Unboundedness of some higher Euler classes

Mann, Kathryn¹

¹Department of Mathematics, Cornell University, Ithaca, NY, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 3, pp. 1221-1234

Résumé

We study Euler classes in groups of homeomorphisms of Seifert-fibered 3–manifolds. In contrast to the familiar Euler class for Homeo0(S1) as a discrete group, we show that these Euler classes for Homeo0(M3) as a discrete group are unbounded classes. In fact, we give examples of flat topological M–bundles over a genus 3 surface whose Euler class takes arbitrary values.

Détail

DOI : 10.2140/agt.2020.20.1221

Classification : 57R20, 57M60, 57S25
Keywords: Euler class, Seifert fibered, $3$–manifold, homeomorphism group

Bibliographie

[1] R H Bamler, B Kleiner, Ricci flow and diffeomorphism groups of 3–manifolds, preprint (2017)

[2] R H Bing, An alternative proof that 3–manifolds can be triangulated, Ann. of Math. 69 (1959) 37 | DOI

[3] M Bucher, T Gelander, Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes, C. R. Math. Acad. Sci. Paris 346 (2008) 661 | DOI

[4] D Calegari, Circular groups, planar groups, and the Euler class, from: "Proceedings of the Casson Fest" (editors C Gordon, Y Rieck), Geom. Topol. Monogr. 7, Geom. Topol. Publ. (2004) 431 | DOI

[5] J L Clerc, B Ørsted, The Gromov norm of the Kaehler class and the Maslov index, Asian J. Math. 7 (2003) 269 | DOI

[6] A Domic, D Toledo, The Gromov norm of the Kaehler class of symmetric domains, Math. Ann. 276 (1987) 425 | DOI

[7] C J Earle, J Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967) 557 | DOI

[8] R D Edwards, R C Kirby, Deformations of spaces of imbeddings, Ann. of Math. 93 (1971) 63 | DOI

[9] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982) 5

[10] A Hatcher, Homeomorphisms of sufficiently large P2–irreducible 3–manifolds, Topology 15 (1976) 343 | DOI

[11] M W Hirsch, W P Thurston, Foliated bundles, invariant measures and flat manifolds, Ann. of Math. 101 (1975) 369 | DOI

[12] S Hong, J Kalliongis, D Mccullough, J H Rubinstein, Diffeomorphisms of elliptic 3–manifolds, 2055, Springer (2012) | DOI

[13] N V Ivanov, Diffeomorphism groups of Waldhausen manifolds, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 66 (1976) 172

[14] N V Ivanov, The second bounded cohomology group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 167 (1988) 117

[15] K Mann, C Rosendal, Large-scale geometry of homeomorphism groups, Ergodic Theory Dynam. Systems 38 (2018) 2748 | DOI

[16] S Matsumoto, S Morita, Bounded cohomology of certain groups of homeomorphisms, Proc. Amer. Math. Soc. 94 (1985) 539 | DOI

[17] D Mccullough, T Soma, The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold, J. Differential Geom. 93 (2013) 327 | DOI

[18] J Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958) 215 | DOI

[19] E E Moise, Affine structures in 3–manifolds, V : The triangulation theorem and Hauptvermutung, Ann. of Math. 56 (1952) 96 | DOI

[20] N Monod, An invitation to bounded cohomology, from: "International Congress of Mathematicians" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc. (2006) 1183

[21] W Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974) 304 | DOI

[22] A V Černavskiĭ, Local contractibility of the group of homeomorphisms of a manifold, Mat. Sb. 79 (121) (1969) 307

[23] J W Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971) 257 | DOI