A short note on 𝜋₁(𝐷𝑖𝑓𝑓_{∂}𝐷^{4𝑘}) for 𝑘≥3

Wang, Wei

doi:10.1090/proc/16908

A short note on $\pi _1(\operatorname {Diff}_{\partial } D^{4k})$ for $k\geq 3$
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by Wei Wang;

Proc. Amer. Math. Soc. 152 (2024), 4067-4073

DOI: https://doi.org/10.1090/proc/16908

Published electronically: August 1, 2024

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Abstract:

Let $\operatorname {Diff}_{\partial }(D^{n})$ be the topological group of diffeomorphisms of $D^{n}$ which agree with the identity near the boundary. In this short note, we compute the fundamental group $\pi _1 \operatorname {Diff}_{\partial }(D^{4k})$ for $k\geq 3$.

References

P. L. Antonelli, D. Burghelea, and P. J. Kahn, The non-finite homotopy type of some diffeomorphism groups, Topology 11 (1972), 1–49. MR 292106, DOI 10.1016/0040-9383(72)90021-3
G. Brumfiel, On the homotopy groups of $\textrm {BPL}$ and $\textrm {PL/O}$, Ann. of Math. (2) 88 (1968), 291–311. MR 234458, DOI 10.2307/1970576
Dan Burghelea and Richard Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. 196 (1974), 37–50. MR 356103, DOI 10.1090/S0002-9947-1974-0356103-2
Dan Burghelea, Richard Lashof, and Melvin Rothenberg, Groups of automorphisms of manifolds, Lecture Notes in Mathematics, Vol. 473, Springer-Verlag, Berlin-New York, 1975. With an appendix (“The topological category”) by E. Pedersen. MR 380841, DOI 10.1007/BFb0079981
Jean Cerf, The pseudo-isotopy theorem for simply connected differentiable manifolds, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Math., Vol. 197, Springer, Berlin-New York, 1971, pp. 76–82. MR 290404
Diarmuid Crowley and Thomas Schick, The Gromoll filtration, $KO$-characteristic classes and metrics of positive scalar curvature, Geom. Topol. 17 (2013), no. 3, 1773–1789. MR 3073935, DOI 10.2140/gt.2013.17.1773
Diarmuid Crowley, Thomas Schick, and Wolfgang Steimle, Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets, J. Topol. 11 (2018), no. 4, 1077–1099. MR 3989438, DOI 10.1112/topo.12081
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
A. E. Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, 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. 3–21. MR 520490
A. Hatcher, A 50-Year View of Diffeomorphism Groups, talk slides, 50th Cornell Topology Festival 2012, available at: https://pi.math.cornell.edu/~hatcher/Papers/Diff(M)2012.pdf
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, $K$-Theory 2 (1988), no. 1-2, vi+355. MR 972368, DOI 10.1007/BF00533643
Daniel C. Isaksen, Guozhen Wang, and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publ. Math. Inst. Hautes Études Sci. 137 (2023), 107–243. MR 4588596, DOI 10.1007/s10240-023-00139-1
Bjørn Jahren, Pseudoisotopies and the Bökstedt trace, Geom. Dedicata 148 (2010), 245–261. MR 2721626, DOI 10.1007/s10711-009-9441-7
Manuel Krannich, A homological approach to pseudoisotopy theory. I, Invent. Math. 227 (2022), no. 3, 1093–1167. MR 4384194, DOI 10.1007/s00222-021-01077-7
A. Kupers, Lectures on diffeomorphism groups of manifolds, https://people.math.harvard.edu/~kupers/teaching/272x/book.pdf, 2019.
S. MacLane, Homology, Reprint of the 1975 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
John Rognes, Two-primary algebraic $K$-theory of pointed spaces, Topology 41 (2002), no. 5, 873–926. MR 1923990, DOI 10.1016/S0040-9383(01)00005-2
John Rognes, The smooth Whitehead spectrum of a point at odd regular primes, Geom. Topol. 7 (2003), 155–184. MR 1988283, DOI 10.2140/gt.2003.7.155
Oscar Randal-Williams, Diffeomorphisms of discs, ICM—International Congress of Mathematicians. Vol. 4. Sections 5–8, EMS Press, Berlin, [2023] ©2023, pp. 2856–2878. MR 4680344
S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR 153022, DOI 10.2307/2372978
T. Watanabe, Some exotic nontrivial elements of the rational homotopy groups of $\operatorname {Diff}(S^4)$, arXiv:1812.02448, 2018.
Michael Weiss, Sphères exotiques et l’espace de Whitehead, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 17, 885–888 (French, with English summary). MR 870913
Michael Weiss and Bruce Williams, Automorphisms of manifolds, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 165–220. MR 1818774

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A short note on $\pi _1(\operatorname {Diff}_{\partial } D^{4k})$ for $k\geq 3$
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Abstract: