On the bordism group for group actions on the torus
Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 989-1009.

In this short note, we study the bordism problem for group actions on the torus and give examples of groups acting on the torus by diffeomorphisms isotopic to the identity that cannot be extended to an action on a bounding 3-manifold. This solves a question raised in the previous work of the authors.

Dans cette courte note, nous étudions le groupe de bordisme pour l’action d’un groupe sur le tore et nous donnons quelques exemples de groupes agissant sur le tore T par difféomorphismes isotopiques à l’identité et n’admettant pas de prolongement à une action sur une variété de dimension 3 avec bord T. Cela répond à une question posée dans les travaux antérieurs des auteurs.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3480
Classification: 57R50, 57R19, 57M60, 19J35, 55R40, 37C85
Keywords: Bordism, Diffeomorphism groups, Thompson’s group, Geometric 3-manifolds, Euler class
Mot clés : Bordisme, groupes de difféomorphisme, groupe de Thompson, variétés géometriques en dimension 3, classe d’Euler

Mann, Kathryn 1; Nariman, Sam 2

1 Department of Mathematics Cornell University Ithaca, NY, 14850 (USA)
2 Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2022__72_3_989_0,
     author = {Mann, Kathryn and Nariman, Sam},
     title = {On the bordism group for group actions on the torus},
     journal = {Annales de l'Institut Fourier},
     pages = {989--1009},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {3},
     year = {2022},
     doi = {10.5802/aif.3480},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3480/}
}
TY  - JOUR
AU  - Mann, Kathryn
AU  - Nariman, Sam
TI  - On the bordism group for group actions on the torus
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 989
EP  - 1009
VL  - 72
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3480/
DO  - 10.5802/aif.3480
LA  - en
ID  - AIF_2022__72_3_989_0
ER  - 
%0 Journal Article
%A Mann, Kathryn
%A Nariman, Sam
%T On the bordism group for group actions on the torus
%J Annales de l'Institut Fourier
%D 2022
%P 989-1009
%V 72
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3480/
%R 10.5802/aif.3480
%G en
%F AIF_2022__72_3_989_0
Mann, Kathryn; Nariman, Sam. On the bordism group for group actions on the torus. Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 989-1009. doi : 10.5802/aif.3480. https://aif.centre-mersenne.org/articles/10.5802/aif.3480/

[1] Bessières, Laurent; Besson, Gérard; Maillot, Sylvain; Boileau, Michel; Porti, Joan Geometrisation of 3-manifolds, EMS Tracts in Mathematics, 13, European Mathematical Society, 2010 | DOI | MR | Zbl

[2] Bonahon, Francis Cobordism of automorphisms of surfaces, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983) no. 2, pp. 237-270 | DOI | Numdam | MR | Zbl

[3] Browder, William Surgery and the theory of differentiable transformation groups, Proc. Conf. on Transformation Groups (New Orleans, La., 1967), Springer, 1968, pp. 1-46 | MR | Zbl

[4] Cerf, Jean Topologie de certains espaces de plongements, Bull. Soc. Math. Fr., Volume 89 (1961), pp. 227-380 | DOI | Numdam | MR | Zbl

[5] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2011 | DOI | MR | Zbl

[6] Ghys, Étienne Prolongements des difféomorphismes de la sphère, Enseign. Math., Volume 37 (1991) no. 1-2, pp. 45-59 | MR | Zbl

[7] Ghys, Étienne; Sergiescu, Vlad Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., Volume 62 (1987) no. 2, pp. 185-239 | DOI | MR | Zbl

[8] Greenberg, Peter Classifying spaces for foliations with isolated singularities, Trans. Am. Math. Soc., Volume 304 (1987) no. 1, pp. 417-429 | DOI | MR | Zbl

[9] Hamstrom, Mary-Elizabeth Homotopy in homeomorphism spaces, TOP and PL, Bull. Am. Math. Soc., Volume 80 (1974) no. 2, pp. 207-230 | DOI | MR | Zbl

[10] Hatcher, Allen E. On the diffeomorphism group of S 1 ×S 2 , Proc. Am. Math. Soc., Volume 83 (1981) no. 2, pp. 427-430 | DOI | MR | Zbl

[11] Hatcher, Allen E. A Proof of the Smale Conjecture, Diff(S 3 )O(4), Ann. Math., Volume 117 (1983), pp. 553-607 | DOI | MR | Zbl

[12] Hatcher, Allen E. Spaces of incompressible surfaces (1999) (https://arxiv.org/abs/math/9906074)

[13] Hatcher, Allen E.; McCullough, Darryl Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds, Geom. Topol., Volume 1 (1997), pp. 91-109 | DOI | MR | Zbl

[14] Jaco, William H.; Shalen, Peter B. Seifert fibered spaces in 3-manifolds, Memoirs of the American Mathematical Society, 21, American Mathematical Society, 1979 no. 220 | DOI | MR | Zbl

[15] Jahren, Bjorn One-parameter families of spheres in 3-manifolds, Ph. D. Thesis, Princeton University, Princeton, USA (1975), 47 pages | MR

[16] Jekel, Solomon Powers of the Euler class, Adv. Math., Volume 229 (2012) no. 3, pp. 1949-1975 | DOI | MR | Zbl

[17] Johannson, Klaus Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, 761, Springer, 1979 | DOI | MR | Zbl

[18] Kreck, Matthias Bordism of diffeomorphisms and related topics, Lecture Notes in Mathematics, 1069, Springer, 1984 (with an appendix by Neal W. Stoltzfus) | DOI | MR | Zbl

[19] Mann, Kathryn; Nariman, Sam Dynamical and cohomological obstructions to extending group actions, Math. Ann., Volume 377 (2020) no. 3-4, pp. 1313-1338 | DOI | MR | Zbl

[20] Meeks, William H. III; Scott, Peter Finite group actions on 3-manifolds, Invent. Math., Volume 86 (1986) no. 2, pp. 287-346 | DOI | MR | Zbl

[21] Melvin, Paul Bordism of diffeomorphisms, Topology, Volume 18 (1979) no. 2, pp. 173-175 | DOI | MR | Zbl

[22] Neumann, Walter D; Notes on geometry and 3-manifolds, Topology Atlas, 1996

[23] Pardon, John Smoothing finite group actions on three-manifolds, Duke Math. J., Volume 170 (2021) no. 6, pp. 1043-1084 | MR | Zbl

[24] Préaux, Jean-Philippe A survey on Seifert fiber space conjecture (2012) (https://arxiv.org/abs/1202.4142)

[25] Scott, Peter The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[26] Zimmermann, Bruno Das Nielsensche Realisierungsproblem für hinreichend große 3-Mannigfaltigkeiten, Math. Z., Volume 180 (1982) no. 3, pp. 349-359 | DOI | MR | Zbl

Cited by Sources: