String Topology and the Based Loop Space Eric J. Malm Simons Center - PowerPoint PPT Presentation

String Topology and the Based Loop Space Eric J. Malm Simons Center for Geometry and Physics Stony Brook University emalm@scgp.stonybrook.edu 2 Aug 2011 Structured Ring Spectra: TNG University of Hamburg

Introduction String Topology Background Hochschild Homology Results and Methods Results String Topology Fix k a commutative ring. Let • M be a closed, k -oriented, smooth manifold of dimension d • LM = Map ( S 1 , M ) Eric J. Malm String Topology and the Based Loop Space 1/12

Introduction String Topology Background Hochschild Homology Results and Methods Results String Topology Fix k a commutative ring. Let • M be a closed, k -oriented, smooth manifold of dimension d • LM = Map ( S 1 , M ) Chas-Sullivan, ’99: H ∗+ d ( LM ) has operations • graded-commutative loop product ○ , from intersection product on M and concatenation product on Ω M • degree-1 cyclic operator ∆ with ∆ 2 = 0, from S 1 rotation Eric J. Malm String Topology and the Based Loop Space 1/12

Introduction String Topology Background Hochschild Homology Results and Methods Results String Topology Fix k a commutative ring. Let • M be a closed, k -oriented, smooth manifold of dimension d • LM = Map ( S 1 , M ) Chas-Sullivan, ’99: H ∗+ d ( LM ) has operations • graded-commutative loop product ○ , from intersection product on M and concatenation product on Ω M • degree-1 cyclic operator ∆ with ∆ 2 = 0, from S 1 rotation Make H ∗+ d ( LM ) a Batalin-Vilkovisky (BV) algebra: • ○ and ∆ combine to produce a degree-1 Lie bracket { , } on H ∗+ d ( LM ) (the loop bracket ) Eric J. Malm String Topology and the Based Loop Space 1/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Hochschild Homology and Cohomology The Hochschild homology and cohomology of an algebra A exhibit similar operations: • HH ∗ ( A ) has a degree-1 Connes operator B with B 2 = 0, • HH ∗ ( A ) has a graded-commutative cup product ∪ and a degree-1 Lie bracket [ , ] compatible with ∪ . Eric J. Malm String Topology and the Based Loop Space 2/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Hochschild Homology and Cohomology The Hochschild homology and cohomology of an algebra A exhibit similar operations: • HH ∗ ( A ) has a degree-1 Connes operator B with B 2 = 0, • HH ∗ ( A ) has a graded-commutative cup product ∪ and a degree-1 Lie bracket [ , ] compatible with ∪ . Goal: Find A so that HH ∗ ( A ) ≅ HH ∗ ( A ) ≅ string topology BV algebra Eric J. Malm String Topology and the Based Loop Space 2/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Hochschild Homology and Cohomology The Hochschild homology and cohomology of an algebra A exhibit similar operations: • HH ∗ ( A ) has a degree-1 Connes operator B with B 2 = 0, • HH ∗ ( A ) has a graded-commutative cup product ∪ and a degree-1 Lie bracket [ , ] compatible with ∪ . Goal: Find A so that HH ∗ ( A ) ≅ HH ∗ ( A ) ≅ string topology BV algebra Candidates: DGAs associated to M 1. C ∗ M , cochains of M : requires M 1-connected Eric J. Malm String Topology and the Based Loop Space 2/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Hochschild Homology and Cohomology The Hochschild homology and cohomology of an algebra A exhibit similar operations: • HH ∗ ( A ) has a degree-1 Connes operator B with B 2 = 0, • HH ∗ ( A ) has a graded-commutative cup product ∪ and a degree-1 Lie bracket [ , ] compatible with ∪ . Goal: Find A so that HH ∗ ( A ) ≅ HH ∗ ( A ) ≅ string topology BV algebra Candidates: DGAs associated to M 1. C ∗ M , cochains of M : requires M 1-connected 2. C ∗ Ω M , chains on the based loop space Ω M Eric J. Malm String Topology and the Based Loop Space 2/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Hochschild Homology and Cohomology The Hochschild homology and cohomology of an algebra A exhibit similar operations: • HH ∗ ( A ) has a degree-1 Connes operator B with B 2 = 0, • HH ∗ ( A ) has a graded-commutative cup product ∪ and a degree-1 Lie bracket [ , ] compatible with ∪ . Goal: Find A so that HH ∗ ( A ) ≅ HH ∗ ( A ) ≅ string topology BV algebra Candidates: DGAs associated to M 1. C ∗ M , cochains of M : requires M 1-connected 2. C ∗ Ω M , chains on the based loop space Ω M Why C ∗ Ω M ? Goodwillie, ’85: H ∗ ( LM ) ≅ HH ∗ ( C ∗ Ω M ) , M connected Eric J. Malm String Topology and the Based Loop Space 2/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Results Theorem (M.) Let M be a connected, k-oriented Poincaré duality space of formal dimension d. Then Poincaré duality induces an isomorphism D ∶ HH ∗ ( C ∗ Ω M ) → HH ∗+ d ( C ∗ Ω M ) . Eric J. Malm String Topology and the Based Loop Space 3/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Results Theorem (M.) Let M be a connected, k-oriented Poincaré duality space of formal dimension d. Then Poincaré duality induces an isomorphism D ∶ HH ∗ ( C ∗ Ω M ) → HH ∗+ d ( C ∗ Ω M ) . Uses “derived” Poincaré duality (Klein, Dwyer-Greenlees-Iyengar) • Generalize (co)homology with local coefficients E to allow C ∗ Ω M -module coefficients Eric J. Malm String Topology and the Based Loop Space 3/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Results Theorem (M.) Let M be a connected, k-oriented Poincaré duality space of formal dimension d. Then Poincaré duality induces an isomorphism D ∶ HH ∗ ( C ∗ Ω M ) → HH ∗+ d ( C ∗ Ω M ) . Uses “derived” Poincaré duality (Klein, Dwyer-Greenlees-Iyengar) • Generalize (co)homology with local coefficients E to allow C ∗ Ω M -module coefficients • Cap product with [ M ] still induces an isomorphism H ∗ ( M ; E ) → H ∗+ d ( M ; E ) . Eric J. Malm String Topology and the Based Loop Space 3/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Results Compatibility of Hochschild operations under D : Theorem (M.) HH ∗ ( C ∗ Ω M ) with the Hochschild cup product and the operator − D − 1 BD is a BV algebra, compatible with the Hochschild Lie bracket. Eric J. Malm String Topology and the Based Loop Space 4/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Results Compatibility of Hochschild operations under D : Theorem (M.) HH ∗ ( C ∗ Ω M ) with the Hochschild cup product and the operator − D − 1 BD is a BV algebra, compatible with the Hochschild Lie bracket. Theorem (M.) When M is a manifold, the composite D Goodwillie HH ∗ ( C ∗ Ω M ) � → HH ∗+ d ( C ∗ Ω M ) → H ∗+ d ( LM ) � � � � � takes this BV structure to that of string topology. Eric J. Malm String Topology and the Based Loop Space 4/12

