are interpolated by a flat Lie-algebra valued 1-form on the disk. (Duistermaat-Kolk 00, section 1.14, see also the example below).

This path method of Lie integration stands out as having natural generalizations to higher Lie theory (Ševera 01).

In its evident generalization from Lie algebra valued differential forms to Lie algebroid valued differential forms this provides a means for Lie integration of Lie algebroids (e.g. Crainic-Fernandes 01).

In another direction, one may observe that L-∞ algebras are formally dually incarnated by their Chevalley-Eilenberg dg-algebras, and that under this identification the evident generalization of the path method to L-∞ algebra valued differential forms is essentially the Sullivan construction, known from rational homotopy theory, applied to these dg-algebras (Hinich 97, Getzler 04). Or rather, the bare such construction gives the geometrically discrete ∞-group underlying what should be the Lie integration to a smooth ∞-group. This is naturally obtained, as in the classical case, by suitably smoothly parameterizing the ∞-Lie algebroid valued differential forms (Henriques 08, Roytenberg 09, FSS 12).

Both these directions may be combined via the evident concept of ∞-Lie algebroid valued differential forms to yield a Lie integration of ∞-Lie algebroids to smooth ∞-groupoids.

While the construction exists and behaves as expected in examples, there is to date no good general theory providing higher analogs of, say, Lie's three theorems. But people are working on it.

Definition

Throughout, let 𝔞\mathfrak{a} be an ∞-Lie algebroid (for instance a Lie algebra, or a Lie algebroid or an L-∞-algebra). Write

CE(𝔞)dgAlg CE(\mathfrak{a}) \in dgAlg

for its Chevalley-Eilenberg algebra, a dg-algebra. Notice that, by the discussion at L-∞ algebra and at ∞-Lie algebroid, the Chevalley-Eilenberg dg-algebras CE(𝔞)CE(\mathfrak{a}) is the formal dual of 𝔞\mathfrak{a}, in that the functor

CE:LieAlgddgAlg op CE \colon \infty LieAlgd \hookrightarrow dgAlg^{op}

is a fully faithful functor. Indeed, the following definition of Lie integration (being just a smooth refinement of the Sullivan construction) makes sense just as well for any dg-algebra, not necessarily in the essential image of this embedding. But only for dg-algebras in the essential image of this embedding do the examples come out as expected for higher Lie theory.

An ∞-Lie algebra is equivalently a pointed ∞-Lie algebroid whose base space is the point. We write 𝔟𝔤LieAlgd\mathfrak{b}\mathfrak{g} \in \infty Lie Algd for objects of this form (“delooping”)

LieAlgd CE dgAlg op b CE LieAlg. \array{ \infty LieAlgd &\stackrel{CE}{\hookrightarrow}& dgAlg^{op} \\ \uparrow^{\mathrlap{b}} & \nearrow_{\mathrlap{CE}} \\ \infty LieAlg \,. }

Notice that this induces some degree shifts that may be a little ambiguous in situations like the line Lie n-algebra: as an L-∞ algebra this is b n1b^{n-1}\mathbb{R}, the corresponding ∞-Lie algebroid is b nb^n \mathbb{R}.

Higher dimensional paths in an \infty-Lie algebroid

Example

For XX a smooth manifold (possibly with boundary and with corners) then its tangent Lie algebroid TXT X is the one whose Chevalley-Eilenberg algebra is the de Rham complex

CE(TX)=(Ω (X),d dR). CE(T X) = (\Omega^\bullet(X), d_{dR}) \,.
Definition

For kk \in \mathbb{N} write Δ k\Delta^k for the standard kk-simplex regarded as a smooth manifold (with boundary and with corners).

A kk-path in the \infty-Lie algebroid 𝔞\mathfrak{a} is a morphism of \infty-Lie algebroids of the form

ΣTΔ k𝔞 \Sigma \; \coloneqq \; T \Delta^k \longrightarrow \mathfrak{a}

from the tangent Lie algebroid TΔ kT \Delta^k of the standard smooth kk-simplex to 𝔞\mathfrak{a}. Dually this is equivalently a homomorphism of dg-algebras

Ω (Δ k)CE(𝔞):Σ * \Omega^\bullet(\Delta^k) \longleftarrow CE(\mathfrak{a}) \;\colon\; \Sigma^*

from the Chevalley-Eilenberg algebra of 𝔞\mathfrak{a} to the de Rham complex of Δ d\Delta^d.

