On the Rational Homology of Spaces of Smooth …sma.epfl.ch/~hessbell/arolla/Lambrechts.pdfOn the...

On the Rational Homology ofSpaces of Smooth Embeddings

Pascal LambrechtsJoint work with Greg Arone and Ismar Volić

Arolla, August 2008

The slogan

Goodwillie embedding calculus+

Kontsevich formality of the little disks operad+

Weiss orthogonal calculus

⇓

Understanding of the rational homology ofthe space of embeddings of a smooth manifold in Rn

Spaces of embeddings: definition

Notation

M is a compact smooth manifold (maybe with boundary)V is a real vector space of dimension v

Simple examples of Emb(M , V ): M = {1, · · · , k}

Embeddings of a discrete finite set M = {1, · · · , k}A typical element f ∈ Emb({1, · · · , k},V ):

M

1 2 3 k· · ·f

↪→

V = R2

x1

x2

x3

xk

f ≡ (x1, · · · , xk) ∈ V k with distinct xi ’s

Example

Emb({1, . . . , k},V ) is the configuration space

Conf(k,V ) := {(x1, · · · , xk) ∈ V : xi 6= xj for i 6= j}

Simple examples of Emb(M , V ): M = D1

Embeddings of the interval M = D1 = [−1, 1]

Example

Emb(D1,V ) is homotopy equivalent to the unit sphere in V :

S(V ) := {v ∈ V : ‖v‖ = 1} ∼= Sv−1

ProofRetract by deformation smooth embeddings of the interval tolinear embeddings.The homotopy equivalence is explicit:

Φ: Emb(D1,V ) → S(V )f 7→ df (0)/‖df (0)‖

Simple examples of Emb(M , V ): M = Dm

Embeddings of the m-dimensional disk M = Dm

Recall the Stiefel manifold:

Stiefelm(V ) := {(v1, ..., vm) m-tuple of lin. indep. elements in V }

Example

Emb(Dm,V ) is homotopy equivalent to a Stiefel manifold:

Emb(Dm,V )'Stiefelm(V )

Proof. The homotopy equivalence is given by f 7→ df (0).

Simple examples of Emb(M , V ): M = qki=1Dm

Embeddings of a disjoint union of k m-dimensional disks:M = Dm q · · · q Dm︸︷︷︸

k

Example

Emb(qkDm,V ) is homotopy equivalent to the product of aconfiguration space and of Stiefel manifolds:

Φ: Emb(qkDm,V )'→ Conf(k,V )× (Stiefelm(V ))k

Proof. Φ(f ) is determined by:

1 The configuration of the centers of the embedded disks

2 The k-tuple of the derivatives at the centers of the disks.

Boring? Actually Emb(qkDm,V ) is a fundamental building blockfor Emb(M,V ) when M is a general smooth manifold.

A more complicated example of Emb(M , V ): M = S1

Embeddings of the circle M = S1 in V = Rv

v = 1 Emb(S1,R) = ∅

v = 2 Emb(S1,R2) ' S1 × Z/2v = 3 π0(Emb(S

1,R3)) = set of isotopy classes of knots !!!

The path components of Emb(Ŝ1,R3) are different.Here is a typical example:' S1 × (Conf(4,R2)×Σ4 (S1)4) .

All path components are constructed in a similar way(Hatcher; Budney 2006).

v ≥ 4 Emb(S1,Rv ) is connected but still complicated.H∗(Emb(Ŝ1,Rv );Q) is explicitely computable as thehomology of a graph complex of generalized chord diagrams(L.-Turchin-Volić).



v = 1 Emb(S1,R) = ∅v = 2 Emb(S1,R2) ' S1 × Z/2

v = 3 π0(Emb(S1,R3)) = set of isotopy classes of knots !!!






v = 1 Emb(S1,R) = ∅v = 2 Emb(S1,R2) ' S1 × Z/2v = 3 π0(Emb(S

1,R3)) = set of isotopy classes of knots !!!




