Stable Homotopy Refinements and Khovanov homology Robert Lipshitz1 and Sucharit Sarkar2 International Congress of Mathematics Rio de Janeiro, Brazil, August 2018 Special thanks to our collaborator Tyler Lawson, whose perspective is reflected throughout. 1 RL was supported by NSF CAREER Grant DMS-1642067 and NSF FRG Grant DMS-1560783 2 SS was supported by NSF CAREER Grant DMS-1643401 and NSF FRG Grant DMS-1563615 • Morse homology • Floer homology and categorification • The Cohen-Jones-Segal question • A theorem of Carlsson's • Applications of spatial refinements • General strategies for spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Floer homology and categorification • The Cohen-Jones-Segal question • A theorem of Carlsson's • Applications of spatial refinements • General strategies for spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Morse homology • The Cohen-Jones-Segal question • A theorem of Carlsson's • Applications of spatial refinements • General strategies for spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Morse homology • Floer homology and categorification • A theorem of Carlsson's • Applications of spatial refinements • General strategies for spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Morse homology • Floer homology and categorification • The Cohen-Jones-Segal question • Applications of spatial refinements • General strategies for spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Morse homology • Floer homology and categorification • The Cohen-Jones-Segal question • A theorem of Carlsson's • General strategies for spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Morse homology • Floer homology and categorification • The Cohen-Jones-Segal question • A theorem of Carlsson's • Applications of spatial refinements • Flow categories and realization Part 1: Stable homotopy refinements • Morse homology • Floer homology and categorification • The Cohen-Jones-Segal question • A theorem of Carlsson's • Applications of spatial refinements • General strategies for spatial refinements Part 1: Stable homotopy refinements • Morse homology • Floer homology and categorification • The Cohen-Jones-Segal question • A theorem of Carlsson's • Applications of spatial refinements • General strategies for spatial refinements • Flow categories and realization X χ(M) = (−1)ind(p) p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) = 1 + (−1) + 1 + 1 = 2: Categorify Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines 1 − 1 1 c ~ a b Homology Z 0 Z of −rf from p to q 1 d Morse homology c d b f a Categorify Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines 1 − 1 1 c ~ a b Homology Z 0 Z of −rf from p to q 1 d X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: f a 1 − 1 1 c a b Homology Z 0 Z 1 d X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: Categorify f Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X a @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines of −r~ f from p to q 1 − 1 1 Homology Z 0 Z 1 X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: Categorify f Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X a @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines c a of −r~ f from p to q b d 1 − 1 Homology Z 0 Z 1 X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: Categorify f Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X a @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines 1 c a of −r~ f from p to q b d 1 − 1 Homology Z 0 Z X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: Categorify f Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X a @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines 1 c a b of −r~ f from p to q 1 d Homology Z 0 Z X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: Categorify f Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X a @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines 1 − 1 1 c a b of −r~ f from p to q 1 d X Morse homology χ(M) = (−1)ind(p) c d p2Crit(f) = (−1)ind(a) + (−1)ind(b) + (−1)ind(c) + (−1)ind(d) b = 1 + (−1) + 1 + 1 = 2: Categorify f Cn(M; f) = Zhp 2 Crit(f) j ind(p) = ni @ : Cn(M; f) ! Cn−1(M; f) X a @(p) = [#M(p; q)]q: ind(q)=n−1 signed count