The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
The Lambrechts–Stanley Model of Configuration
Spaces
Najib Idrissi
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Configuration spaces
M: smooth closed n-manifold
(+ future adjectives)
Conf k (M) = {(x1 , . . . , xk ) ∈ M ×k | xi 6= xj ∀i 6= j}
4
2
1
3
Goal
Obtain a CDGA model of Conf k (M) from a CDGA model of M
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Plan
1
The model
2
Action of the Fulton–MacPherson operad
3
Sketch of proof through Kontsevich formality
4
Computing factorization homology
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Models
We are interested in rational/real models
A ' Ω∗ (M) “forms on M”
(de Rham, piecewise polynomial...)
where A is an “explicit” CDGA
M simply connected =⇒ A contains all the rational/real homotopy
type of M
Conf k (M) smooth (but noncompact); we’re looking for a CDGA
' Ω∗ (Conf k (M)) built from A
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Poincaré duality models
Poincaré duality CDGA (A, d, ε) (example: A = H ∗ (M))
• (A, d): finite type connected CDGA;
• ε : An → k such that ε ◦ d = 0;
• Ak ⊗ An−k → k, a ⊗ b 7→ ε(ab) non degenerate.
Theorem (Lambrechts–Stanley 2008)
Any simply connected manifold has such
a model
∼
Ω∗ (M)
R
M
·
k
∼
∃A
∃ε
Remark
Reasonable assumption: ∃ non simply-connected L ' L0 but
Conf k (L) 6' Conf k (L0 ) for k ≥ 2 [Longoni–Salvatore].
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Diagonal class
In cohomology, diagonal class
[M] ∈ Hn (M) 7→ δ∗ [M] ∈ Hn (M × M)
δ(x ) = (x , x )
↔ ∆M ∈ H 2n−n (M × M)
Representative in a Poincaré duality model (A, d, ε):
∆A =
X
(−1)|ai | ai ⊗ ai∨ ∈ (A ⊗ A)n
{ai }: graded basis and ε(ai aj∨ ) = δij (independent of chosen basis)
Properties
• (a ⊗ 1)∆A = (1 ⊗ a)∆A “concentrated around the diagonal”
µ
A
• A ⊗ A −→
A, ∆A 7→ e(A) = χ(A) · volA
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
The Lambrechts–Stanley model
Conf k (Rn ) is a formal space, with cohomology [Arnold–Cohen]:
H ∗ (Conf k (Rn )) = S(ωij )1≤i6=j≤k /I,
deg ωij = n − 1
I = hωji = ±ωij , ωij2 = 0, ωij ωjk + ωjk ωki + ωki ωij = 0i.
GA (k) conjectured model of Conf k (M) = M ×k \
S
i6=j
∆ij
• “Generators”: A⊗k ⊗ S(ωij )1≤i6=j≤k
• Relations:
• Arnold relations for the ωij
• pi∗ (a) · ωij = pj∗ (a) · ωij . (pi∗ (a) = 1 ⊗ · · · ⊗ 1 ⊗ a ⊗ 1 ⊗ · · · ⊗ 1)
• dωij = (pi∗ · pj∗ )(∆A ).
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
First examples
GA (k) = (A⊗k ⊗ S(ωij )1≤i<j≤k /J, dωij = (pi∗ · pj∗ )(∆A ))
GA (0) = R: model of Conf 0 (M) = {∅}
GA (1) = A: model of Conf 1 (M) = M
X
X
A ⊗ A ⊗ 1 ⊕ A ⊗ A ⊗ ω12
GA (2) =
, dω12 = ∆A ⊗ 1
1 ⊗ a ⊗ ω12 ≡ a ⊗ 1 ⊗ ω12
∼
= (A ⊗ A ⊗ 1 ⊕ A ⊗A A ⊗ ω12 , dω12 = ∆A ⊗ 1)
∼
= (A ⊗ A ⊗ 1 ⊕ A ⊗ ω12 , dω12 = ∆A ⊗ 1)
∼
−
→ A⊗2 /(∆A )
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Brief history of GA
1969 [Arnold–Cohen] H ∗ (Conf k (Rn )) ≈ “GH ∗ (Rn ) (k)”
1978 [Cohen–Taylor] E 2 = GH ∗ (M) (k) =⇒ H ∗ (Conf k (M))
~1994 For smooth projective complex manifolds (=⇒ Kähler):
• [Kříž] GH ∗ (M) (k) model of Conf k (M)
• [Totaro] The Cohen–Taylor SS collapses
2004 [Lambrechts–Stanley] A⊗2 /(∆A ) model of Conf 2 (M) for a
2-connected manifold
~2004 [Félix–Thomas, Berceanu–Markl–Papadima] G∨ ∗ (k) ∼
=
H (M)
page E 2 of Bendersky–Gitler SS for H ∗ (M ×k , i6=j ∆ij )
2008 [Lambrechts–Stanley] H ∗ (GA (k)) ∼
=Σk −gVect H ∗ (Conf k (M))
S
2015 [Cordova Bulens] A⊗2 /(∆A ) model of Conf 2 (M) for
dim M = 2m
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
First part of the theorem
Theorem (I.)
Let M be a smooth, closed, simply connected manifold of dimension
≥ 4. Then GA (k) is a model over R of Conf k (M) for all k ≥ 0.
dim M ≥ 3 =⇒ Conf k (M) is simply connected when M is (cf.
Fadell–Neuwirth fibrations).
Corollary
All the real homotopy type of Conf k (M) is contained in (A, d, ε).
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Fulton–MacPherson compactification
FMn (k): Fulton–MacPherson compactification of Conf k (Rn )
7 6
8
3
5
4
2
1
(+ normalization to deal with Rn being noncompact)
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Fulton–MacPherson compactification (2)
FMM (k): similar compactification of Conf k (M)
5
3
6
4
7
1
2
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Operads
Idea
Study all of {Conf k (M)}k≥0 =⇒ more structure.
FMn = {FMn (k)}k≥0 is an operad: we can insert an infinitesimal
configuration into another
2
1
2
1
◦2
2
=
◦
i
FMn (k) × FMn (l) −
→
FMn (k + l − 1),
Najib Idrissi
3
1
1≤i ≤k
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Structure de module
M framed =⇒ FMM = {FMM (k)}k≥0 is a right FMn -module: we
can insert an infinitesimal configuration into a configuration on M
4
3
5
3
1
2
◦3
1
3
=
1
2
2
◦
i
FMM (k) × FMn (l) −
→
FMM (k + l − 1),
Najib Idrissi
1≤i ≤k
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Cohomology of FMn and coaction on GA
H ∗ (FMn ) inherits a Hopf cooperad structure
One can rewrite:
GA (k) = (A⊗k ⊗ H ∗ (FMn (k))/relations, d)
Proposition
χ(M) = 0 =⇒ GA = {GA (k)}k≥0 Hopf right H ∗ (FMn )-comodule
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Motivation
We are looking for something to put here:
∼
∼
GA (k) ←
−?−
→ Ω∗ (FMM (k))
Hunch: if true, then hopefully it fits in something like this!
∼
GA
?
H ∗ (FM
∼
Ω∗ (FMM )
∼
Ω∗ (FM
n)
∼
?
n)
Fortunately, the bottom row is already known: formality of FMn
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Kontsevich’s graph complexes
[Kontsevich] Hopf cooperad Graphsn = {Graphsn (k)}k≥0
1
∈ Graphsn (3)
2








