Geodesic
The Disc-structure space
Forum of Mathematics, Pi, Tome 12 (2024)

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We study the $\mathscr {D}\mathrm {isc}$-structure space $S^{\mathscr {D}\mathrm {isc}}_\partial (M)$ of a compact smooth manifold M. Informally speaking, this space measures the difference between M, together with its diffeomorphisms, and the diagram of ordered framed configuration spaces of M with point-forgetting and point-splitting maps between them, together with its derived automorphisms. As the main results, we show that in high dimensions, the $\mathscr {D}\mathrm {isc}$-structure space a) only depends on the tangential $2$-type of M, b) is an infinite loop space, and c) is nontrivial as long as M is spin. The proofs involve intermediate results that may be of independent interest, including an enhancement of embedding calculus to the level of bordism categories, results on the behaviour of derived mapping spaces between operads under rationalisation, and an answer to a question of Dwyer and Hess in that we show that the map $\mathrm {BTop}(d)\rightarrow \mathrm {BAut}(E_d)$ is an equivalence if and only if d is at most $2$. 
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Manuel Krannich; Alexander Kupers. The Disc-structure space. Forum of Mathematics, Pi, Tome 12 (2024). doi: 10.1017/fmp.2024.25

[AF15] Ayala, D. and Francis, J., ‘Factorization homology of topological manifolds’, J. Topol. 8(4) (2015), 1045–1084.Google Scholar | DOI

[AFT17] Ayala, D., Francis, J. and Tanaka, H. L., ‘Factorization homology of stratified spaces’, Selecta Math. (N.S.) 23(1) (2017), 293–362.Google Scholar | DOI

[And10] Andrade, R., ‘From manifolds to invariants of En-algebras’, PhD dissertation, 2010, Massachusetts Institute of Technology.Google Scholar

[BBP+17] Basterra, M., Bobkova, I., Ponto, K., Tillmann, U. and Yeakel, S., ‘Infinite loop spaces from operads with homological stability’, Adv. Math. 321 (2017), 391–430.Google Scholar | DOI

[BdBW13] Boavida De Brito, P. and Weiss, M., ‘Manifold calculus and homotopy sheaves’, Homology Homotopy Appl. 15(2) (2013), 361–383.Google Scholar | DOI

[BdBW18] Boavida De Brito, P. and Weiss, M., ‘Spaces of smooth embeddings and configuration categories’, J. Topol. 11(1) (2018), 65–143.Google Scholar | DOI

[BK72] Bousfield, A. K. and Kan, D. M., Homotopy Limits, Completions and Localizations (Lecture Notes in Mathematics) vol. 304 (Springer-Verlag, Berlin-New York, 1972).Google Scholar | DOI

[Bou75] Bousfield, A. K., ‘The localization of spaces with respect to homology’, Topology 14 (1975), 133–150.Google Scholar | DOI

[BW24] Brun, S. and Willwacher, T., ‘Graph homology computations’, New York J. Math. 30 (2024), 58–92.Google Scholar

[Čer69] Černavskiĭ, A. V., ‘Local contractibility of the group of homeomorphisms of a manifold’, Mat. Sb. (N.S.) 79 (121) (1969), 307–356.Google Scholar

[Che08] Chernavskiĭ, A. V., ‘Local contractibility of the homeomorphism group of Rn’, Tr. Mat. Inst. Steklova 263 (2008), no. Geometriya, Topologiya i Matematicheskaya Fizika. I, 201–215.Google Scholar

[CHH18] Chu, H., Haugseng, R. and Heuts, G., ‘Two models for the homotopy theory of ∞-operads’, J. Topol. 11(4) (2018), 857–873.Google Scholar | DOI

[DI04] Dugger, D. and Isaksen, D. C., ‘Topological hypercovers and A1-realizations’, Math. Z. 246(4) (2004), 667–689.Google Scholar | DOI

[DK80a] Dwyer, W. G. and Kan, D. M., ‘Function complexes in homotopical algebra’, Topology 19(4) (1980), 427–440.Google Scholar | DOI

