Embeddings of manifolds in Euclidean space and Feynman diagrams

Embeddings of manifolds in Euclidean space and Feynman diagrams

Thomas Willwacher

Algebra, Geometry and Physics Seminar HU/MPIM, 09.03.2021

Goal

LetM,Nbe smooth manifolds of dimensionm,n. Long standing problems: Understand

Embedding spaces (knot spaces)

Emb(M,N) ={f :M→N |f smooth embedding} ⊂C^∞(M,N) Diffeomorphism groupsDiff(M)

with theC^∞topology.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Goal

Concrete questions:

π₀(Emb(M,N)) =?, i.e., classify embeddings modulo isotopy.

(knot theory) Higherπk(−) =?

Simpler question: Rational homotopy groupsπ_k(−)⊗Q=?for k ≥2, or rational homotopy type.

Hope: Possible for wide class of manifolds in a few years.

Rational homotopy theory I

One has a (Quillen) equivalence

| − |:sSetTop:S

and a (Quillen) adjunction

Ω :sSetdgca^op:G

between the categories of topological spaces, simplicial sets and differential graded commutative algebras /Q.

In particular, forX a topological space the differential graded commutative algebraΩ(X)are the (PL) differential forms on X.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Rational homotopy theory II

LetX,Y be (simply connected) spaces. A mapf :X →Y is a weak homotopy equivalence iffinduces bijections πk(X)→πk(Y).

a rational (homotopy) equivalence iffinduces bijections π^Q_k(X)→π^Q_k(Y), with

π^Q_k(X) :=πk(X)⊗_ZQ.

AmodelforX is a dg comm. alg.A that is connected toΩ(X) via a chain of quasi-isomorphisms.

A −→ · · ·^∼ ←^∼−Ω(X).

(Quasi-isomorphism:=dg comm. alg. morphism inducing isom. on cohomology)

For goodX (e.g. simply connected) one can recoverπ^Q_k(X) from a model forX, andX,Y are rationally equivalent iff they have quasi-isomorphic models.

Mapping spaces

ForX,Y topological spaces (simplicial sets) we may consider the mapping spaceMap(X,Y) ={f :X →Y |f continuos}. Sincedgcais a model category, we may also define the (derived) mapping spaceMap(A,B)forA,B ∈dgca. By functoriality we have a map

Map(X,Y)→Map(Ω(Y),Ω(X)).

(Haefliger, Sullivan ’80) For goodX,Y the above map induces componentwise rational homotopy equivalences.

Map(X,Y)_f −−→^∼Q Map(Ω(Y),Ω(X))_Ω(f), and a finite-to-one map onπ₀.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Mapping spaces – Example

Map(S¹,S¹)'Z×S¹. Model forS¹:

A =Q[ω]−→^∼ Ω(S¹),

withωa variable of degree 1.

Map(A,A)' |Hom(A,A⊗Ω(∆^•))|

Any dgca morphismA →B is determined by image ofω, hence one can show

π_k(Map(A,A)) =















Q k =0 Q k =1

∗ k ≥2.

Towards Emb( M , N )

We ”can understand” mapping spaces.

We would like to seeEmb(M,N)as an upgraded version of Map(M,N).

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Goodwillie-Weiss embedding calculus

Let

conf^m−fr_M (r) ={(x₁,F₁, . . . ,x_r,F_r)|x_j ∈M,x_i,^xjfori ,^j} the space of configurations ofr points onM, with anm-frame F_jin the tangent space atx_j.

Any embeddingf :M→Ninduces maps conf^m−fr_M (r)→conf^m−fr_N (r).

Main idea: StudyEmb(M,N)via

Emb(M,N)→ {Map(conf^m_M^−fr(r),conf^m_N^−fr(r))}_r≥1. Problem: ...still need coherences between the variousr and the points.

Fulton-MacPherson operad and action

The framed Fulton-MacPherson–Axelrod-Singer operad FM^fr_m is a compactification

conf^m−fr_Rm →FM^fr_m(r).

1

2 3

3 4 5

Gluing yields operations

FM^fr_m(r)×FM^fr_m(s)→FM^fr_m(r+s−1)

that assemble into an operad structure. (FM^fr_m is equivalent to the framed littlem-disks operad.)

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Fulton-MacPherson operad and action

Similarly one has a compactification

conf^m_Rm^−fr →FM^fr_M(r) =FM^m_M^−fr(r).

1

2 3

3 4 5

Gluing produces right actions

FM^fr_M(r)×FM^fr_m(s)→FM^fr_M(r+s−1)

that assemble into a right operadic FM^fr_m-module structure on FM^fr_M.

Goodwillie-Weiss embedding calculus III

For an embeddingf :M→Nthe induced map FM^fr_M →FM^m−fr_N is compatible with the right FM^fr_m-actions.

