Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-18T02:13:16.542Z Has data issue: false hasContentIssue false
  • English
  • Français

ZERO-POINTED MANIFOLDS

Part of: Generalized manifolds Topological manifolds

Published online by Cambridge University Press:  02 July 2019

David Ayala  and
John Francis
David Ayala
Affiliation:
Department of Mathematics, Montana State University, Bozeman, MT59717, USA (david.ayala@montana.edu)
John Francis
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL60208-2370, USA (jnkf@northwestern.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin–Thom collapse maps, so as to present a common generalization of PoincarĂ© duality in topology and Koszul duality in ${\mathcal{E}}_{n}$-algebra.

Keywords

Koszul dualityAtiyah dualityfactorization algebras𝓔n-algebrasfactorization homologytopological chiral homologylittle n-disks operad∞-categories

MSC classification

Primary: 57P05: Local properties of generalized manifolds
Secondary: 57N80: Stratifications 57N65: Algebraic topology of manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 785 - 858
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

DA was partially supported by ERC adv.grant no.228082, and by the National Science Foundation under Award 0902639 and Award 1507704. JF was supported by the National Science Foundation under Award 1207758 and Award 1508040.

References

Ayala, D. and Francis, J., Factorization homology of topological manifolds, J. Topol. 8(4) (2015), 1045–1084.CrossRefGoogle Scholar
Ayala, D. and Francis, J., PoincarĂ©/Koszul duality, Comm. Math. Phys. 365(3) (2019), 847–933.CrossRefGoogle Scholar
Ayala, D., Francis, J. and Tanaka, H. L., Local structures on stratified spaces, Adv. Math. 307 (2017), 903–1028.CrossRefGoogle Scholar
Ayala, D., Francis, J. and Tanaka, H. L., Factorization homology of stratified spaces, Selecta Math. (N.S.) 23(1) (2017), 293–362.CrossRefGoogle Scholar
Bandklayder, L., The Dold–Thom theorem via factorization homology, J. Homotopy Relat. Struct. 14(2) (2019), 579–593.CrossRefGoogle Scholar
Beilinson, A. and Drinfeld, V., Chiral Algebras, American Mathematical Society Colloquium Publications, Volume 51 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Boardman, J. M. and Vogt, R., Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics, Volume 347 (Springer, Berlin–New York, 1973). x+257 pp.CrossRefGoogle Scholar
Bödigheimer, C.-F., Stable splittings of mapping spaces, in Algebraic Topology (Seattle, WA, 1985), Lecture Notes in Mathematics, Volume 1286, pp. 174–187 (Springer, Berlin, 1987).Google Scholar
Burghelea, D. and Vigu-Poirrier, M., Cyclic homology of commutative algebras. I, in Algebraic Topology Rational Homotopy, 5172 (Louvain-la-Neuve, 1986), Lecture Notes in Mathematics, Volume 1318 (Springer, Berlin, 1988).Google Scholar
Costello, K. and Gwilliam, O., Factorization Algebras in Quantum Field Theory, 1, New Mathematical Monographs, Volume 31 (Cambridge University Press, Cambridge, 2017).CrossRefGoogle Scholar
Dwyer, W., Weiss, M. and Williams, B., A parametrized index theorem for the algebraic K-theory Euler class, Acta Math. 190(1) (2003), 1–104.CrossRefGoogle Scholar
Farrell, F. T. and Hsiang, W.-C., H-cobordant manifolds are not necessarily homeomorphic, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 741–744.CrossRefGoogle Scholar
Feigin, B. and Tsygan, B., Additive K-theory and crystalline cohomology, Funktsional. Anal. i Prilozhen. 19(2) (1985), 5262, 96.CrossRefGoogle Scholar
Fresse, B., Koszul duality of 𝓔n-operads, Selecta Math. (N.S.) 17(2) (2011), 363–434.CrossRefGoogle Scholar