[DK80b] Dwyer, W. G. and Kan, D. M., ‘Simplicial localizations of categories’, J. Pure Appl. Algebra 17(3) (1980), 267–284.Google Scholar | DOI

[DS78] Dydak, J. and Segal, J., Shape Theory (Lecture Notes in Mathematics) vol. 688 (Springer, Berlin, 1978). An introduction.Google Scholar | DOI

[DT22] Ducoulombier, J. and Turchin, V., ‘Delooping the functor calculus tower’, Proc. Lond. Math. Soc. (3) 124(6) (2022), 772–853.Google Scholar | DOI

[Dug78] Dugundji, J., Topology (Allyn and Bacon Series in Advanced Mathematics) (Allyn and Bacon, Inc., Boston, London-Sydney, 1978). Reprinting of the 1966 original.Google Scholar

[Dwy14] Dwyer, W., ‘Introduction to operads’, Lecture at the MSRI ‘Algebraic topology introductory workshop’, 2014, https://www.msri.org/workshops/685/schedules/17876.Google Scholar

[EK71] Edwards, R. D. and Kirby, R. C., ‘Deformations of spaces of imbeddings’, Ann. of Math. (2) 93 (1971), 63–88.Google Scholar | DOI

[ERW22] Ebert, J. and Randal-Williams, O., ‘The positive scalar curvature cobordism category’, Duke Math. J. 171(11) (2022), 2275–2406.Google Scholar | DOI

[EW24] Ebert, J. and Wiemeler, M., ‘On the homotopy type of the space of metrics of positive scalar curvature’, J. Eur. Math. Soc. (JEMS) 26(9) (2024), 3327–3363.Google Scholar | DOI

[Far96] Farjoun, E. D., Cellular Spaces, Null Spaces and Homotopy Localization (Lecture Notes in Mathematics) vol. 1622 (Springer-Verlag, Berlin, 1996).Google Scholar | DOI

[FQ90] Freedman, M. H. and Quinn, F., Topology of 4-Manifolds (Princeton Mathematical Series) vol. 39 (Princeton University Press, Princeton, NJ, 1990).Google Scholar

[Fra13] Francis, J., ‘The tangent complex and Hochschild cohomology of En-rings’, Compos. Math. 149(3) (2013), 430–480.Google Scholar | DOI

[Fre17] Fresse, B., Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 2 (Mathematical Surveys and Monographs) vol. 217 (American Mathematical Society, Providence, RI, 2017).Google Scholar

[FTW17] Fresse, B., Turchin, V. and Willwacher, T., ‘The rational homotopy of mapping spaces of En operads’, Preprint, 2017, .Google Scholar | arXiv

[FTW18] Fresse, B., Turchin, V. and Willwacher, T., ‘The homotopy theory of operad subcategories’, J. Homotopy Relat. Struct. 13(4) (2018), 689–702.Google Scholar | DOI

[GK15] Goodwillie, T. G. and Klein, J. R., ‘Multiple disjunction for spaces of smooth embeddings’, J. Topol. 8(3) (2015), 651–674.Google Scholar | DOI

[GP57] Gleason, A. M. and Palais, R. S., ‘On a class of transformation groups’, Amer. J. Math. 79 (1957), 631–648.Google Scholar | DOI

[GRW14] Galatius, S. and Randal-Williams, O., ‘Stable moduli spaces of high-dimensional manifolds’, Acta Math. 212(2) (2014), 257–377.Google Scholar | DOI

[GRW17] Galatius, S. and Randal-Williams, O., ‘Homological stability for moduli spaces of high dimensional manifolds. II’, Ann. of Math. (2) 186(1) (2017), 127–204.Google Scholar | DOI

[GW99] Goodwillie, T. G. and Weiss, M., ‘Embeddings from the point of view of immersion theory. II’, Geom. Topol. 3 (1999), 103–118.Google Scholar | DOI

[GW24] Göppl, F. and Weiss, M., ‘A spectral sequence for spaces of maps between operads’, Algebr. Geom. Topol. 24(3) (2024), 1655–1690.Google Scholar | DOI

[Har85] Harer, J. L., ‘Stability of the homology of the mapping class groups of orientable surfaces’, Ann. of Math. (2) 121(2) (1985), 215–249.Google Scholar | DOI