Our main theorem

Theorem (Arone - L. -Volić)

Assume that dim(V ) > 4 dim(M).There is an “orthogonal spectral sequence”

E 1p,q =⇒ H∗(Emb(M,V );Q)which collapses at page E 1.

Corollary

If M and M ′ are H∗(−;Q)-equivalent manifolds thenH∗(Emb(M,V );Q) ∼= H∗(Emb(M ′,V );Q)

when dim(V ) > 4 ·max(dim(M), dim(M ′)).

Examples

Emb(RP(2n),V ) is a torsion space.

H∗(Emb(S3 × S6,V );Q)

∼= H∗(Emb(S3 × S6#RP(9),V );Q)∼= H∗(Emb(RP(3)× S6 × RP(12),V );Q)

Spaces of immersions: definition and examples

Definition

Define the space of immersions

Imm(M,V ) := {f : M # V | f is a smooth immersion}

with a suitable topology. There is a continuous inclusion

ι : Emb(M,V ) ↪→ Imm(M,V ).

ExamplesM Emb(M,V ) Imm(M,V )

Dm Stiefelm(V ) Stiefelm(V )qkDm Conf(k,V )× (Stiefelm(V ))k (Stiefelm(V ))kS1 complicated... map(S1,S(V )))

(free loop space on the sphere)

Smale’s cutting method to determine Imm(S1, V )

Immersions of the circle M = S1

O1 := thickened northern hemicircle ∼= [−1, 1]O2 := thickened southern hemicircle ∼= [−1, 1]O1 ∩ O2 = thickened equator ∼= S0 × [−�, �]S1 =(homotopy)-push-out of {O1 ←↩ O1 ∩ O2 ↪→ O2}

O1 ∩ O2 � //

_

��(h.)p.o.

O1 _

��O2

� // S1

Imm(−,V ) Imm(O1 ∩ O2,V ) Imm(O1,V )oooo

Imm(O2,V )

OOOO

Imm(S1,V )oo

OO

h.p.b.

Theorem (Smale, 1957)

Assume that dim V > dim M.An inclusion of compact codim 0 submanifolds j : O ↪→ O ′induces a fibration of immersions spaces

j∗ = restr : Imm(O ′,V )� Imm(O,V )

Imm(S1, V ) is the free loop space on the sphere S(V )

Theorem (Smale)

Imm(S1,V ) ' map(S1,S(V ))(the free loop space on the sphere Sv−1)

Proof.

S(V )× S(V ) S(V )∆oo

Imm(O1 ∩ O2,V )

'iiRRRRRRRRRRRRR

Imm(O1,V )oooo

'

OO

S(V )

∆

OO

Imm(O2,V )'oo

OOOO

Imm(S1,V )oo

OO

h.p.b.

The eversion of the sphere

Exercise. Use the same method to prove that Imm(S2,R3) isconnected (Smale, 1958).

Eversion of the sphere (twice)

eversionsphere.qtMedia File (video/quicktime)

Smale-Hirsh’s cutting method to determine Imm(M , V )

The method of Smale works as well for manifolds M other thanspheres.We have a contravariant functor

Imm: O(M) := {open subsets of M} −→ TopO 7→ Imm(O,V )

(j : O ↪→ O ′) 7→(j∗ = restr : Imm(O ′,V )� Imm(O,V )

)Theorem (Smale-Hirsh)

Assume that dim V > dim M.Let I 7→ O(I ) be a diagram of open sets of M. Then

Imm(hocolimI

O(I ) , V ) ' holimI

Imm(O(I ) , V )

Smale-Hirsh method as a right Kan extension

O1(M) := a suitable subcategory of O(M) consisting of opensets that are either empty or diffeo to a single open m-disk Ḋm.

Theorem (Smale-Hirsh)

If dim V > dim M then

Imm(M,V ) ' holimO∈O1(M)

Imm(O,V )

O ∈ O1(M) =⇒ Imm(O,V ) ' Stiefelm(V ) or Imm(O,V ) = ∗.