See also at differential forms on simplices.

Remark

A kk-path in 𝔞\mathfrak{a}, def. 1, is equivalently

  1. a flat 𝔞\mathfrak{a}-valued differential form on Δ k\Delta^k;

  2. a Maurer-Cartan element in Ω (Δ k)𝔞\Omega^\bullet(\Delta^k)\otimes \mathfrak{a}.

The Lie integration of 𝔞\mathfrak{a} is essentially the simplicial object whose kk-cells are the dd-paths in 𝔞\mathfrak{a}. However, in order for this to be well-behaved, it is possible and useful to restrict to dd-paths that are sufficiently well-behaved towards the boundary of the simplex:

Definition

Regard the smooth simplex Δ k\Delta^k as embedded into the Cartesian space k+1\mathbb{R}^{k+1} in the standard way, and equip Δ k\Delta^k with the metric space structure induced this way.

A smooth differential form ω\omega on Δ k\Delta^k is said to have sitting instants along the boundary if, for every (r<k)(r \lt k)-face FF of Δ k\Delta^k there is an open neighbourhood U FU_F of FF in Δ k\Delta^k such that ω\omega restricted to UU is constant in the directions perpendicular to the rr-face on its value restricted to that face.

More generally, for any UU \in CartSp a smooth differential form ω\omega on U×Δ kU \times\Delta^k is said to have sitting instants if there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that for all points u:*Uu : * \to U the pullback along (u,Id):Δ kU×Δ k(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k is a form with sitting instants on ϵ\epsilon-neighbourhoods of faces.

Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write Ω si (U×Δ k)\Omega^\bullet_{si}(U \times \Delta^k) for this sub-dg-algebra.

We write Ω si,vert (U×Δ k)\Omega_{si,vert}^\bullet(U \times \Delta^k) for the further sub-dg-algebra of vertical differential forms with respect to the projection p:U×Δ kUp : U \times \Delta^k \to U, hence the coequalizer

Ω (U)0p *Ω si (U×Δ k)Ω si,vert (U×Δ k). \Omega^\bullet(U) \stackrel{\stackrel{p^*}{\longrightarrow}}{\underset{0}{\longrightarrow}} \Omega^\bullet_{si}(U \times \Delta^k) \to \Omega^\bullet_{si, vert}(U \times \Delta^k) \,.
Remark

The dimension of the normal direction to a face depends on the dimension of the face: there is one perpendicular direction to a codimension-1 face, and kk perpendicular directions to a vertex.

Examples
  • A smooth 0-form (a smooth function) has sitting instants on Δ 1\Delta^1 if in a neighbourhood of the endpoints it is constant.

    A smooth function f:U×Δ 1f : U \times \Delta^1 \to \mathbb{R} is in Ω vert 0(U×Δ 1)\Omega^0_{\mathrm{vert}}(U \times \Delta^1) if there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that for each uUu \in U the function f(u,):Δ 1[0,1]f(u,-) : \Delta^1 \simeq [0,1] \to \mathbb{R} is constant on [0,ϵ)(1ϵ,1)[0,\epsilon) \coprod (1-\epsilon,1).

  • A smooth 1-form has sitting instants on Δ 1\Delta^1 if in a neighbourhood of the endpoints it vanishes.

  • Let XX be a smooth manifold, ωΩ (X)\omega \in \Omega^\bullet(X) be a smooth differential form. Let

    ϕ:Δ kX \phi \colon \Delta^k \to X

    be a smooth function that has sitting instants as a function: towards any jj-face of Δ k\Delta^k it eventually becomes perpendicularly constant.

    Then the pullback form ϕ *ωΩ (Δ k)\phi^* \omega \in \Omega^\bullet(\Delta^k) is a form with sitting instants.

