On diffeomorphisms of even-dimensional discs

Kupers, Alexander; Randal-Williams, Oscar

doi:10.1090/jams/1040

On diffeomorphisms of even-dimensional discs
HTML articles powered by AMS MathViewer

by Alexander Kupers and Oscar Randal-Williams

J. Amer. Math. Soc.

DOI: https://doi.org/10.1090/jams/1040

Published electronically: January 18, 2024

HTML | PDF

Abstract:

We determine $\pi _*(B\operatorname {Diff}_\partial (D^{2n})) \otimes \mathbb {Q}$ for $2n \geq 6$ completely in degrees $* \leq 4n-10$, far beyond the pseudoisotopy stable range. Furthermore, above these degrees we discover a systematic structure in these homotopy groups: we determine them outside of certain “bands” of degrees.

References

V. I. Arnol′d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231 (Russian). MR 242196
Alexander Berglund, Koszul spaces, Trans. Amer. Math. Soc. 366 (2014), no. 9, 4551–4569. MR 3217692, DOI 10.1090/S0002-9947-2014-05935-7
Alexander Berglund and Jonas Bergström, Hirzebruch $L$-polynomials and multiple zeta values, Math. Ann. 372 (2018), no. 1-2, 125–137. MR 3856808, DOI 10.1007/s00208-018-1647-2
Alexander Berglund and Ib Madsen, Rational homotopy theory of automorphisms of manifolds, Acta Math. 224 (2020), no. 1, 67–185. MR 4086715, DOI 10.4310/acta.2020.v224.n1.a2
A. J. Berrick, Group epimorphisms preserving perfect radicals, and the plus-construction, Algebraic $K$-theory, number theory, geometry and analysis (Bielefeld, 1982) Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 1–12. MR 750672, DOI 10.1007/BFb0072013
R. Bezrukavnikov, Koszul DG-algebras arising from configuration spaces, Geom. Funct. Anal. 4 (1994), no. 2, 119–135. MR 1262702, DOI 10.1007/BF01895836
Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 387496, DOI 10.24033/asens.1269
Armand Borel, Stable real cohomology of arithmetic groups. II, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, MA, 1981, pp. 21–55. MR 642850
A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault., DOI 10.1007/BF02566134
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 365573, DOI 10.1007/978-3-540-38117-4
Angeline Brandt, The free Lie ring and Lie representations of the full linear group, Trans. Amer. Math. Soc. 56 (1944), 528–536. MR 11305, DOI 10.1090/S0002-9947-1944-0011305-0
E. S. Chibrikov, The right-normed basis of a free Lie superalgebra and Lyndon-Shirshov words, Algebra Logika 45 (2006), no. 4, 458–483, 504 (Russian, with Russian summary); English transl., Algebra Logic 45 (2006), no. 4, 261–276. MR 2287651, DOI 10.1007/s10469-006-0024-5
F. R. Cohen, On configuration spaces, their homology, and Lie algebras, J. Pure Appl. Algebra 100 (1995), no. 1-3, 19–42. MR 1344842, DOI 10.1016/0022-4049(95)00054-Z
F. R. Cohen and S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748. MR 1881013, DOI 10.1090/S0002-9947-02-02948-3
F. R. Cohen and L. R. Taylor, Computations of Gel′fand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 657, Springer, Berlin-New York, 1978, pp. 106–143. MR 513543
James Conant and Karen Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003), 1167–1224. MR 2026331, DOI 10.2140/agt.2003.3.1167
Pierre Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A203–A208 (French, with English summary). MR 507760
V. G. Drinfel′d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989), no. 6, 114–148 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 6, 1419–1457. MR 1047964
W. G. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255–265. MR 394663, DOI 10.1016/0040-9383(74)90018-4
F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, RI, 1978, pp. 325–337. MR 520509
Benoit Fresse, Modules over operads and functors, Lecture Notes in Mathematics, vol. 1967, Springer-Verlag, Berlin, 2009. MR 2494775, DOI 10.1007/978-3-540-89056-0
Benoit Fresse, Homotopy of operads and Grothendieck-Teichmüller groups. Part 2, Mathematical Surveys and Monographs, vol. 217, American Mathematical Society, Providence, RI, 2017. The applications of (rational) homotopy theory methods. MR 3616816, DOI 10.1090/surv/217.2
Søren Galatius and Oscar Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014), no. 2, 257–377. MR 3207759, DOI 10.1007/s11511-014-0112-7
Søren Galatius and Oscar Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds. II, Ann. of Math. (2) 186 (2017), no. 1, 127–204. MR 3665002, DOI 10.4007/annals.2017.186.1.4