Corollary

If M is parallelizable then Imm(M,V ) ' map(M,Stiefelm(V )).

Remarks

Imm(M,V ) is very well understood

Imm(−,V ) is not a homotopy functor of M.

Goodwillie’s cutting method for Emb(M , V )

Naive guess: Emb(M,V ) ' holimO∈O1(M) Emb(O,V )?

WRONG!

O ∈ O1(M) =⇒ ι : Emb(O,V )'↪→ Imm(O,V )

Smale-Hirsh=⇒ holim

O∈O1(M)Emb(O,V ) ' Imm(M,V )

It does not work because being an embedding is not a localproperty and a single disk is too local.O∞(M) := {open subsets diffeo to a finite disjoint union of disks}

Theorem (Goodwillie-Klein)

If dimV > dim M + 2 then

Emb(M,V ) ' holimO∈O∞(M)

Emb(O,V ).

O ∈ O∞(M) =⇒ Emb(O,V ) ' Conf(k,V )× (Stiefelm(V ))k .


Naive guess: Emb(M,V ) ' holimO∈O1(M) Emb(O,V )? WRONG!

O ∈ O1(M) =⇒ ι : Emb(O,V )'↪→ Imm(O,V )



It does not work because being an embedding is not a localproperty and a single disk is too local.

O∞(M) := {open subsets diffeo to a finite disjoint union of disks}




Emb(O,V ).



Naive guess: Emb(M,V ) ' holimO∈O1(M) Emb(O,V )? WRONG!

O ∈ O1(M) =⇒ ι : Emb(O,V )'↪→ Imm(O,V )



It does not work because being an embedding is not a localproperty and a single disk is too local.O∞(M) := {open subsets diffeo to a finite disjoint union of disks}




Emb(O,V ).


Embeddings modulo immersions: definition and examples

We can get rid of the Stiefel manifold factor by using the followingvariation of Emb

Definition (Embeddings modulo immersions)

The space of embeddings modulo immersions is

Emb(M,V ) := hofibre(ι : Emb(M,V ) ↪→ Imm(M,V ))

ExamplesM Emb(M,V ) Emb(M,V ) Imm(M,V )

qkDm Conf(k,V ) Conf(k,V )× (Stiefelm(V ))k×(Stiefelm(V ))k

S1 complicated... complicated... (S(V ))S1

Why to look at Emb(M , V )?

Emb(M,V ) is a homotopy invariant of M fordim V > 2 dim M

Imm(M,V ) is very well understood.Therefore any knowledge on Emb(M,V ) transposes to asmuch knowledge on Emb(M,V ).

Goodwillie-Klein theorem works exactly the same forEmb(M,V ) as for Emb(M,V ).

ConclusionWe loose nothing by working with Emb(M,V )...

... we only gain some simplicity.From now on, we work only with Emb.

O ∈ O∞(M) =⇒ Emb(O,V ) ' Conf(k,V )

A concrete example: the space of long knots

Consider M = Ŝ1.The combinatorics of the category O∞(Ŝ1) is very simple: itconsists of unions of intervals on the ĉırcle.

Theorem (Sinha)

There is a cosimplicial space {Conf[k,V ]}k≥0 whose totalizationgives the space of long knots modulo immersions:

Tot(Conf[•,V ]) ' holimO∈O∞(Ŝ1)

Emb(O,V )) ' Emb(Ŝ1,V )

Philosophy: An embedded circle is approximated by a configurationof k � 1 distinct points (x1, · · · , xk), with xi ≈ xi+1 and xk ≈ x1.Analogy: Tot(X×•) ' ΩX .

