Algebraic Structures on Symplectic Homology and Their Applications

• Applicants: Professor Dr. Kai Cieliebak; Professor Dr. Janko Latschev
• Subject Area: Mathematics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 227710160

• Final Report Year: 2018

Final Report Abstract

The goal of this project was to introduce new algebraic structures on the symplectic homology of Liouville domains and apply these structures to problems in symplectic topology. After a broadening of the scope of algebraic structures to be considered, the following results were obtained: A general setup for obtaining operadic structures on Floer theories from moduli spaces of holomorphic maps (in progress); • a new approach to transversality of holomorphic curves based on scaled Fréchet spaces; • new algebraic structures such as Tate versions of equivariant symplectic homology and a secondary product on positive symplectic cohomology; • Viterbo transfer maps for exact symplectic embeddings of Liouville cobordisms and applications, such as obstructions to the existence of Liouville cobordisms between given contact manifolds; • computations of symplectic homology of Brieskorn manifolds and constructions of new infinite families of exotic contact structures; • proof that symplectic homology is finitely generated as an algebra for a large class of Liouville domains with periodic Reeb flow on the boundary.

Publications

• (2018) Symplectic homology and the Eilenberg–Steenrod axioms. Algebr. Geom. Topol. (Algebraic & Geometric Topology) 18 (4) 1953–2130
  Cieliebak, Kai; Oancea, Alexandru
  (See online at https://doi.org/10.2140/agt.2018.18.1953)
• A version of scale calculus and the associated Fredholm theory
  A. Gerstenberger
  
• Periodic Reeb flows and products in symplectic homology
  P. Uebele
  
• Free loop spaces in geometry and topology, IRMA Lect. Math. Theor. Phys. 24, 500p., Eur. Math. Soc. (2015)
  J. Latschev, A Oancea
  (See online at https://doi.org/10.4171/153)
• Homological algebra related to surfaces with boundary
  K. Cieliebak, K. Fukaya, J. Latschev
  
• Symplectic Tate homology, Proc. Lond. Math. Soc. (3) 112 (2016), no. 1, 169–205
  P. Albers, K. Cieliebak, U. Frauenfelder
  (See online at https://doi.org/10.1112/plms/pdv065|)
• Knot contact homology, ´ string topology, and the cord algebra, J. Ec. polytech. Math. 4 (2017), 661–780
  K. Cieliebak, T. Ekholm, J. Latschev, L. Ng
  (See online at https://doi.org/10.5802/jep.55)
• Punctured holomorphic curves and Lagrangian embeddings, Invent. Math. 212 (2018), no. 1, 213–295
  K. Cieliebak, K. Mohnke
  (See online at https://doi.org/10.1007/s00222-017-0767-8)