Remark

The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of Δ k\Delta^k. Notably for ω jΩ (Δ k1)\omega_j \in \Omega^\bullet(\Delta^{k-1}) a collection of forms with sitting instants on the (k1)(k-1)-cells of a horn Λ i k\Lambda^k_i that coincide on adjacent boundaries, and for

p:Δ kΛ i k1 p \colon \Delta^k \to \Lambda^{k-1}_i

a standard piecewise smooth retract, the pullbacks

p *ω i p^* \omega_i

glue to a single smooth differential form (with sitting instants) on Δ k\Delta^k.

Remark

That ωΩ (Δ k)\omega \in \Omega^\bullet(\Delta^k) having sitting instants does not imply that there is a neighbourhood of the boundary of Δ k\Delta^k on which ω\omega is entirely constant. It is important for the following constructions that in the vicinity of the boundary ω\omega is allowed to vary parallel to the boundary, just not perpendicular to it.

Integration to a discrete \infty-groupoid

Here we discuss the discrete ∞-groupoids underlying the smooth ∞-groupoids to which an ∞-Lie algebroid integrates.

For 𝔞\mathfrak{a} an \infty-Lie algebroid, the dd-paths in 𝔞\mathfrak{a} naturally form a simplicial set as dd varies:

exp(𝔞) bare (Hom LieAlgd(TΔ 2,𝔞)Hom LieAlgd(TΔ 1,𝔞)Hom LieAlgd(TΔ 0,𝔤)) =(Hom dgAlg(CE(𝔞),Ω (Δ 2))Hom dgAlg(CE(𝔞),Ω (Δ 1))Hom dgAlg(CE(𝔞),Ω (Δ 0))). \begin{aligned} \exp(\mathfrak{a})_{bare} & \coloneqq \left( \cdots Hom_{\infty LieAlgd}(T \Delta^2, \mathfrak{a}) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{\infty LieAlgd}(T \Delta^1, \mathfrak{a}) \stackrel{\longrightarrow}{\longrightarrow} Hom_{\infty LieAlgd}(T \Delta^0, \mathfrak{g}) \right) \\ & = ( \cdots Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^2)) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^1)) \stackrel{\longrightarrow}{\longrightarrow} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^0)) ) \end{aligned} \,.

(We are indicating only the face maps, not the degeneracy maps, just for notational simplicity).

If here instead of smooth differential forms one uses polynomial differential forms then this is precisely the Sullivan construction of rational homotopy theory applied to CE(𝔞)CE(\mathfrak{a}). We next realize smooth structure on this and hence realize this as an object in higher Lie theory.

Remark

(spurious homotopy groups)

For 𝔞\mathfrak{a} a Lie n-algebroid (an nn-truncated \infty-Lie algebroid) this construction will not yield in general an nn-truncated ∞-groupoid exp(𝔞)\exp(\mathfrak{a}).

To see this, consider the example (discussed in detail below) that 𝔞=𝔤\mathfrak{a} = \mathfrak{g} is an ordinary Lie algebra. Then exp(𝔤) n\exp(\mathfrak{g})_n is canonically identified with the set of smooth based maps Δ nG\Delta^n \to G into the simply connected Lie group that integrates 𝔤\mathfrak{g} in ordinary Lie theory. This means that the simplicial homotopy groups of exp(𝔤)\exp(\mathfrak{g}) are the topological homotopy groups of GG, which in general (say for GG the orthogonal group or unitary group) will be non-trivial in arbitrarily higher degree, even though 𝔤\mathfrak{g} is just a Lie 1-algebra. This phenomenon is well familiar from rational homotopy theory, where a classical theorem asserts that the rational homotopy groups of exp(𝔤)\exp(\mathfrak{g}) are generated from the generators in a minimal Sullivan model resolution of 𝔤\mathfrak{g}.

For the purposes of higher Lie theory therefore instead one wants to truncate exp(𝔤)\exp(\mathfrak{g}) to its (n+1)(n+1)-coskeleton

cosk n+1exp(𝔞) bare. \mathbf{cosk}_{n+1}\exp(\mathfrak{a})_{bare} \,.

This divides out n-morphisms by (n+1)(n+1)-morphisms and forgets all higher higher nontrivial morphisms, hence all higher homotopy groups.