The rational version of Goodwillie-Klein theorem

Our goal: understand the stable rational homotopy type ofembedding spaces.Two Quillen equivalent viewpoints:

C∗(−;Q) : Top∗ → ChQX 7→ C∗(−;Q)

Σ∞Q∼= HQ ∧ (−)+ : Top∗ → SpectraQ

X 7→ Σ∞Q X

Goodwillie-Klein theorem “stably rationalizes” in the stable range:

Theorem (Weiss)

If dim V > 2 dim(M) + 1 then

C∗(Emb(M,V );Q) ' holimO∈O∞(M)

C∗(Emb(O,V );Q)

Σ∞Q Emb(M,V ) ' holimO∈O∞(M)

Σ∞Q Emb(O,V )

Stable rational formality of the functor Emb(−, V )

Theorem (Arone - L. - Volić)

If dim V > 4 dim M then

Emb(−,V ) : O∞(M)→ Top

is stably rationally formal, i.e. there is a weak equivalence

C∗(Emb(O,V );Q) ' H∗(Emb(O,V );Q)

natural in O ∈ O∞(M).

Proof Use a relative version (in the stable range) of Kontsevich’sformality of the little disks operad.

A first application of the formality of Emb(−, V )

The slogan: Spectral Sequence + Formality =⇒ collapse of SSReason: because an internal differential is zero.

An example: M = Ŝ1

Recall Sinha’s cosimplicial model Conf[•,V ] for Emb(Ŝ1,V ).

Theorem (L.-Turchin-Volić)

Assume that dim V ≥ 4.The cosimplicial space Conf[•,V ] is sort of rationally formal.Therefore its H∗(−;Q)-BKSS collapses at page E2.

Proof Formality can be deduced from the previous theorem.The BKSS is associated to the double complex

⊕k≥0 C∗(Conf[k,V ];Q).By formality it is equivalent to ⊕k≥0 H∗(Conf[k,V ];Q).The vertical differential vanishes.

A first application of the formality of Emb(−, V )

The slogan: Spectral Sequence + Formality =⇒ collapse of SSReason: because an internal differential is zero.An example: M = Ŝ1

Recall Sinha’s cosimplicial model Conf[•,V ] for Emb(Ŝ1,V ).

Theorem (L.-Turchin-Volić)

Assume that dim V ≥ 4.The cosimplicial space Conf[•,V ] is sort of rationally formal.Therefore its H∗(−;Q)-BKSS collapses at page E2.

Proof Formality can be deduced from the previous theorem.The BKSS is associated to the double complex

⊕k≥0 C∗(Conf[k,V ];Q).By formality it is equivalent to ⊕k≥0 H∗(Conf[k,V ];Q).The vertical differential vanishes.

Formality as a splitting of the Postnikov tower

What are we running after ? For a general manifold M,we want a computable spectral sequenceconverging to H∗(Emb(M,V );Q)and collapsing “by a formality argument”.

Reinterpretation of formality: Formality is equivalent to a naturalhomogeneous splitting:

C∗(Emb(O,V );Q) ' H∗(Emb(O,V );Q)∼= ⊕∞p=0 Hp(Emb(O,V );Q)or

Σ∞Q Emb(O,V ) '∞∏

p=0

LPop Σ∞Q Emb(O,V )

Natural strategy: Study carefully the Postnikov tower ofΣ∞Q Emb(O,V ) ' Σ∞Q Conf(k,V ).

The Postnikov tower

The Postnikov tower of a space/spectrum X is a tower offibrations:

LPo2 X

��

LPo1 X

��

LPo0 X

X' // Po∞(X ) // · · · // Po2(X ) // Po1(X ) // Po0(X )

The layers LPop X := hofibre(Pop(X )→ Pop−1(X )) areEilenberg-MacLane spaces/spectra, in other words:the layers are “homogeneous for π∗ in degree p”.

The tower converges: X'→ Po∞(X ) := holimn→∞ Pon(X ).

The Postnikov tower of Σ∞Q Conf(k , V )

What are the layers of the Postnikov tower of Σ∞Q Conf(k,V ) ?

πp(Σ∞Q Conf(k,V ))

∼= Hp(Conf(k,V );Q)The homology of Conf(k,V ) is concentrated in degreesp = n(dim V − 1) for n = 0, 1, 2, 3, · · · (F. Cohen)The Eilenberg-Mac Lane rational spectra are ∨jΣ∞Q Sp.

