Tree- versus graph-level quasilocal Poincaré duality on S1

Theo Johnson-Freyd*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Among its many corollaries, Poincaré duality implies that the de Rham cohomology of a compact oriented manifold is a commutative Frobenius algebra. Focusing on the case of S1, this paper studies the question of whether this commutative Frobenius algebra structure lifts to a “homotopy” commutative Frobenius algebra structure at the cochain level, under a mild locality-type condition called “quasilocality”. The answer turns out to depend on the choice of context in which to do homotopy algebra—there are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations. If one works at “tree level”, we prove that there is a homotopically-unique quasilocal cochain-level homotopy Frobenius algebra structure lifting the Frobenius algebra structure on cohomology. However, if one works instead at “graph level”, we prove that a quasilocal lift does not exist.

Original languageEnglish (US)
Pages (from-to)333-374
Number of pages42
JournalJournal of Homotopy and Related Structures
Volume11
Issue number2
DOIs
StatePublished - Jun 1 2016

Keywords

  • Homotopy algebra
  • Locality
  • Obstruction complexes
  • Poincare duality
  • Properads
  • dioperads

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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