of flowlines 1 − 1 1 c ~ a b Homology Z 0 Z of −rf from p to q 1 d χ Ozsv´ath-Szab´o'01 Heegaard Floer homology Turaev torsion χ Hutchings '02 Embedded contact homology Turaev torsion χ Kronheimer-Mrowka '07 Monopole Floer homology Turaev torsion χ Morse homology (PDE) Semi-infinite dimensional Ozsv´ath-Szab´oRasmussen '03 Knot Floer homology Alexander polynomial χ Khovanov '99 sl2 Khovanov homology Jones polynomial χ Khovanov-Rozansky '08 HOMFLY-PT homology HOMFLY-PT polynomial χ Rep. theory Seidel-Smith '06 Symplectic Khovanov homology Knot determinant (Combinatorial) (and many others. ) Floer homology and categorification χ Floer '88 Lagrangian Floer homology Intersection number χ Floer '88 Instanton Floer homology Casson invariant χ Morse homology (PDE) Semi-infinite dimensional Ozsv´ath-Szab´oRasmussen '03 Knot Floer homology Alexander polynomial χ Khovanov '99 sl2 Khovanov homology Jones polynomial χ Khovanov-Rozansky '08 HOMFLY-PT homology HOMFLY-PT polynomial χ Rep. theory Seidel-Smith '06 Symplectic Khovanov homology Knot determinant (Combinatorial) (and many others. ) Floer homology and categorification χ Floer '88 Lagrangian Floer homology Intersection number χ Floer '88 Instanton Floer homology Casson invariant χ Ozsv´ath-Szab´o'01 Heegaard Floer homology Turaev torsion χ Hutchings '02 Embedded contact homology Turaev torsion χ Kronheimer-Mrowka '07 Monopole Floer homology Turaev torsion χ Khovanov '99 sl2 Khovanov homology Jones polynomial χ Khovanov-Rozansky '08 HOMFLY-PT homology HOMFLY-PT polynomial χ Rep. theory Seidel-Smith '06 Symplectic Khovanov homology Knot determinant (Combinatorial) (and many others. ) Floer homology and categorification χ Floer '88 Lagrangian Floer homology Intersection number χ Floer '88 Instanton Floer homology Casson invariant χ Ozsv´ath-Szab´o'01 Heegaard Floer homology Turaev torsion χ Hutchings '02 Embedded contact homology Turaev torsion χ Kronheimer-Mrowka '07 Monopole Floer homology Turaev torsion χ Morse homology (PDE) Semi-infinite dimensional Ozsv´ath-Szab´oRasmussen '03 Knot Floer homology Alexander polynomial χ Seidel-Smith '06 Symplectic Khovanov homology Knot determinant (and many others. ) Floer homology and categorification χ Floer '88 Lagrangian Floer homology Intersection number χ Floer '88 Instanton Floer homology Casson invariant χ Ozsv´ath-Szab´o'01 Heegaard Floer homology Turaev torsion χ Hutchings '02 Embedded contact homology Turaev torsion χ Kronheimer-Mrowka '07 Monopole Floer homology Turaev torsion χ Morse homology (PDE) Semi-infinite dimensional Ozsv´ath-Szab´oRasmussen '03 Knot Floer homology Alexander polynomial χ Khovanov '99 sl2 Khovanov homology Jones polynomial χ Khovanov-Rozansky '08 HOMFLY-PT homology HOMFLY-PT polynomial Rep. theory (Combinatorial) Floer homology and categorification χ Floer '88 Lagrangian Floer homology Intersection number χ Floer '88 Instanton Floer homology Casson invariant χ Ozsv´ath-Szab´o'01 Heegaard Floer homology Turaev torsion χ Hutchings '02 Embedded contact homology Turaev torsion χ Kronheimer-Mrowka '07 Monopole Floer homology Turaev torsion χ Morse homology (PDE) Semi-infinite dimensional Ozsv´ath-Szab´oRasmussen '03 Knot Floer homology Alexander polynomial χ Khovanov '99 sl2 Khovanov homology Jones polynomial χ Khovanov-Rozansky '08 HOMFLY-PT homology HOMFLY-PT polynomial χ Rep. theory Seidel-Smith '06 Symplectic Khovanov homology Knot determinant (Combinatorial) (and many others. ) Spatial Refinement Problem. Given a chain complex C∗ with distinguished basis, arising in an interesting way, construct a CW cell ∼ spectrum X with C∗ (X) = C∗ with the distinguished basis given by the cells. The Cohen-Jones-Segal realization question Question. (Cohen-Jones-Segal) Are these Floer homologies the homologies of naturally associated spaces? Seems not have a natural cup product, so perhaps a spectrum (or, sometimes, pro-spectrum) instead of space? The Cohen-Jones-Segal realization question Question. (Cohen-Jones-Segal) Are these Floer homologies the homologies of naturally associated spaces? Seems not have a natural cup product, so perhaps a spectrum (or, sometimes, pro-spectrum) instead of space? Spatial Refinement Problem.