Therefore

LPop Σ∞Q Conf(k,V ) '

{∨jΣ∞Q S(V )∧n if p = n(dim V − 1)∗ otherwise

Key point V 7→ ∨jΣ∞Q S(V )∧n is a functor which is “homogeneousin V of degree n”.

The orthogonal tower of a functor on vector spaces

The Weiss orthogonal tower of a continuous functorX : {vector spaces} → Top / Spectra , V 7→ X (V )

is a tower of functors:

LW2 X

��

LW1 X

��

LW0 X

X // W∞(X ) // · · · // W2(X ) // W1(X ) // W0(X )

The layers LWn X := hofibre(Wn X →Wn−1 X ) are functors ofV that are “homogeneous in degree n”.Example: V 7→ Σ∞Q S(V )∧n

In good cases the tower converges for dim V � 1:X (V )

'→ (W∞ X )(V ) := holimn→∞(Wn X )(V )

The splitting of Σ∞Q Emb(M , V ) in orthogonal layers

Theorem

Assume that dim V > 4 dim M. We have the following splitting asthe product of the orthogonal layers:

Σ∞Q Emb(M,V ) '∏∞

n=0 LWn (Σ

∞Q Emb(M,−))(V )

Corollary

Assume that dim V > 4 dim M.The spectral sequence associated to the orthogonal tower ofΣ∞Q Emb(M,V )

E 1n,q = πn+q(LWn Σ

∞Q Emb(M,V )) =⇒ Hn+q(Emb(M,V );Q)

collapses at E 1-page.

Usefull?... Yes, because the layers of the orthogonal tower are wellunderstood thanks to a previous work of Arone.

The orthogonal layers of Σ∞Emb(M ,−)

Theorem (Arone)

The layers of the orthogonal tower of V 7→ Σ∞(Q)Emb(M,V )

LWn Σ∞(Q)Emb(M,V )

can be made explicit as (rational) homotopy functors of M.In particular they are rational homology invariants of M.

These layers are fairly explicitely described as certain equivariantmapping spaces associated to spaces of generalized diagrams ofchords attached on M.

Corollary

H∗(Emb(M,V );Q) is a rational homology invariant of M.

Moreover the E 1 page “should be” computable.

Proof of our main theorem

Set EQ(−) := Σ∞Q Emb(−,V ) : O(M)→ Spectra .

EQ(M)

Goodwillie-Klein-Weiss ' holimO∈O∞(M)

EQ(O)

formality of EQ(−) on O∞ ' holimO∈O∞(M)

∏p≥0

LPop EQ(O)

orthogonal regrading of Postnikov ' holimO∈O∞(M)

∏n≥0

LWn EQ(O)

holim commutes with products '∏n≥0

holimO∈O∞(M)

LWn EQ(O)

emb. and orth. calculi commute '∏n≥0

LWn holimO∈O∞(M)

EQ(O)

Goodwillie-Klein-Weiss '∏n≥0

LWn EQ(M)

QED

Where can we go from here ?

Compute E 1pq explicitely.

The little disks operad is also coformal. We have deduced acollapsing of a SS computing π∗(Emb(Ŝ1,V ))⊗Q.Conjecture. A SS for π∗(Emb(M,V ))⊗Q collapses.Easy generalization to “long planes”, i.e. Emb(Rk ,Rn).Interesting double periodicity phenomenon (work in progresswith Arone and Turchin).

Is the (unstable) rational homotopy type of Emb(M,V ) aninvariant of the rational homotopy type of Emb(M,V )?

What about Emb(M,N) for a general smooth manifold N ?Problem: N 7→ Emb(M,N) is not a homotopy functor evenwhen M is a discrete manifold of dimension 0 !

The E 1 page of the orthogonal SS seems closely related to theVassiliev SS. Understand better this connection.

It seems that the “embedding SS” does not collapse.Understand better that one.

etc, etc, etc...

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