The Goldman-Millson theorem revisited

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1 The Goldman-Millson theorem revisited Vasily Dolgushev Temple University Based on joint work arxiv: with Christopher L. Rogers. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 1 / 14

2 An L -algebra is... Definition An L -algebra is a cochain complex (L, ) equipped with symmetric multi-brackets of degree 1 (m 2) which satisfy {v 1, v 2,..., v m } m + m 1 + k=2 {,,..., } m : S m (L) L m ±{v 1,..., v i 1, v i, v i+1,..., v m } m i=1 σ Sh k,m k ±{{v σ(1),..., v σ(k) } k, v σ(k+1),..., v σ(m) } m k+1 = 0. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 2 / 14

3 An L -algebra is... Definition An L -algebra is a cochain complex (L, ) equipped with symmetric multi-brackets of degree 1 (m 2) which satisfy {v 1, v 2,..., v m } m + m 1 + k=2 {,,..., } m : S m (L) L m ±{v 1,..., v i 1, v i, v i+1,..., v m } m i=1 σ Sh k,m k ±{{v σ(1),..., v σ(k) } k, v σ(k+1),..., v σ(m) } m k+1 = 0. The base field k has characteristic zero. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 2 / 14

4 The dg cocommutative coalgebra corresponding to L Let (L, ) be a cochain complex and S(L) = L m 2 S m (L) be the space of the truncated symmetric algebra. We view S(L) as the cocommutative coalgebra with the standard comultiplication. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 3 / 14

5 The dg cocommutative coalgebra corresponding to L Let (L, ) be a cochain complex and S(L) = L m 2 S m (L) be the space of the truncated symmetric algebra. We view S(L) as the cocommutative coalgebra with the standard comultiplication. An L -structure on L is a degree 1 coderivation Q on the coalgebra S(L) which satisfies the Maurer-Cartan (MC) equation Q Q = 0 and the condition Q(v) = (v) v L. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 3 / 14

6 The dg cocommutative coalgebra corresponding to L Let (L, ) be a cochain complex and S(L) = L m 2 S m (L) be the space of the truncated symmetric algebra. We view S(L) as the cocommutative coalgebra with the standard comultiplication. An L -structure on L is a degree 1 coderivation Q on the coalgebra S(L) which satisfies the Maurer-Cartan (MC) equation Q Q = 0 and the condition Q(v) = (v) v L. The multi-brackets {,,..., } m are related to Q by the formula where p L is the projection S(L) L. {v 1, v 2,..., v m } m = p L Q(v 1 v 2... v m ), Vasily Dolgushev (Temple University) The Goldman-Millson theorem 3 / 14

7 The dg cocommutative coalgebra corresponding to L Let (L, ) be a cochain complex and S(L) = L m 2 S m (L) be the space of the truncated symmetric algebra. We view S(L) as the cocommutative coalgebra with the standard comultiplication. An L -structure on L is a degree 1 coderivation Q on the coalgebra S(L) which satisfies the Maurer-Cartan (MC) equation Q Q = 0 and the condition Q(v) = (v) v L. The multi-brackets {,,..., } m are related to Q by the formula where p L is the projection S(L) L. {v 1, v 2,..., v m } m = p L Q(v 1 v 2... v m ), To every L -algebra L, we assign the dg cocomm. coalgebra (S(L), Q). Vasily Dolgushev (Temple University) The Goldman-Millson theorem 3 / 14

8 An L -morphism from L to L is... Definition An L -morphism from an L -algebra L to an L -algebra L is a homomorphism U of dg cocommutative coalgebras (S(L), Q) (S( L), Q). Vasily Dolgushev (Temple University) The Goldman-Millson theorem 4 / 14

9 An L -morphism from L to L is... Definition An L -morphism from an L -algebra L to an L -algebra L is a homomorphism U of dg cocommutative coalgebras (S(L), Q) (S( L), Q). Recall that every homomorphism U : S(L) S( L) is uniquely determined by its composition U := p L U : S(L) L with the projection p L : S( L) L and the linear term U (1) := U L : L L is always a chain map from (L, ) to ( L, ). Vasily Dolgushev (Temple University) The Goldman-Millson theorem 4 / 14

10 An L -morphism from L to L is... Definition An L -morphism from an L -algebra L to an L -algebra L is a homomorphism U of dg cocommutative coalgebras (S(L), Q) (S( L), Q). Recall that every homomorphism U : S(L) S( L) is uniquely determined by its composition U := p L U : S(L) L with the projection p L : S( L) L and the linear term U (1) := U L : L L is always a chain map from (L, ) to ( L, ). An L -morphism U is called a quasi-isomorphism if its linear term U (1) induces an isomorphism H (L, ) H ( L, ). Vasily Dolgushev (Temple University) The Goldman-Millson theorem 4 / 14

11 Filtered L -algebras Definition An L -algebra L is filtered if it is equipped with a descending filtration L = F 1 L F 2 L F 3 L, such that L = lim k L/F k L, (F i L) F i L, {F i1 L, F i2 L,..., F im L} m F i1 +i 2 + +i m L m 2. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 5 / 14

