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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M.: Thom complexes. Proc. Lond. Math. Soc. 3(11), 291–310 (1961)
Ayala, D., Francis, J.: Factorization homology of topological manifolds. J. Topol. 8(4), 1045–1084 (2015)
Ayala, D., Francis, J.: Zero-pointed manifolds. Preprint
Ayala, D., Francis, J., Rozenblyum, N.: A stratified homotopy hypothesis. Preprint
Ayala, D., Francis, J., Tanaka, H.L.: Local structures on stratified spaces. Preprint
Barwick, C.: On the Q-construction for exact \(\infty \)-categories. http://math.mit.edu/~clarkbar/papers.html
Baez, J., Dolan, J.: Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36(11), 6073–6105 (1995)
Beilinson, A., Drinfeld, V.: Chiral algebras. American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI (2004)
Boardman, J.M., Vogt, R.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347. Springer, Berlin (1973)
Costello, K.: Renormalization and Effective Field Theory. Mathematical Surveys and Monographs, 170. American Mathematical Society, Providence, RI (2011)
Costello, K., Gwilliam, O.: Factorization algebras in perturbative quantum field theory. Preprint. http://www.math.northwestern.edu/~costello/renormalization
Dugger, D., Isaksen, D.: Topological hypercovers and \({\mathbb{A}}^1\)-realizations. Math. Z. 246(4), 667–689 (2004)
Francis, J.: The tangent complex and Hochschild cohomology of \({\cal {E}}_n\)-rings. Compos. Math. 149(3), 430–480 (2013)
Francis, J., Gaitsgory, D.: Chiral Koszul duality. Sel. Math. (N.S.) 18(1), 27–87 (2012)
Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19(2), 135–162 (1980)
Goresky, M., MacPherson, R.: Intersection homology. II. Invent. Math. 72(1), 77–129 (1983)
Joyal, A.: Quasi-categories and Kan complexes. Special volume celebrating the 70th birthday of Professor Max Kelly. J. Pure Appl. Algebra 175(1–3), 207–222 (2002)
Longoni, R., Salvatore, P.: Configuration spaces are not homotopy invariant. Topology 44(2), 375–380 (2005)
Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, 170. Princeton University Press, Princeton (2009)
Lurie, J.: Higher algebra. Preprint, Sept 2014. http://www.math.harvard.edu/~lurie/
Lurie, J.: On the Classification of Topological Field Theories. Current Developments in Mathematics 2008. International Press, Somerville, pp 129-280 (2009)
McDuff, D.: Configuration spaces of positive and negative particles. Topology 14, 91–107 (1975)
Salvatore, P.: Configuration spaces with summable labels. Cohomological methods in homotopy theory (Bellaterra, 1998). Progr. Math. 196, 375–395 (2001)
Segal, G.: Configuration-spaces and iterated loop-spaces. Invent. Math. 21, 213–221 (1973)
Segal, G.: Locality of Holomorphic Bundles, and Locality in Quantum Field Theory. The Many Facets of Geometry. Oxford University Press, Oxford (2010)
Thomas, J.: Kontsevich’s Swiss Cheese Conjecture. Ph.D. Thesis. Northwestern University (2010)
Voronov, A.: The Swiss-cheese operad. Homotopy invariant algebraic structures (Baltimore, MD 1998). Contemp. Math. 239, 365–373 (1999)
Acknowledgments
We are indebted to Kevin Costello for many conversations and his many insights which have motivated and informed the greater part of this work. We also thank Jacob Lurie for illuminating discussions, his inspirational account of topological field theories, and his substantial contribution to the theory of \(\infty \)-categories. J.F. thanks Alexei Oblomkov for helpful conversations on knot homology.
Author information
Authors and Affiliations
Corresponding author
Additional information
D.A. was partially supported by ERC Adv.Grant No. 228082 and by the National Science Foundation under Award No. 0902639. J.F. was supported by the National Science Foundation under Award No. 0902974 and Award No. 1207758. H.L.T. was supported by a National Science Foundation Graduate Research Fellowship, by the Northwestern University Office of the President, by the Centre for Quantum Geometry of Moduli Spaces, and by the National Science Foundation under Award No. DMS-1400761.
Rights and permissions
About this article
Cite this article
Ayala, D., Francis, J. & Tanaka, H.L. Factorization homology of stratified spaces. Sel. Math. New Ser. 23, 293–362 (2017). https://doi.org/10.1007/s00029-016-0242-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-016-0242-1
Keywords
- Factorization homology
- Topological quantum field theory
- Topological chiral homology
- Knot homology
- Configuration spaces
- Operads
- \({{\mathrm{\infty }}}\)-Categories