[Hat83] Hatcher, A. E., ‘A proof of the Smale conjecture, Diff(S3) ≃ O(4)’, Ann. of Math. (2) 117(3) (1983), 553–607.Google Scholar | DOI

[Hau17] Haugseng, R., ‘The higher Morita category of En-algebras’, Geom. Topol. 21(3) (2017), 1631–1730.Google Scholar | DOI

[Hau18] Haugseng, R., ‘Iterated spans and classical topological field theories’, Math. Z. 289(3–4) (2018), 1427–1488.Google Scholar | DOI

[Hau21] Haugseng, R., ‘Segal spaces, spans, and semicategories’, Proc. Amer. Math. Soc. 149(3) (2021), 961–975.Google Scholar | DOI

[Hau23] Haugseng, R., ‘Some remarks on higher Morita categories, Preprint, 2023, .Google Scholar | arXiv

[HJ20] Hebestreit, F. and Joachim, M., ‘Twisted spin cobordism and positive scalar curvature’, J. Topol. 13(1) (2020), 1–58.Google Scholar | DOI

[HM22] Heuts, G. and Moerdijk, I., Simplicial and Dendroidal Homotopy Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]) vol. 75 (Springer, Cham, 2022).Google Scholar | DOI

[HMS20] Haugseng, R., Melani, V. and Safronov, P., ‘Shifted coisotropic correspondences’, J. Inst. Math. Jussieu (2020), 1–65.Google Scholar

[Hor17] Horel, G., ‘Profinite completion of operads and the Grothendieck-Teichmüller group’, Adv. Math. 321 (2017), 326–390.Google Scholar | DOI

[JK15] Jahren, B. and Kwasik, S., ‘How different can h-cobordant manifolds be?’, Bull. Lond. Math. Soc. 47(4) (2015), 617–630.Google Scholar | DOI

[Kir69] Kirby, R. C., ‘Stable homeomorphisms and the annulus conjecture’, Ann. of Math. (2) 89 (1969), 575–582.Google Scholar | DOI

[KK24a] Knudsen, B. and Kupers, A., ‘Embedding calculus and smooth structures’, Geom. Topol. 28(1) (2024), 353–392.Google Scholar | DOI

[KK24b] Krannich, M. and Kupers, A., ‘Embedding calculus for surfaces’, Algebr. Geom. Topol. 24(2) (2024), 981–1018.Google Scholar | DOI

[KK24c] Krannich, M. and Kupers, A., ‘∞-operadic foundations for embedding calculus’, Preprint, 2024, .Google Scholar | arXiv

[Kre99] Kreck, M., ‘Surgery and duality’, Ann. of Math. (2) 149(3) (1999), 707–754.Google Scholar | DOI

[KuRW25] Kupers, A. and Randal-Williams, O., ‘On diffeomorphisms of even-dimensional discs’, J. Amer. Math. Soc. 38 (2025), 63–178.Google Scholar | DOI

[KrRW21] Krannich, M. and Randal-Williams, O., ‘Diffeomorphisms of discs and the second Weiss derivative of BTop(-)’, Preprint, 2021, .Google Scholar | arXiv

[KS75] Kirby, R. C. and Siebenmann, L. C., ‘Normal bundles for codimension 2 locally flat imbeddings’, in Geometric Topology. (Proc Conf., Park City, Utah, 1974) (Lecture Notes in Math.) vol. 438 (1975), 310–324.Google Scholar | DOI

[KS77] Kirby, R. C. and Siebenmann, L. C., Foundational Essays on Topological Manifolds, Smoothings, and Triangulations (Annals of Mathematics Studies) no. 88 (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977). With notes by John Milnor and Michael Atiyah.Google Scholar | DOI

[Lur09a] Lurie, J., Higher Topos Theory (Annals of Mathematics Studies) vol. 170 (Princeton University Press, Princeton, NJ, 2009).Google Scholar | DOI

[Lur09b] Lurie, J., ‘On the classification of topological field theories’, in Current Developments in Mathematics (Int. Press, Somerville, MA, 2009), 129–280.Google Scholar

