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Journal of Homotopy and Related Structures

, Volume 11, Issue 2, pp 333–374 | Cite as

Tree- versus graph-level quasilocal Poincaré duality on \(S^{1}\)

  • Theo Johnson-FreydEmail author
Article
  • 62 Downloads

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 \(S^1\), 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.

Keywords

Homotopy algebra Poincare duality Locality Properads dioperads Obstruction complexes 

Notes

Acknowledgments

Gabriel C. Drummond-Cole provided enlightening discussions about both the general meaning and the onerous details of the calculations in this paper. Many of those conversations occurred while I was a visitor at the IBS Center for Geometry and Physics (Korea), where I was also provided generous hospitality and a comfortable work environment. The anonymous referee provided many valuable comments improving the content and exposition of this paper. This work is supported by the NSF Grant DMS-1304054.

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2015

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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