Stability in the Homology of Torelli Groups Jenny Wilson (Michigan) - PowerPoint PPT Presentation

Stability in the Homology of Torelli Groups Jenny Wilson (Michigan) joint with Jeremy Miller (Purdue) and Peter Patzt (Purdue) International Conference on Manifolds, Groups, and Homotopy 18–22 June 2018 Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 1 / 14

Stability in the homology of Torelli groups Σ g , 1 = compact orientable smooth genus- g surface with 1 boundary } component g Today’s goal: Theorem (Miller–Patzt–Wilson) Let I g , 1 denote the Torelli group of Σ g , 1 . The sequence of Sp 2 g p Z q –reps t H 2 p I g , 1 ; Z qu g is centrally stable for g ě 45 . Analogous results (Miller–Patzt–Wilson): IA n Ď Aut p F n q , congruence subgroups of GL n p R q Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 2 / 14

The Mapping Class Group Definition (Mapping Class Group Mod p Σ q ) Surface Σ . Mod p Σ q : “ Diffeo ` p Σ , B Σ q / (isotopy fixing B Σ ). Example (Dehn Twist about γ ) γ – simple closed curve in Σ Theorem (Dehn, Mumford, Lickorish, Humphries) Mod p Σ g , 1 q is f.g. by p 2 g ` 1 q Dehn twists. Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 3 / 14

Action on Homology Mod p Σ g , 1 q ü H 1 p Σ g , 1 , Z q – Z 2 g Mod p Σ g , 1 q ։ Sp 2 g p Z q ù Example (Closed Torus T 2 ) T α β T α p β q “ α ` β α β – Mod p T 2 q Ý Ñ Sp 2 p Z q – SL 2 p Z q „ 1  1 T α ÞÝ Ñ 0 1 „ 1  0 T β ÞÝ Ñ ´ 1 1 Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 4 / 14

The Torelli Group Definition (Torelli group I g , 1 ) Torelli group I g , 1 = kernel of the symplectic representation 1 Ý Ñ I g , 1 Ý Ñ Mod p Σ g , 1 q Ý Ñ Sp 2 g p Z q Ý Ñ 1 Examples of mapping classes in I g , 1 : α separating curve homologous curves T α T ´ 1 T γ P I g , 1 P I g , 1 β γ β Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 5 / 14

Finitness Properties of Torelli Finiteness Properties of Torelli Theorem (McCullough–Miller). I 2 , 1 is not f.g. Theorem (Johnson). I g , 1 is f.g. for g ě 3. Major Open Question . Is I g , 1 finitely presentable for g ě 3? Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 6 / 14

Homology of the Torelli Group Finiteness Properties of Torelli Open Question . Which groups H i p I g , 1 q are f.g.? H i p I g , 1 q – known not f.g. for certain i [Mess, Johnson–Millson–Mess, Hain, Akita, Bestvina–Bux–Margalit] Little is known about H 2 p I g , 1 q . Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 7 / 14

Action on t H ˚ p I g , 1 qu g Key : The sequence t H 2 p I g , 1 qu g has more structure. 1 Ý Ñ I g , 1 Ý Ñ Mod p Σ g , 1 q Ý Ñ Sp 2 g p Z q Ý Ñ 1 Sp 2 g p Z q ü H ˚ p I g , 1 q . ù extend by id Ñ Σ g ` 1 , 1 Σ g , 1 ã { Mod p Σ g , 1 q Ñ Mod p Σ g ` 1 , 1 q respects Torelli ù H ˚ p I g , 1 q Ñ H ˚ p I g ` 1 , 1 q Sp 2 g p Z q –equivariant ù Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 8 / 14

t H 2 p I g , 1 qu as an SI–module Key : Realize t H 2 p I g , 1 qu g as a functor SI Ñ AbGp. Category SI (Putman–Sam) objects = Z 2 g with symplectic structure morphisms = symplectic embeddings Z 2 Z 4 Z 6 0 ¨ ¨ ¨ Sp 2 p Z q Sp 4 p Z q Sp 6 p Z q H 2 p I 1 , 1 q H 2 p I 2 , 1 q H 2 p I 3 , 1 q ¨ ¨ ¨ 0 Sp 2 p Z q Sp 4 p Z q Sp 6 p Z q Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 9 / 14