Integration to a smooth \infty-groupoid

We now discuss Lie integration of \infty-Lie algebroids to smooth ∞-groupoids, presented by the model structure on simplicial presheaves [CartSp smooth op,sSet] proj,loc[CartSp_{smooth}^{op}, sSet]_{proj,loc} over the site CartSp smooth{}_{smooth}.

For the following definition recall the presentation of smooth ∞-groupoids by the model structure on simplicial presheaves over the site CartSp smooth{}_{smooth}.

Definition

For 𝔞\mathfrak{a} an L-∞ algebra of finite type with Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) define the simplicial presheaf

exp(𝔞):CartSp smooth opsSet \exp(\mathfrak{a}) \;\colon\; CartSp_{smooth}^{op} \to sSet

by

exp(𝔞):(U,[k])Hom dgAlg(CE(𝔞),Ω (U×Δ k) si,vert), \exp(\mathfrak{a}) \;\colon\; (U,[k]) \mapsto Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(U \times \Delta^k)_{si,vert}) \,,

for all UU \in CartSp and [k]Δ[k] \in \Delta.

Remark

Compared to the integration to discrete ∞-groupoids above this definition knows about UU-parametrized smooth families of kk-paths in 𝔤\mathfrak{g}.

The underlying discrete ∞-groupoid is recovered as that of the 0=*\mathbb{R}^0 = *-parameterized family:

exp(𝔞): 0exp(𝔞) disc. \exp(\mathfrak{a}) \colon \mathbb{R}^0 \mapsto \exp(\mathfrak{a})_{disc} \,.
Proposition

The objects exp(𝔤)\exp(\mathfrak{g}) are indeed Kan complexes over each UU \in CartSp.

Proof

Observe that the standard continuous horn retracts f:Δ kΛ i kf : \Delta^k \to \Lambda^k_i are smooth away from the preimages of the (r<k)(r \lt k)-faces of Λ[k] i\Lambda[k]^i.

For ωΩ si,vert (U×Λ[k] i)\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i) a differential form with sitting instants on ϵ\epsilon-neighbourhoods, let therefore KΔ kK \subset \partial \Delta^k be the set of points of distance ϵ\leq \epsilon from any subface. Then we have a smooth function

f:Δ kKΛ i kK. f : \Delta^k \setminus K \to \Lambda^k_i \setminus K \,.

The pullback f *ωΩ (Δ kK)f^* \omega \in \Omega^\bullet(\Delta^k \setminus K) may be extended constantly back to a form with sitting instants on all of Δ k\Delta^k.

The resulting assignment

(CE(𝔤)AΩ si,vert (U×Λ i k))(CE(𝔤)AΩ si,vert (U×Λ i k)f *Ω si,vert (U×Δ n)) (CE(\mathfrak{g}) \stackrel{A}{\longrightarrow} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i)) \mapsto (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i) \stackrel{f^*}{\to} \Omega^\bullet_{si,vert}(U \times \Delta^n))

provides fillers for all horns over all UU \in CartSp.

Definition

Write cosk n+1exp(a)\mathbf{cosk}_{n+1} \exp(a) for the simplicial presheaf obtained by postcomposing exp(𝔞):CartSp opsSet\exp(\mathfrak{a}) : CartSp^{op} \to sSet with the (n+1)(n+1)-coskeleton functor cosk n+1:sSettr nsSet n+1cosk n+1sSet\mathbf{cosk}_{n+1} : sSet \stackrel{tr_n}{\longrightarrow} sSet_{\leq n+1} \stackrel{cosk_{n+1}}{\to} sSet.

Properties

Homotopy groups

…(Henriques, theorem 6.4)… remark 5

Examples

See also at smooth ∞-groupoid -- structures the section Exponentiated ∞-Lie algebras.

Interating Lie algebras to Lie groups

Let 𝔤L \mathfrak{g} \in L_\infty be an ordinary (finite dimensional) Lie algebra. Standard Lie theory (see Lie's three theorems) provides a simply connected Lie group GG integrating 𝔤\mathfrak{g}.