Theorem (Goodwillie, Weiss, Klein, Boavida) The map

Emb(M,N)→Map_FMfr

m−mod(FM^fr_M,FM^m_N^−fr) is a weak homotopy equivalence if dimN−dimM≥3.

HereMap_FM^fr

m−mod(−)is the derived mapping space in the model category of right operadic FM^fr_m-modules.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Our results

To state our results, we will slightly adjust the setting as follows:

Specialize toN =Rⁿ.

M⊂R^m is the complement of a compact submanifold (with boundary), i.e., open, extending to∞. ⇒incorporate mixed dimensions by taking tubular neighborhood, and ”long”

objects.

ConsiderEmb∂(M,Rⁿ)⊂Emb(M,Rⁿ)be the embeddings that agree with the given embedding outside a compact.

fixed outside compact

Our results

LetImm_∂(M,Rⁿ)be the immersions (supported on a compact) and consider the homotopy fiber:

Emb_∂(M,Rⁿ) =hofiber(Emb_∂(M,Rⁿ)→Imm_∂(M,Rⁿ)) Advantage 1: ForM=Tube(M⁰)(tubular neighborhood), Emb(M,Rⁿ) =Emb(M⁰,Rⁿ).

Advantage 2: SinceM,Rⁿare framed, we can consider the non-framed analogs FM_M ⊂FM^fr_M,FM_n ⊂FM^fr_n.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Our results II

Theorem (Fresse-Turchin-W., arxiv:2008.08146) For M⊂R^m as above, n≥m+3:

Emb_∂(M,Rⁿ)−−→^'

(1) Map_FMm−bmod(FM_M,FM_n)

'_Q

−−→

(2) Map_{Ω(FMm)−bmod}(Ω(FM_n),Ω(FM_M))−−→^'

(3) |MC•(HGCA,n¯ )|, with A a model for M∪ {∞}and_A¯ ⊂A the augmentation ideal.

(1) is due to Goodwillie, Weiss, Klein, Arone, Turchin (2) is an operadic extension of Haefliger’s result (Fresse, Mienn ´e), and a rational weak equivalence componentwise, finite-to-one onπ₀.

(3) is our main result and holds ifn−m≥2 ... with the right-hand side to be defined...

Hairy graph L

-algebra HGC

ForA a dgca,nan integer:

HGC_A,n =span{isom. classes of admissibleA-decorated hairy graphs}

a₁

,

a₁

a₂ a₃ ,

a₁ a₂

, a₁

. Vertices have degree−n, edgesn−1,a_j ∈A carry their

(homological, non-positive) degree.

Admissible: (i) Valence of vertices≥3. (ii) No odd symmetries.

Carries natural homotopy Lie- (L_∞-)algebra structure.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

L

-algebra structure

Differentialδ=d_A +δsplit+δjoin, δ_splitΓ = X

vvertex

±Γsplitv 7→X

δ_join Γ a₁a₂. . .a_k

= X

S⊂hairs

|S|≥2

± Γ

a₁. . .

Q

j∈Saj

.

Lie bracket:













Γ , Γ⁰













=X Γ Γ⁰

,

HigherL∞-operations[−, . . . ,−]are defined similarly.

Interlude: dg Lie algebras and Maurer-Cartan spaces

Maurer-Cartan elements ofL∞-algebrag MC(g) =

(

x∈g₀ |dx+¹

2[x,x] + ¹

3![x,x,x] +· · ·=0 )

Maurer-Cartan space (simplicial set)

MC•(g) =MC(g⊗Ωˆ poly(∆^•))

Forx ∈MC(g)we can consider the twistedL∞-algebra with operations

[a₁, . . . ,a_r]^x =X

k≥0

1

k![x| {z }, . . . ,x

k×

,a₁, . . . ,a_r]

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Interlude: Two important results about MC spaces

Theorem (Berglund)

g: pro-nilpotent L∞-algebra, x ∈MC(g). Then for k ≥1 π_k(MC_•(g),x)^Hk(g^x),

where for k =1the rhs. is equipped with the PBW group structure.

Remark: Furthermore, in good cases that cohomologyH(MC•(g)x) is computed by the Chevalley complex of the truncation

H(MC•(g)_x)^H(C(g^x_trunc)).

Consequences

Recall:Emb∂(M,Rⁿ)'_Q |MC•(HGCA,n¯ )|forA a model forM∪ {∗}. Consequences:

Forn≥m+3,k ≥1 one can computeπ^Q_kEmb(M,Rⁿ)in terms of diagrams.