12 Filtered L -algebras Definition An L -algebra L is filtered if it is equipped with a descending filtration L = F 1 L F 2 L F 3 L, such that L = lim k L/F k L, (F i L) F i L, {F i1 L, F i2 L,..., F im L} m F i1 +i 2 + +i m L m 2. Assumption: all -morphisms U in question are compatible with the filtrations in the sense that U (F i1 L F i2 L F im L) F i1 +i 2 + +i m L m 1. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 5 / 14

13 Examples: Let A be an associative algebra over k and ε be a formal deformation parameter. Then L A := εc (A, A)[[ε]] (with the Hochschild differential and the Gerstenhaber bracket) is a filtered dg Lie algebra with F k L A := ε k C (A, A)[[ε]]. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 6 / 14

14 Examples: Let A be an associative algebra over k and ε be a formal deformation parameter. Then L A := εc (A, A)[[ε]] (with the Hochschild differential and the Gerstenhaber bracket) is a filtered dg Lie algebra with F k L A := ε k C (A, A)[[ε]]. If O is a dg operad and P is a pseudo-cooperad with P(0) = P(1) = 0 then Conv(P, O) := n 2 Hom Sn (P(n), O(n)) is a filtered dg Lie algebra with F k Conv(P, O) := Hom Sn (P(n), O(n)) n k+1 Vasily Dolgushev (Temple University) The Goldman-Millson theorem 6 / 14

15 Examples: Let A be an associative algebra over k and ε be a formal deformation parameter. Then L A := εc (A, A)[[ε]] (with the Hochschild differential and the Gerstenhaber bracket) is a filtered dg Lie algebra with F k L A := ε k C (A, A)[[ε]]. If O is a dg operad and P is a pseudo-cooperad with P(0) = P(1) = 0 then Conv(P, O) := n 2 Hom Sn (P(n), O(n)) is a filtered dg Lie algebra with F k Conv(P, O) := Hom Sn (P(n), O(n)) n k+1 Let X be a simply-connected space and L X := π i (X) Z Q i 2 be the minimal L -algebra representing the rational homotopy type of X. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 6 / 14

16 MC elements and the de Rham-Sullivan algebra Ω n Recall that a MC element of a (filtered) L -algebra L is a degree 0 element α L which satisfies α + m=2 1 m! {α, α,..., α} m = 0. Denote by MC(L) the set of MC elements of an L -algebra L. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 7 / 14

17 MC elements and the de Rham-Sullivan algebra Ω n Recall that a MC element of a (filtered) L -algebra L is a degree 0 element α L which satisfies α + m=2 1 m! {α, α,..., α} m = 0. Denote by MC(L) the set of MC elements of an L -algebra L. Let Ω n be the de Rham-Sullivan algebra of polynomial differential forms on the geometric simplex n : Ω n := k[t 0, t 1,..., t n, dt 0, dt 1,..., dt n ] / (t t n 1, dt dt n ) Each t i has degree 0, each dt i has degree 1, d(t i ) := dt i. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 7 / 14

18 MC elements and the de Rham-Sullivan algebra Ω n Recall that a MC element of a (filtered) L -algebra L is a degree 0 element α L which satisfies α + m=2 1 m! {α, α,..., α} m = 0. Denote by MC(L) the set of MC elements of an L -algebra L. Let Ω n be the de Rham-Sullivan algebra of polynomial differential forms on the geometric simplex n : Ω n := k[t 0, t 1,..., t n, dt 0, dt 1,..., dt n ] / (t t n 1, dt dt n ) Each t i has degree 0, each dt i has degree 1, d(t i ) := dt i. To a filtered L -algebra L, we assign the simplicial set MC (L) with MC n (L) := MC(L ˆ Ω n ). Vasily Dolgushev (Temple University) The Goldman-Millson theorem 7 / 14

19 Deligne-Getzler-Hinich (DGH) -groupoid A straightforward generalization of Prop 4.7 from E. Getzler, 2009 implies that Proposition For every filtered L -algebra L, the simplicial set MC (L) is a Kan complex. We call MC (L) the DGH -groupoid corresponding to L. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 8 / 14

20 Deligne-Getzler-Hinich (DGH) -groupoid A straightforward generalization of Prop 4.7 from E. Getzler, 2009 implies that Proposition For every filtered L -algebra L, the simplicial set MC (L) is a Kan complex. We call MC (L) the DGH -groupoid corresponding to L. For example, if A is an associative algebra and L A := εc (A, A)[[ε]] then π 0 ( MC (L A ) ) is the set of equivalence classes of 1-parameter formal deformations of A. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 8 / 14

21 More examples: If O is a dg operad, P is a pseudo-cooperad, and L = Conv(P, O) then ( ( ) ) π 0 MC Conv(P, O) is the set of homotopy classes of operad maps Cobar(C) O, where C is obtained from P via adjoining the counit. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 9 / 14