With GG regarded as a smooth ∞-group write BG\mathbf{B}G \in Smooth∞Grpd for its delooping. The standard presentation of this on [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet] is by the simplicial presheaf

BG c:UN(C (U,G)*)*. \mathbf{B}G_c \colon U \mapsto N(C^\infty(U,G) \stackrel{\longrightarrow}{\longrightarrow} *) * \,.

See at smooth infinity-groupoid – structures – Lie groups for more details.

Proposition

The operation of parallel transport Pexp():Ω 1([0,1],𝔤)GP \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G yields a weak equivalence (in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj})

Pexp():cosk 3exp(𝔤)cosk 2exp(𝔤)BG c. P \exp(\int - ) \;\colon\; \mathbf{cosk}_3 \exp(\mathfrak{g}) \;\simeq\; \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_c \,.

This follows from the Steenrod-Wockel approximation theorem and the following observation.

Lemma

For XX a simply connected smooth manifold and x 0Xx_0 \in X a basepoint, there is a canonical bijection

Ω flat 1(X,𝔤)C * (X,G) \Omega^1_{flat}(X,\mathfrak{g}) \simeq C^\infty_*(X,G)

between the set of Lie-algebra valued 1-forms on XX whose curvature 2-form vanishes, and the set of smooth functions XGX\to G that take x 0x_0 to the neutral element eGe \in G.

Proof

The bijection is given as follows. For AΩ flat 1(X,𝔤)A \in \Omega^1_{flat}(X,\mathfrak{g}) a flat 1-form, the corresponding function f A:XGf_A : X \to G sends xXx \in X to the parallel transport along any path x 0xx_0 \to x from the base point to xx

f A:xtra A(x 0x). f_A : x \mapsto tra_A(x_0 \to x) \,.

Because of the assumption that the curvature 2-form of AA vanishes and the assumption that XX is simply connected, this assignment is independent of the choice of path.

Conversely, for every such function f:XGf : X \to G we recover AA as the pullback of the Maurer-Cartan form on GG

A=f *θ. A = f^* \theta \,.

From this we obtain

Proof of prop. 2.

The \infty-groupoid cosk 2exp(𝔤)\mathbf{cosk}_2 \exp(\mathfrak{g}) is equivalent to the groupoid with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths Δ 1G\Delta^1 \to G (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopy D 2GD^2 \to G (with sitting instant) between them.

Since GG is simply connected, these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with GG.

Remark

We do not need to fall back to classical Lie theory to obtain GG in the above argument. A detailed discussion of how to find GG with its group structure and smooth structure from dd-paths in 𝔤\mathfrak{g} is in (Crainic).

Integrating to line/circle Lie nn-groups

Definition

For n,n1n \in \mathbb{N}, n \geq 1 write b n1b^{n-1} \mathbb{R} for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree nn and vanishing differential. We may call this the line Lie n-algebra.

Write B n\mathbf{B}^{n} \mathbb{R} for the smooth line (n+1)-group.

Observation

The discrete ∞-groupoid underlying exp(b n1)\exp(b^{n-1} \mathbb{R}) is given by the Kan complex that in degree kk has the set of closed differential nn-forms (with sitting instants) on the kk-simplex

exp(b n1) disc:[k]Ω si,cl n(Δ k) \exp(b^{n-1} \mathbb{R})_{disc} : [k] \mapsto \Omega^n_{si,cl}(\Delta^k)
Proposition

The \infty-Lie integration of b n1b^{n-1} \mathbb{R} is the circle n-group B n\mathbf{B}^{n} \mathbb{R}.

Moreover, with B n chn[CartSp smooth op,sSet]\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet] the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree nn on C (,)C^\infty(-, \mathbb{R}) the equivalence is induced by the fiber integration of differential nn-forms over the nn-simplex:

Δ :exp(b n1)B n chn. \int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \stackrel{\simeq}{\longrightarrow} \mathbf{B}^{n} \mathbb{R}_{chn} \,.
Proof

First we observe that the map

Δ :(ωΩ si,vert,cl n(U×Δ k)) Δ kωC (U,) \int_{\Delta^\bullet} : (\omega \in \Omega^n_{si,vert,cl}(U \times \Delta^k)) \mapsto \int_{\Delta^k} \omega \in C^\infty(U, \mathbb{R})

is a morphism of simplicial presheaves exp(b n1)B n chn\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn} on CartSp smooth{}_{smooth}. Since it goes between presheaves of abelian simplicial groups by the Dold-Kan correspondence it is sufficient to check that we have a morphism of chain complexes of presheaves on the corresponding normalized chain complexes.

