Factorization homology of stratified spaces

David Ayala, John Francis*, Hiro Lee Tanaka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg–Steenrod axioms; it can also be viewed as an analogue of the Baez–Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topological quantum field theory. Using these axioms, we extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of stratified spaces—this is a nonabelian version of the Poincaré duality given by intersection homology. We pay special attention to the simple case of singular manifolds whose singularity datum is a properly embedded submanifold, and give a further simplified algebraic characterization of these homology theories. In the case of 3-manifolds with one-dimensional submanifolds, these structures give rise to knot and link homology theories.

Original languageEnglish (US)
Pages (from-to)293-362
Number of pages70
JournalSelecta Mathematica, New Series
Volume23
Issue number1
DOIs
StatePublished - Jan 1 2017

Keywords

  • Configuration spaces
  • Factorization homology
  • Knot homology
  • Operads
  • Topological chiral homology
  • Topological quantum field theory
  • ∞-Categories

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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