22 More examples: If O is a dg operad, P is a pseudo-cooperad, and L = Conv(P, O) then ( ( ) ) π 0 MC Conv(P, O) is the set of homotopy classes of operad maps Cobar(C) O, where C is obtained from P via adjoining the counit. Theorem (A. Berglund, 2011) For every nilpotent L -algebra L and α MC(L), we have the isomorphism of groups H i (L α ) = π i (MC (L), α), i 1, where L α is obtained from L via twisting the L -structure by α and the group structure on H 1 (L α ) is given the Campbell-Hausdorff formula. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 9 / 14

23 More examples: If O is a dg operad, P is a pseudo-cooperad, and L = Conv(P, O) then ( ( ) ) π 0 MC Conv(P, O) is the set of homotopy classes of operad maps Cobar(C) O, where C is obtained from P via adjoining the counit. Theorem (A. Berglund, 2011) For every nilpotent L -algebra L and α MC(L), we have the isomorphism of groups H i (L α ) = π i (MC (L), α), i 1, where L α is obtained from L via twisting the L -structure by α and the group structure on H 1 (L α ) is given the Campbell-Hausdorff formula. In particular, if MC (L) is simply-connected then MC (L) is a rational space. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 9 / 14

24 The main result is... Observation: Every L -morphism U : L L compatible with filtrations induces the map of simplicial sets MC (U) : MC (L) MC ( L). This way, the construction MC upgrades to a functor from the category of (filtered) L -algebras with L -morphisms to the category of simplicial sets. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 10 / 14

25 The main result is... Observation: Every L -morphism U : L L compatible with filtrations induces the map of simplicial sets MC (U) : MC (L) MC ( L). This way, the construction MC upgrades to a functor from the category of (filtered) L -algebras with L -morphisms to the category of simplicial sets. Theorem (Christopher L. Rogers and V.D.) Let L and L be filtered L -algebras and U be an -morphism from L to L compatible with the filtrations. If the linear term U (1) : L L gives us a quasi-isomorphism U FmL (1) : F ml F m L for every m 1 then MC (U) : MC (L) MC ( L) is a weak equivalence of simplicial sets. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 10 / 14

26 Applications Let ϕ : O Õ be a quasi-isomorphisms of dg operads and C be a dg operad satisfying C(0) = 0 and C(1) = k. Then for every operad map f : Cobar(C) Õ there exists an operad map f : Cobar(C) O such that the diagram Cobar(C) f f O Õ ϕ commutes up to homotopy. The homotopy class of f : Cobar(C) O is uniquely determined by the homotopy class of f. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 11 / 14

27 Applications Let ϕ : O Õ be a quasi-isomorphisms of dg operads and C be a dg operad satisfying C(0) = 0 and C(1) = k. Then for every operad map f : Cobar(C) Õ there exists an operad map f : Cobar(C) O such that the diagram Cobar(C) f f O Õ ϕ commutes up to homotopy. The homotopy class of f : Cobar(C) O is uniquely determined by the homotopy class of f. Please, come to the next talk! Vasily Dolgushev (Temple University) The Goldman-Millson theorem 11 / 14

28 Does the homotopy type of MC (L) depend on the choice of filtration? Let G L be another descending filtration on L such that L = G 1 L, L is complete with respect to this filtration, the multi-brackets are compatible with G L. Let J L be the filtration on L obtained by intersecting J m L := F m L G m L. Then we have the DGH -groupoids MC F (L), MC G (L), and MC J (L) of L constructed with the help of the filtrations F, G and J, respectively. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 12 / 14

29 Does the homotopy type of MC (L) depend on the choice of filtration? Let G L be another descending filtration on L such that L = G 1 L, L is complete with respect to this filtration, the multi-brackets are compatible with G L. Let J L be the filtration on L obtained by intersecting J m L := F m L G m L. Then we have the DGH -groupoids MC F (L), MC G (L), and MC J (L) of L constructed with the help of the filtrations F, G and J, respectively. Claim (based on a discussion with T. Willwacher) The canonical maps of simplicial sets are weak homotopy equivalences. MC F (L) MC J (L) MC G (L) Vasily Dolgushev (Temple University) The Goldman-Millson theorem 12 / 14

30 References [1] A. Berglund, Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras, arxiv: [2] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan, Deligne groupoid revisited, Theory Appl. Categ. 30 (2015), [3] V.A. Dolgushev, A.E. Hoffnung, and C.L. Rogers, What do homotopy algebras form? Adv. Math. 274 (2015) [4] V.A. Dolgushev and C.L. Rogers, A Version of the Goldman-Millson theorem for filtered L -algebras, J. Algebra 430 (2015) [5] E. Getzler, Lie theory for nilpotent L -algebras, Ann. of Math. (2) 170, 1 (2009) [6] M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy type, arxiv: [7] T. Willwacher, private communication. Vasily Dolgushev (Temple University) The Goldman-Millson theorem 13 / 14

31 THANK YOU! Vasily Dolgushev (Temple University) The Goldman-Millson theorem 14 / 14

Erratum to: A Proof of Tsygan s Formality Conjecture for an Arbitrary Smooth Manifold

Erratum to: A Proof of Tsygan s Formality Conjecture for an Arbitrary Smooth Manifold Vasiliy A. Dolgushev arxiv:math/0703113v1 [math.qa] 4 Mar 2007 Abstract Boris Shoikhet noticed that the proof of lemma