Søren Galatius and Oscar Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds. I, J. Amer. Math. Soc. 31 (2018), no. 1, 215–264. MR 3718454, DOI 10.1090/jams/884
Søren Galatius and Oscar Randal-Williams, Moduli spaces of manifolds: a user’s guide, Handbook of homotopy theory, CRC Press/Chapman Hall Handb. Math. Ser., CRC Press, Boca Raton, FL, [2020] ©2020, pp. 443–485. MR 4197992
Søren Galatius and Oscar Randal-Williams, Algebraic independence of topological Pontryagin classes, J. Reine Angew. Math. 802 (2023), 287–305. MR 4635347, DOI 10.1515/crelle-2023-0051
S. Garoufalidis and E. Getzler, Graph complexes and the symplectic character of the Torelli group. https://arxiv.org/abs/1712.03606, 2017.
Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MR 1301191, DOI 10.1215/S0012-7094-94-07608-4
Thomas G. Goodwillie and John R. Klein, Multiple disjunction for spaces of smooth embeddings, J. Topol. 8 (2015), no. 3, 651–674. MR 3394312, DOI 10.1112/jtopol/jtv008
Thomas G. Goodwillie and Michael Weiss, Embeddings from the point of view of immersion theory. II, Geom. Topol. 3 (1999), 103–118. MR 1694808, DOI 10.2140/gt.1999.3.103
P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 68218, DOI 10.1112/jlms/s1-30.2.154
Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, $K$-Theory 2 (1988), no. 1-2, vi+355. MR 972368, DOI 10.1007/BF00533643
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
Robion C. Kirby and Laurence C. Siebenmann, 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. MR 645390, DOI 10.1515/9781400881505
Toshitake Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 4, 139–160 (English, with French summary). MR 927394, DOI 10.5802/aif.1114
Kazuhiko Koike and Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B_n,\;C_n,\;D_n$, J. Algebra 107 (1987), no. 2, 466–511. MR 885807, DOI 10.1016/0021-8693(87)90099-8
A. Kosiński, On the inertia group of $\pi$-manifolds, Amer. J. Math. 89 (1967), 227–248. MR 214085, DOI 10.2307/2373121
M. Krannich and A. Kupers, The disc-structure space. https://arxiv.org/abs/2205.01755, 2022.
M. Kreck, Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978) Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 643–663. MR 561244
Igor Kříž, On the rational homotopy type of configuration spaces, Ann. of Math. (2) 139 (1994), no. 2, 227–237. MR 1274092, DOI 10.2307/2946581
N. H. Kuiper, Problems concerning manifolds, Manifolds — Amsterdam 1970, 1971, pp. 220–230.
Alexander Kupers, Some finiteness results for groups of automorphisms of manifolds, Geom. Topol. 23 (2019), no. 5, 2277–2333. MR 4019894, DOI 10.2140/gt.2019.23.2277
Alexander Kupers and Oscar Randal-Williams, The cohomology of Torelli groups is algebraic, Forum Math. Sigma 8 (2020), Paper No. e64, 52. MR 4190064, DOI 10.1017/fms.2020.41
Alexander Kupers and Oscar Randal-Williams, On the cohomology of Torelli groups, Forum Math. Pi 8 (2020), e7, 83. MR 4089394, DOI 10.1017/fmp.2020.5
Alexander Kupers and Oscar Randal-Williams, Framings of $W_{g,1}$, Q. J. Math. 72 (2021), no. 3, 1029–1053. MR 4310307, DOI 10.1093/qmath/haaa057
Alexander Kupers and Oscar Randal-Williams, On the Torelli Lie algebra, Forum Math. Pi 11 (2023), Paper No. e13, 47. MR 4575357, DOI 10.1017/fmp.2023.10
G. I. Lehrer and Louis Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra 104 (1986), no. 2, 410–424. MR 866785, DOI 10.1016/0021-8693(86)90225-5
J. Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group” [Algebr. Geom. Topol. 1 (2001), 243–270], Algebr. Geom. Topol. 2 (2002), 1197–1204., DOI 10.2140/agt.2002.2.1197
Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392, DOI 10.1007/978-3-642-30362-3
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144, DOI 10.1093/oso/9780198534891.001.0001
A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8(50) (1940), 405–422 (Russian, with English summary). MR 3420
J. P. May and K. Ponto, More concise algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012. Localization, completion, and model categories. MR 2884233
John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
Jeremy Miller and Martin Palmer, A twisted homology fibration criterion and the twisted group-completion theorem, Q. J. Math. 66 (2015), no. 1, 265–284. MR 3356291, DOI 10.1093/qmath/hau030