The only nontrivial degree to check is degree nn. Let λΩ si,vert,cl n(Δ n+1)\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1}). The differential of the normalized chains complex sends this to the signed sum of its restrictions to the nn-faces of the (n+1)(n+1)-simplex. Followed by the integral over Δ n\Delta^n this is the piecewise integral of λ\lambda over the boundary of the nn-simplex. Since λ\lambda has sitting instants, there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that there are no contributions to this integral in an ϵ\epsilon-neighbourhood of the (n1)(n-1)-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the (n+1)(n+1)-simplex, as indicated in the following diagram

<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ --> Layer 1

Since λ\lambda is a closed form on the nn-simplex, this surface integral vanishes, by the Stokes theorem. Hence Δ \int_{\Delta^\bullet} is indeed a chain map.

It remains to show that Δ :exp(b n1)B n chn\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn} is an isomorphism on all the simplicial homotopys group over each UCartSpU \in CartSp. This amounts to the statement that

  1. a smooth family of closed nn-forms with sitting instants on the boundary of Δ n+1\Delta^{n+1} may be extended to a smooth family of closed forms with sitting instants on Δ n+1\Delta^{n+1} precisely if their smooth family of integrals over the boundary vanishes;

  2. Any smooth family of closed n<kn \lt k-forms with sitting instants on the boundary of Δ k+1\Delta^{k+1} may be extended to a smooth family of closed nn-forms with sitting instants on Δ k+1\Delta^{k+1}.

To demonstrate this, we want to work with forms on the (k+1)(k+1)-ball instead of the (k+1)(k+1)-simplex. To achieve this, choose again 0<ϵ0 \lt \epsilon \in \mathbb{R} and construct the diffeomorphic image of S k×[1,1ϵ]S^k \times [1,1-\epsilon] inside the (k+1)(k+1)-simplex as indicated in the above diagram: outside an ϵ\epsilon-neighbourhood of the corners the image is a rectangular ϵ\epsilon-thickening of the faces of the simplex. Inside the ϵ\epsilon-neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this ϵ\epsilon-thickening of the smoothed boundary of the simplex to S k×[1ϵ,1]S^k \times [1-\epsilon,1] extends to a smooth function from the (k+1)(k+1)-simplex to the (k+1)(k+1)-ball.

By choosing ϵ\epsilon smaller than each of the sitting instants of the given nn-form on Δ k+1\partial \Delta^{k+1}, we have that this nn-form vanishes on the ϵ\epsilon-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the (k+1)(k+1)-ball.

It is now sufficient to show: a smooth family of smooth nn-forms ωΩ vert,cl n(U×S k)\omega \in \Omega^n_{vert,cl}(U \times S^k) extends to a smooth family of closed nn-forms ω^Ω vert,cl n(U×B k+1)\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1}) that is radially constant in a neighbourhood of the boundary for all n<kn \lt k and for k=nk = n precisely if its smooth family of integrals vanishes, S kω=0C (U,)\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R}).

Notice that over the point this is a direct consequence of the de Rham theorem: an nn-form ω\omega on S kS^k is exact precisely if n<kn \lt k or if n=kn = k and its integral vanishes. In that case there is an (n1)(n-1)-form AA with ω=dA\omega = d A. Choosing any smoothing function f:[0,1][0,1]f : [0,1] \to [0,1] (smooth, surjective, non,decreasing and constant in a neighbourhood of the boundary) we obtain an nn-form fAf \wedge A on (0,1]×S k(0,1] \times S^k, vertically constant in a neighbourhood of the ends of the interval, equal to AA at the top and vanishing at the bottom. Pushed forward along the canonical (0,1]×S kD k+1(0,1] \times S^k \to D^{k+1} this defines a form on the (k+1)(k+1)-ball, that we denote by the same symbol fAf \wedge A. Then the form ω^:=d(fA)\hat \omega := d (f \wedge A) solves the problem.

