Symplectic cohomology and Viterbo's theorem

By M. Abouzaid. 2013

127 citations recorded by the source corpus.

This is a research monograph on symplectic cohomology (disguised as an advanced graduate textbook), which provides a construction of this version of Hamiltonian Floer cohomology for cotangent bundles of closed manifolds. The focus is on the aspects of the theory that have been neglected in the literature: (1) the base is not assumed to be orientable or Spin, (2) local systems on the free loop space are used to define twisted versions of Floer cohomology, (3) a (twisted) Batalin-Vilkovisky structure is constructed, and (4) the BV relation is verified. In this setting (i.e. with all the appropriate twists), a proof of Viterbo's theorem relating symplectic cohomology to the homology of the free loop space is provided, and we show that this map respects the BV structure. Viterbo's theorem is proved by constructing three different maps relating the two sides, and proving that two of the compositions are isomorphisms by using degenerations of moduli spaces of genus 0 Riemann surfaces with boundary. Two of the maps constructed are new, and use ideas inspired by a Lagrangian version of family Floer cohomology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e702f86db434190f6c536a6c07ec2dc825fa3f78

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Citations in the corpus

Covariantly functorial wrapped Floer theory on Liouville sectors (2017, background): “rds the veri- cation of Abouzaid’s generation criterion, which we turn to next. The idea of localizing open-closed maps has appeared earlier in unpublished work of Abouzaid [2], and Abouzaid’s proof [6] of Viterbo’s theorem also served as an inspiration for our work. 1.3 Local-to-global principle fo”
Bounds on spectral norms and barcodes (2018, background): “However for part of our examples, including the case of absolute Hamiltonian Floer homology, one could in fact work with an arbitrary base field K, when a proper system of coherent or canonical (see [119], [100], [1] and references therein) orientations has been set up.”
Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “where K : C(α(f))→ C(α(f))[1] is a homotopy given in matrix form by”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, methodology): “That the latter were insufficient for linearized contact homology to be defined was pointed out to us independently by Janko Latschev and Mohammed Abouzaid.”
Contact homology and virtual fundamental cycles (2015, background): “omology of cotangent bundles should be given by CHlin • (T ∗N) = HS1 • (LN,N) (S1-equivariant homology of the loop space LNrelative to the constant loops, possibly with twisted coeﬃcients asinAbouzaid[Abo14]). This isoriginallydue toCieliebak– Latschev [CL09], and is consistent with the expectation ”
Brieskorn manifolds in contact topology (2013, methodology): “To make this precise, it is convenient to use Abouzaid’s description of orientations in Floer homology; see [2], Sections 1.”