Introduction String Topology Background Hochschild Homology Results and Methods Results Results Compatibility of Hochschild operations under D : Theorem (M.) HH ∗ ( C ∗ Ω M ) with the Hochschild cup product and the operator − D − 1 BD is a BV algebra, compatible with the Hochschild Lie bracket. Theorem (M.) When M is a manifold, the composite D Goodwillie HH ∗ ( C ∗ Ω M ) � → HH ∗+ d ( C ∗ Ω M ) → H ∗+ d ( LM ) � � � � � takes this BV structure to that of string topology. Generalizes results of Abbaspour-Cohen-Gruher (’05) and Vaintrob (’06) when M ≃ K ( G , 1 ) , so C ∗ Ω M ≃ kG . Eric J. Malm String Topology and the Based Loop Space 4/12

Introduction Background Derived Poincaré Duality Results and Methods Derived Poincaré Duality Replace Ω M with an equivalent top group so C ∗ Ω M a DGA Eric J. Malm String Topology and the Based Loop Space 5/12

Introduction Background Derived Poincaré Duality Results and Methods Derived Poincaré Duality Replace Ω M with an equivalent top group so C ∗ Ω M a DGA • (Co)homology with local coefficients: for E a k [ π 1 M ] -module, H ∗ ( M ; E ) ≅ Ext ∗ H ∗ ( M ; E ) ≅ Tor C ∗ Ω M ( E , k ) , C ∗ Ω M ( k , E ) ∗ Eric J. Malm String Topology and the Based Loop Space 5/12

Introduction Background Derived Poincaré Duality Results and Methods Derived Poincaré Duality Replace Ω M with an equivalent top group so C ∗ Ω M a DGA • (Co)homology with local coefficients: for E a k [ π 1 M ] -module, H ∗ ( M ; E ) ≅ Ext ∗ H ∗ ( M ; E ) ≅ Tor C ∗ Ω M ( E , k ) , C ∗ Ω M ( k , E ) ∗ • For E a C ∗ Ω M -module, take E ⊗ L C ∗ Ω M k and R Hom C ∗ Ω M ( k , E ) as “derived” (co)homology with local coefficients Eric J. Malm String Topology and the Based Loop Space 5/12