Joan Millès, The Koszul complex is the cotangent complex, Int. Math. Res. Not. IMRN 3 (2012), 607–650. MR 2885984, DOI 10.1093/imrn/rnr034
Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki, Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras, Adv. Math. 282 (2015), 291–334. MR 3374528, DOI 10.1016/j.aim.2015.06.017
Brian A. Munson and Ismar Volić, Cubical homotopy theory, New Mathematical Monographs, vol. 25, Cambridge University Press, Cambridge, 2015. MR 3559153, DOI 10.1017/CBO9781139343329
Dan Petersen, Cohomology of generalized configuration spaces, Compos. Math. 156 (2020), no. 2, 251–298. MR 4045070, DOI 10.1112/s0010437x19007747
V. M. Petrogradsky, On Witt’s formula and invariants for free Lie superalgebras, Formal power series and algebraic combinatorics (Moscow, 2000) Springer, Berlin, 2000, pp. 543–551. MR 1798248
Claudio Procesi, Lie groups, Universitext, Springer, New York, 2007. An approach through invariants and representations. MR 2265844
Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
Oscar Randal-Williams, ‘Group-completion’, local coefficient systems and perfection, Q. J. Math. 64 (2013), no. 3, 795–803. MR 3094500, DOI 10.1093/qmath/hat024
Oscar Randal-Williams, An upper bound for the pseudoisotopy stable range, Math. Ann. 368 (2017), no. 3-4, 1081–1094. MR 3673647, DOI 10.1007/s00208-016-1504-0
O. Randal-Williams, The family signature theorem, Proc. Roy. Soc. Edinburgh Sect. A (2023), 1–44.
Oscar Randal-Williams and Nathalie Wahl, Homological stability for automorphism groups, Adv. Math. 318 (2017), 534–626. MR 3689750, DOI 10.1016/j.aim.2017.07.022
Steven V. Sam and Andrew Snowden, Stability patterns in representation theory, Forum Math. Sigma 3 (2015), Paper No. e11, 108. MR 3376738, DOI 10.1017/fms.2015.10
Kevin P. Scannell and Dev P. Sinha, A one-dimensional embedding complex, J. Pure Appl. Algebra 170 (2002), no. 1, 93–107. MR 1896343, DOI 10.1016/S0022-4049(01)00078-0
Pavol Ševera and Thomas Willwacher, Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175–206. MR 2838354, DOI 10.1215/00127094-1443502
Dev Sinha and Benjamin Walter, Lie coalgebras and rational homotopy theory, I: graph coalgebras, Homology Homotopy Appl. 13 (2011), no. 2, 263–292. MR 2861231, DOI 10.4310/HHA.2011.v13.n2.a16
Dmitry E. Tamarkin, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1-2, 65–72. MR 2064592, DOI 10.1023/B:MATH.0000017651.12703.a1
R. M. Thrall, On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64 (1942), 371–388. MR 6149, DOI 10.2307/2371691
Burt Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057–1067. MR 1404924, DOI 10.1016/0040-9383(95)00058-5
Bena Tshishiku, Borel’s stable range for the cohomology of arithmetic groups, J. Lie Theory 29 (2019), no. 4, 1093–1102. MR 4022145
C. T. C. Wall, The action of $\Gamma _{2n}$ on $(n-1)$-connected $2n$-manifolds, Proc. Amer. Math. Soc. 13 (1962), 943–944. MR 143223, DOI 10.1090/S0002-9939-1962-0143223-8
Tadayuki Watanabe, On Kontsevich’s characteristic classes for higher dimensional sphere bundles. I. The simplest class, Math. Z. 262 (2009), no. 3, 683–712. MR 2506314, DOI 10.1007/s00209-008-0396-4
Tadayuki Watanabe, On Kontsevich’s characteristic classes for higher-dimensional sphere bundles. II. Higher classes, J. Topol. 2 (2009), no. 3, 624–660. MR 2546588, DOI 10.1112/jtopol/jtp024
T. Watanabe, Some exotic nontrivial elements of the rational homotopy groups of $\mathrm {Diff}({S}^4)$. https://arxiv.org/abs/1812.02448, 2018.
T. Watanabe, Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $\mathrm {Diff}({S}^4)$ (homological interpretation). https://arxiv.org/abs/2109.01609, 2021.
Michael Weiss, Embeddings from the point of view of immersion theory. I, Geom. Topol. 3 (1999), 67–101. MR 1694812, DOI 10.2140/gt.1999.3.67
Michael Weiss, Erratum to the article Embeddings from the point of view of immersion theory: Part I, Geom. Topol. 15 (2011), no. 1, 407–409. MR 2776849, DOI 10.2140/gt.2011.15.407
Michael S. Weiss, Rational Pontryagin classes of Euclidean fiber bundles, Geom. Topol. 25 (2021), no. 7, 3351–3424. MR 4372633, DOI 10.2140/gt.2021.25.3351
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0
Donald Yau, Lambda-rings, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. MR 2649360, DOI 10.1142/7664