Freudenthal, H., Über die enden topologischer rĂ€ume und gruppen, Math. Z. 33(1) (1931), 692–713.CrossRefGoogle Scholar
Gaitsgory, D. and Lurie, J., Weil’s conjecture for function fields, preprint, Available at http://www.math.harvard.edu/~lurie/.Google Scholar
Gerstenhaber, M. and Schack, S., A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra 48(3) (1987), 229–247.CrossRefGoogle Scholar
Getzler, E. and Jones, J., Operads, homotopy algebra and iterated integrals for double loop spaces. Unpublished work, preprint, 1994, available at arXiv:hep-th/9403055.Google Scholar
Ginzburg, V. and Kapranov, M., Koszul duality for operads, Duke Math. J. 76(1) (1994), 203–272.CrossRefGoogle Scholar
Glasman, S., A spectrum-level Hodge filtration on topological Hochschild homology, preprint.Google Scholar
Goodwillie, T., Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711. (electronic).CrossRefGoogle Scholar
Joyal, A., Quasi-categories and Kan complexes. Special volume celebrating the 70th birthday of Professor Max Kelly, J. Pure Appl. Algebra 175(1–3) (2002), 207–222.10.1016/S0022-4049(02)00135-4CrossRefGoogle Scholar
Kallel, S., Spaces of particles on manifolds and generalized PoincarĂ© dualities, Q. J. Math. 52(1) (2001), 45–70.CrossRefGoogle Scholar
Kirby, R. and Siebenmann, L., Foundational essays on topological manifolds, smoothings, and triangulations, in With notes by John Milnor and Michael Atiyah, Annals of Mathematics Studies, Volume 88 (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977). vii+355 pp.Google Scholar
Lewis, G., The stable category and generalized Thom spectra. Thesis (PhD)–The University of Chicago. 1978.Google Scholar
Loday, J.-L., OpĂ©rations sur l’homologie cyclique des algĂšbres commutatives, Invent. Math. 96(1) (1989), 205–230.CrossRefGoogle Scholar
Lurie, J., Higher Topos Theory, Annals of Mathematics Studies, Volume 170 (Princeton University Press, Princeton, NJ, 2009). xviii+925 pp.CrossRefGoogle Scholar
Lurie, J., Higher algebra, preprint. available at http://www.math.harvard.edu/~lurie/.Google Scholar
Lurie, J., DAG 10. Preprint. Available at http://www.math.harvard.edu/~lurie/.Google Scholar
May, J. P., The Geometry of Iterated Loop Spaces, Lectures Notes in Mathematics, Vol. 271, (Springer, Berlin-New York, 1972). viii+175 p.CrossRefGoogle Scholar
McDuff, D., Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107.CrossRefGoogle Scholar
Pirashvili, T., Hodge decomposition for higher order Hochschild homology, Ann. Sci. Éc. Norm. SupĂ©r. (4) 33(2) (2000), 151–179.CrossRefGoogle Scholar
Rezk, C., A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353(3) (2001), 973–1007.CrossRefGoogle Scholar
Salvatore, P., Configuration spaces with summable labels, in Cohomological Methods in Homotopy Theory (Bellaterra, 1998), Progress in Mathematics, Volume 196, pp. 375–395 (BirkhĂ€user, Basel, 2001).CrossRefGoogle Scholar
Segal, G., Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221.CrossRefGoogle Scholar
Segal, G., The topology of spaces of rational functions, Acta. Math. 143 (1979), 39–72.CrossRefGoogle Scholar
Segal, G., Locality of holomorphic bundles, and locality in quantum field theory, in The Many Facets of Geometry, pp. 164–176 (Oxford University Press, Oxford, 2010).CrossRefGoogle Scholar
Siebenmann, L., Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136. ibid. 47 (1972), 137–163.CrossRefGoogle Scholar
Weiss, M., Embeddings from the point of view of immersion theory. I, Geom. Topol. 3 (1999), 67–101.10.2140/gt.1999.3.67CrossRefGoogle Scholar