## 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.

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## 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.

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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.

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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

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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