Results: stability for t H 2 p I g , 1 qu Theorem (Boldsen–Hauge Dollerup) For g ą 6 , Sp 2 g p Z q ¨ im H 2 p I g ´ 1 , 1 ; Q q “ H 2 p I g , 1 ; Q q Theorem (Miller–Patzt–Wilson) H 2 p I g , 1 ; Z q is centrally stable as an SI–module in degree ď 45 . Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 10 / 14

Consequences: stability for t H 2 p I g , 1 qu Corollary (Miller–Patzt–Wilson) The sequence t H 2 p I g , 1 qu g is presentable as an SI–module in degree ď 45 . Corollary (Miller–Patzt–Wilson) The sequence t H 2 p I g , 1 qu g and all maps are determined by 0 Ý Ñ H 2 p I 1 , 1 q Ý Ñ H 2 p I 2 , 1 q Ý Ñ ¨ ¨ ¨ Ý Ñ H 2 p I 45 , 1 q Corollary (Miller–Patzt–Wilson) For g ą 45 , there is a partial resolution Sp 2 g p Z q Sp 2 g p Z q Ind Sp 2 g ´ 4 p Z q H 2 p I g ´ 2 , 1 q Ý Ñ Ind Sp 2 g ´ 2 p Z q H 2 p I g ´ 1 , 1 q Ý Ñ H 2 p I g , 1 q Ý Ñ 0 Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 11 / 14

Proof Ingredients • For an SI–module t V g u , construct a chain complex Sp 2 g p Z q Sp 2 g p Z q ¨ ¨ ¨ Ý Ñ Ind Sp 2 g ´ 4 p Z q V g ´ 2 Ý Ñ Ind Sp 2 g ´ 2 p Z q V g ´ 1 Ý Ñ V g Ý Ñ 0 Main Lemma. If t V g u is a polynomial functor , the homology satisfies a certain regularity result. • Theorem (Hatcher–Vogtmann). The space of tethered chains in ´ ¯ g ´ 3 Σ g , 1 is –connected. 2 Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 12 / 14

Proof Ingredients • spectral sequence analysis (Quillen homological stability argument) H Sp p Z q H Sp p Z q H Sp p Z q H Sp p Z q r r r r 3 p H 3 p I qq 2 g p H 3 p I qq 2 g p H 3 p I qq 2 g p H 3 p I qq 2 g ´ 1 0 1 2 H Sp p Z q H Sp p Z q H Sp p Z q H Sp p Z q r r r r 2 p H 2 p I qq 2 g p H 2 p I qq 2 g p H 2 p I qq 2 g p H 2 p I qq 2 g ´ 1 0 1 2 H Sp p Z q H Sp p Z q H Sp p Z q H Sp p Z q r r r r 1 p H 1 p I qq 2 g p H 1 p I qq 2 g p H 1 p I qq 2 g p H 1 p I qq 2 g ´ 1 0 1 2 H Sp p Z q H Sp p Z q H Sp p Z q H Sp p Z q r r r r 0 p H 0 p I qq 2 g p H 0 p I qq 2 g p H 0 p I qq 2 g p H 0 p I qq 2 g ´ 1 0 1 2 ´ 1 0 1 2 3 Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 13 / 14

Proof Ingredients • spectral sequence analysis (Quillen homological stability argument) 3 ‹ ‹ ‹ ‹ ‹ H Sp p Z q H Sp p Z q H Sp p Z q H Sp p Z q r r r r 2 p H 2 p I qq 2 g p H 2 p I qq 2 g p H 2 p I qq 2 g p H 2 p I qq 2 g ´ 1 0 1 2 d 2 d 2 1 0 0 0 0 ‹ d 3 d 3 0 0 0 0 0 0 ´ 1 0 1 2 3 Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 13 / 14

Thank you! Miller–Patzt–Wilson Stability in the Homology of Torelli Groups Manifolds,Groups,&Homotopy 14 / 14