To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the (n1)(n-1)-form AA in a way depending smoothly on the the nn-form ω\omega.

One way of achieving this is using Hodge theory. Fix a Riemannian metric on S nS^n, and let Δ\Delta be the corresponding Laplace operator, and π\pi the projection on the space of harmonic forms. Then the central result of Hodge theory for compact Riemannian manifolds states that the operator π\pi, seen as an operator from the de Rham complex to itself, is a cochain map homotopic to the identity, via an explicit homotopy P:=d *GP := d^* G expressed in terms of the adjoint d *d^* of the de Rham differential and of the Green operator GG of Δ\Delta. Since the kk-form ω\omega is exact its projection on harmonic forms vanishes. Therefore

ω =(Idπ)ω =d(Pω)+P(dω) =d(Pω). \begin{aligned} \omega & = (Id-\pi)\omega \\ & = d (P\omega)+P (d\omega) \\ & = d (P\omega). \end{aligned}

Hence A:=PωA := P\omega is a solution of the differential equation dA=ωd A=\omega depending smoothly on ω\omega.

Integrating the string Lie 2-algebra to the string Lie 2-group

Let 𝔰𝔱𝔯𝔦𝔫𝔤=𝔤 μ\mathfrak{string} = \mathfrak{g}_\mu be the string Lie 2-algebra.

Then cosk 3exp(𝔤 μ)\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu) is equivalent to the 2-groupoid BString\mathbf{B}String

  • with a single object;

  • whose morphisms are based paths in GG;

  • whose 2-morphisms are equivalence class of pairs (Σ,c)(\Sigma,c), where

    • Σ:D * 2G\Sigma : D^2_* \to G is a smooth based map (where we use a homeomorphism D 2Δ 2D^2 \simeq \Delta^2 which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of D * 2D^2_* is the 0-vertex of Δ 2\Delta^2)

    • and cU(1)c \in U(1), and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball ϕ:D 3G\phi : D^3 \to G filling them the labels c 1,c 2U(1)c_1, c_2 \in U(1) differ by the integral D 3ϕ *μ(θ)mod\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z},,

where θ\theta is the Maurer-Cartan form, μ(θ)=θ[θθ]\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle the 3-form obtained by plugging it into the cocycle.

This is the string Lie 2-group. It’s construction in terms of integration by paths is due to (Henriques)

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

The “path method” of integrating Lie algebras to simply connected Lie groups appears in

The idea of identifying the Sullivan construction applied to Chevalley-Eilenberg algebras as Lie integration to discrete ∞-groupoids appears in

and for general ∞-Lie algebras in

(whose main point is the discussion of a gauge condition applicable for nilpotent L L_\infty-algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model) .

This was refined from integration to bare \infty-groupoids to an integration to internal ∞-groupoids in Banach manifolds in

(whose origin possibly preceeds that of Getzler’s article).

For general ∞-Lie algebroids the general idea of the integration process by “dd-paths” had been indicated in

Discussion of Lie integration of Lie algebroids by the path method is due to

following (Duistermaat-Kolk 00, section 1.14) and following the discussion of the special case of Lie integration of Poisson Lie algebroids to symplectic groupoids in

A description of Lie integration with values in smooth ∞-groupoids regarded as simplicial presheaves on CartSp (and further the Lie integration of L-infinity cocycles) is in

Essentially the same integration prescription is considered in

The Lie integration- of Lie algebroid representations 𝔞end(V)\mathfrak{a} \to end(V) to morphisms of ∞-categories ACh A \to Ch_\bullet^\circ / higher parallel transport is discussed in

Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in

A generalization of Lie integration to conjectural Leibniz groups has been conjectured by J-L. Loday. A local version via local Lie racks has been proposed in

  • Simon Covez, The local integration of Leibniz algebras, arXiv:1011.4112; On the conjectural cohomology for groups, arXiv:1202.2269; L’intégration locale des algèbres de Leibniz, Thesis (2010), pdf

category: Lie theory

Revised on June 17, 2015 10:07:33 by Zoran Škoda (161.53.130.104)