[Lur17] Lurie, J., Higher Algebra, version September 2017.Google Scholar

[Mil09] Milnor, J., ‘Collected papers of John Milnor. IV’, in Homotopy, Homology and Manifolds (American Mathematical Society, Providence, RI, 2009). Edited by Mccleary, John.Google Scholar

[MM92] Mcgibbon, C. A. and Møller, J. M., ‘On spaces with the same n-type for all n’, Topology 31(1) (1992), 177–201.Google Scholar | DOI

[MP12] May, J. P. and Ponto, K., More Concise Algebraic Topology: Localization, Completion, and Model Categories (Chicago Lectures in Mathematics) (University of Chicago Press, Chicago, IL, 2012)Google Scholar

[Pal60] Palais, R. S., ‘Local triviality of the restriction map for embeddings’, Comment. Math. Helv. 34 (1960), 305–312.Google Scholar | DOI

[PS18] Pavlov, D. and Scholbach, J., ‘Admissibility and rectification of colored symmetric operads’, J. Topol. 11(3) (2018), 559–601.Google Scholar | DOI

[Rie14] Riehl, E., Categorical Homotopy Theory (New Mathematical Monographs) vol. 24 (Cambridge University Press, Cambridge, 2014).Google Scholar | DOI

[Sal01] Salvatore, P., ‘Configuration spaces with summable labels’, in Cohomological Methods in Homotopy Theory (Bellaterra, 1998) (Progr. Math.) vol. 196 (Birkhäuser, Basel, 2001), 375–395.Google Scholar | DOI

[Sch14] Scheimbauer, C., ‘Factorization homology as a fully extended topological field theory’, PhD dissertation, 2014.Google Scholar

[Sin04] Sinha, D. P., ‘Manifold-theoretic compactifications of configuration spaces’, Selecta Math. (N.S.) 10(3) (2004), 391–428.Google Scholar | DOI

[Sma59] Smale, S., ‘Diffeomorphisms of the 2-sphere’, Proc. Amer. Math. Soc. 10 (1959), 621–626.Google Scholar | DOI

[Ste21] Steimle, W., ‘An additivity theorem for cobordism categories’, Algebr. Geom. Topol. 21(2) (2021), 601–646.Google Scholar | DOI

[Til00] Tillmann, U., ‘Higher genus surface operad detects infinite loop spaces’, Math. Ann. 317(3) (2000), 613–628.Google Scholar | DOI

[Wal71] Wall, C. T. C., ‘Geometrical connectivity. I’, J. London Math. Soc. (2) 3 (1971), 597–604.Google Scholar | DOI

[Wat09] Watanabe, T., ‘On Kontsevich’s characteristic classes for higher-dimensional sphere bundles. II. Higher classes’, J. Topol. 2(3) (2009), 624–660.Google Scholar | DOI

[Wat22] Watanabe, T., ‘Corrigendum: On Kontsevich’s characteristic classes for higher-dimensional sphere bundles ii: Higher classes’, J. Topol. 15(1) (2022), 347–357.Google Scholar | DOI

[Wei99] Weiss, M., ‘Embeddings from the point of view of immersion theory. I’, Geom. Topol. 3 (1999), 67–101.Google Scholar | DOI

[Wei05] Weiss, M., ‘What does the classifying space of a category classify?’, Homology Homotopy Appl. 7(1) (2005), 185–195.Google Scholar | DOI

[Wei11] Weiss, M., ‘Erratum to the article Embeddings from the point of view of immersion theory: Part I’, Geom. Topol. 15(1) (2011), 407–409.Google Scholar | DOI

[Wei21] Weiss, M., ‘Rational Pontryagin classes of Euclidean fiber bundles’, Geom. Topol. 25(7) (2021), 3351–3424.Google Scholar | DOI

[WW01] Weiss, M. and Williams, B., ‘Automorphisms of manifolds’, in Surveys on Surgery Theory, Vol. 2 (Ann. of Math. Stud.) vol. 149 (Princeton Univ. Press, Princeton, NJ, 2001), 165–220.Google Scholar

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