d

3
 
 
·
 

1
2
 
1
2
3
3


=±


1
2
3
1


=

1
2
1
1
±
2
3
3
±
2
3
Theorem (Kontsevich 1999, Lambrechts–Volić 2014)
Najib Idrissi
2
3
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Labeled graph complexes
Recall:
Ω∗PA (M)
∼
R
∼
A
ε
R
k
labeled graph complex GraphsR :
M
x
1
d
x
1
y
2 ∈ GraphsR (1)
y
2
!
=
dx
1
+
X
±
x ∆0R
(∆R )
Najib Idrissi
xy
dy
±
1
2
x
y
±
1
2

x
1
(where x , y ∈ R)
y
2
!
≡
1

y ∆00R 
2
x
σ(y ) · 1
M
Z
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Complete version of the theorem
Theorem (I., complete version)
∼
GA
GraphsR
†
†
H ∗ (FMn )
†
When χ(M) = 0
Najib Idrissi
∼
‡
∼
Graphsn
Ω∗PA (FMM )
‡
∼
Ω∗PA (FMn )
When M is framed
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Factorization homology
FMn -algebra: space B + maps
FMn ◦ B =
G
FMn (k) × B ×k → B
k≥0
→ “homotopy commutative” (up to degree n) algebra
Factorization homology of M with coefficients in B:
Z
M
B := FMM ◦LFMn B = “ TorFMn (FMM , B)”
= hocoeq(FMM ◦ FMn ◦ B ⇒ FMM ◦ B)
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Factorization homology (2)
In chain complexes over R:
Z
M
B := C∗ (FMM ) ◦LC∗ (FMn ) B.
Formality C∗ (FMn ) ' H∗ (FMn ) =⇒
Ho(C∗ (FMn )-Alg) ' Ho(H∗ (FMn )-Alg)
B ↔ B̃
Full theorem + abstract nonsense =⇒
Z
M
L
B ' G∨
A ◦H∗ (FMn ) B̃
much more computable (as soon as B̃ is known)
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Comparison with a theorem of Knudsen
Theorem (Knudsen, 2016)
`
∃Un
Lie-Alg forgetful FMn -Alg
R
M
Un (g) ' C∗CE (A−∗
PL (M) ⊗ g)
Abstract nonsense =⇒
C∗ (FMn )-Alg ←→ H∗ (FMn )-Alg
Un (g) ←→ S(Σ1−n g)
Proposition
∼
L
1−n
1−n
∼ C CE (A−∗ ⊗ g)
G∨
g) −
→ G∨
g) =
A ◦H∗ (FMn ) S(Σ
A ◦H∗ (FMn ) S(Σ
∗
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces
The model
Fulton–MacPherson operad
Sketch of proof
Factorization homology
Thanks!
Thank you for your attention!
arXiv:1608.08054
Najib Idrissi
The Lambrechts–Stanley Model of Configuration Spaces