One has characterization ofπ₀Emb(M,Rⁿ)(invariants) extending classical Vassiliev invariants, that are complete up to finite ambiguity ifn≥m+3.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Example I - long knots

M=R^m, thenEmb∂(R^m,Rⁿ)is the space ofm-dimensional long knots inRⁿ.

fixed outside compact

ThenR^m ∪ {∞}^Sm.

We can takeA =Q[ω]withωof degreem,ω²=0.

A¯ is one-dimensional, spanned byω, hence all hairs carry same decoration.

Example I - long knots, ctd.

ω ,

ω

ω ω , ω ω

, ω

.

AllL∞-operations joining hairs are 0⇒abelianL∞-algebra.

⇒π₀(MC•(HGCA,n¯ ))^H0(HGCA,n¯ ).

In the casen=3,m=1 one recovers the diagrams enumerating Vassiliev knot inveriants, i.e., uni-trivalent diagrams modulo IHX.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Example II - string links

M=Tube(R^m1tR^m2t · · · tR^mr)⊂R^m (tubular neighborhod), then Emb∂(M,Rⁿ)is (essentially) the space of string links withk

components of dimensionsm₁, . . . ,m_r. (see ”board”) M∪ {∞}^Sm1∨ · · · ∨S^mr wedge product of spheres.

A =Q[ω₁, . . . , ω_r]/hω_iω_j =0i, and_A¯ =span(ω₁, . . . , ω_r). HGCA,n¯ is given by hairy graphs with hairs ofr”colors”, and is still abelian.

ω_i

ω_j ω_k

Recovers results of Turchin-Tsopm ´en ´e

General remark

LetM⁰⊂R^m be anm⁰-dimensional submanifold.

Then we can consider a tubular neighborhood M=Tube(M⁰)⊂R^m, an open submanifold.

The embedding spaceEmb(M,Rⁿ)is identified with the space of framed embeddings ofM⁰inR^m.

However,Emb(M,Rⁿ)Emb(M⁰,Rⁿ), that is, the homotopy fiber ”eats” the framing.

⇒our result applies to embeddings of general compact submanifoldsM⁰ ⊂R^m, with the choiceM=Tube(M⁰t {∞}), thenEmb∂(M,Rⁿ)Emb(M⁰,Rⁿ)

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

MC elements and trees

Letm⁰the cohomological dimension ofM, i.e., the smallest number such thatH^k(M) =0 for allk ≤m⁰, and suppose m⁰≤n−3.

Degree counting⇒all elements of HGC_H(M),n¯ of non-positive degree are trees.

Henceπ₀MC•(HGCH(M),n¯ )is determined by tree diagrams.

Example - Unknotting

Classical unknotting theorem:

Theorem (Whitney-Wu)

M⁰compact k -connected of dimension m⁰, n≥2m⁰+1, then π₀(Emb(M⁰,Rⁿ)) =∗.

We can ”see” this on graphs: In casen>2m⁰+1 there are no graphs of degree≤0, hence we can conclude from our result that π₀(Emb(M⁰,Rⁿ))is finite.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Example - Nonlinear MC equation

M⁰ =S²×S²⊂R⁵,n=7.

A =Q[ω1, ω2]/hω²₁ =ω²₂=0i, with|ω1|=|ω2|=2.

List of (relevant) degree 0 graphs:

L₁ = ω₁ ω₁∧ω₂ ; L₂ = ω₂ ω₁∧ω₂ . MC equation forx=λ₁L₁+λ₂L₂:

0= ¹

2[x,x] =λ₁λ₂

ω₁∧ω₂ ω₁∧ω₂ ω₁∧ω₂ . Thus

π_o(MC•(HGCA,n¯ )) ={λ₁L₁+λ₂L₂|λ₁=0 orλ₂ =0}.

Example - with gauge transformations

We considerM⁰ =S¹×S²,n=6.

A =H^∗(S¹×S²) =Q[α, β].

hβ²=0i, with|α|=1,|β|=2.

Relevant graphs:

L_α=α α∧β

L_β=β α∧β ^Tα∧β =

α∧β α∧β α∧β

withL_αof degree 1,L_β,T_α∧βof degree−1.

AnyλL_β+µT_α∧β is an MC element.

Gauge equivalenceλL_β+µT_α∧β ∼ λL_βifλ,^{0. Hence} π₀(MC_•(HGC_A,n¯ )){λL_β+µT_α∧β|µ=0 orλ=0}.

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams

Future directions

GeneralizeRⁿtoN. Probably possible with similar techniques.

Expected result: Need to decorate vertices of graphs by H(N), with more complicatedL∞-structure.

Attack codimension restrictionn−m≥3, and target in particularDiff(M)(n=m): Can likely be done using tricks akin Weiss fiber sequence.

The End

Thanks for listening!

Thomas Willwacher Embeddings of manifolds in Euclidean space and Feynman diagrams