Fixed-point free circle actions on 4–manifolds

Chen, Weimin¹

¹Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3253-3303

Résumé

This paper is concerned with fixed-point free S1–actions (smooth or locally linear) on orientable 4–manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4–manifolds. In particular, it is shown that for any finitely presented group with infinite center there are at most finitely many distinct smooth (resp. topological) 4–manifolds which support a fixed-point free smooth (resp. locally linear) S1–action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free S1–actions under some further conditions on the fundamental group. The connection between the classification of the S1–manifolds and the fundamental group is given by a certain decomposition, called a fiber-sum decomposition, of the S1–manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z–splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z–splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free S1–action on the 4–manifold must be infinite, unless the 4–manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3–manifold.

Détail

DOI : 10.2140/agt.2015.15.3253

Classification : 57S15, 57M07, 57M50
Keywords: four-manifolds, circle actions, Rips–Sela theory, geometrization of $3$–orbifolds

Bibliographie

[1] K Asano, Homeomorphisms of prism manifolds, Yokohama Math. J. 26 (1978) 19

[2] S J Baldridge, Seiberg–Witten invariants, orbifolds, and circle actions, Trans. Amer. Math. Soc. 355 (2003) 1669

[3] S Baldridge, Seiberg–Witten vanishing theorem for $S^1$–manifolds with fixed points, Pacific J. Math. 217 (2004) 1

[4] M Boileau, B Leeb, J Porti, Geometrization of $3$–dimensional orbifolds, Ann. of Math. 162 (2005) 195

[5] M Boileau, S Maillot, J Porti, Three-dimensional orbifolds and their geometric structures, Panoramas et Synthèses 15, Soc. Math. France (2003)

[6] M Boileau, J P Otal, Scindements de Heegaard et groupe des homéotopies des petites variétés de Seifert, Invent. Math. 106 (1991) 85

[7] F Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983) 305

[8] S Cappell, S Weinberger, M Yan, Closed aspherical manifolds with center, J. Topol. 6 (2013) 1009

[9] A Casson, D Jungreis, Convergence groups and Seifert fibered $3$–manifolds, Invent. Math. 118 (1994) 441

[10] W Chen, On a notion of maps between orbifolds, II: Homotopy and CW–complex, Commun. Contemp. Math. 8 (2006) 763

[11] W Chen, Seiberg–Witten invariants of $3$–orbifolds and non-Kähler surfaces, J. Gökova Geom. Topol. GGT 6 (2012) 1

[12] W Chen, Hurwitz-type bound, knot surgery, and smooth $\mathbb{S}^1$–four-manifolds, Math. Z. 276 (2014) 267

[13] W Dicks, M J Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics 17, Cambridge Univ. Press (1989)

[14] J Dinkelbach, B Leeb, Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3–manifolds, Geom. Topol. 13 (2009) 1129

[15] W D Dunbar, Nonfibering spherical $3$–orbifolds, Trans. Amer. Math. Soc. 341 (1994) 121

[16] R Fintushel, Locally smooth circle actions on homotopy $4$–spheres, Duke Math. J. 43 (1976) 63

[17] R Fintushel, Circle actions on simply connected $4$–manifolds, Trans. Amer. Math. Soc. 230 (1977) 147

[18] R Fintushel, Classification of circle actions on $4$–manifolds, Trans. Amer. Math. Soc. 242 (1978) 377

[19] R Fintushel, R J Stern, Immersed spheres in $4$–manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995) 145

[20] D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447

[21] A Haefliger, Homotopy and integrability, from: "Manifolds" (editor N H Kuiper), Lecture Notes in Mathematics 197, Springer (1971) 133

[22] A Haefliger, Groupoïdes d'holonomie et classifiants, from: "Transversal structure of foliations" (1984) 70

[23] A Hatcher, On the diffeomorphism group of $S^{1}\times S^{2}$, Proc. Amer. Math. Soc. 83 (1981) 427

[24] J Hempel, $3$–manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)

[25] J A Hillman, Four-manifolds, geometries and knots, Geometry Topology Monographs 5 (2002)

[26] C Hodgson, J H Rubinstein, Involutions and isotopies of lens spaces, from: "Knot theory and manifolds" (editor D Rolfsen), Lecture Notes in Math. 1144, Springer (1985) 60

[27] B Kleiner, J Lott, Notes on Perelman's papers,

[28] K B Lee, F Raymond, The role of Seifert fiber spaces in transformation groups, from: "Group actions on manifolds" (editor R Schultz), Contemp. Math. 36, Amer. Math. Soc. (1985) 367

[29] A M Macbeath, Geometrical realization of isomorphisms between plane groups, Bull. Amer. Math. Soc. 71 (1965) 629

[30] W Magnus, A Karrass, D Solitar, Combinatorial group theory: presentations of groups in terms of generators and relations, Interscience (1966)

[31] D Mccullough, A Miller, Manifold covers of $3$–orbifolds with geometric pieces, Topology Appl. 31 (1989) 169

[32] W H Meeks Iii, P Scott, Finite group actions on $3$–manifolds, Invent. Math. 86 (1986) 287

[33] W H Meeks Iii, S T Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. 112 (1980) 441

[34] G Meng, C H Taubes, $\underline{\mathrm{SW}}=$ Milnor torsion, Math. Res. Lett. 3 (1996) 661

[35] P S Pao, Nonlinear circle actions on the $4$–sphere and twisting spun knots, Topology 17 (1978) 291

[36] G Perelman, The entropy formula for the Ricci flow and its geometric applications,

[37] C Petronio, Spherical splitting of $3$–orbifolds, Math. Proc. Cambridge Philos. Soc. 142 (2007) 269

[38] E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997) 53

[39] J H Rubinstein, On $3$–manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979) 129

[40] J H Rubinstein, J S Birman, One-sided Heegaard splittings and homeotopy groups of some $3$–manifolds, Proc. London Math. Soc. 49 (1984) 517

[41] P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401

[42] P Scott, T Wall, Topological methods in group theory, from: "Homological group theory" (editor C T C Wall), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137

[43] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups, II, Geom. Funct. Anal. 7 (1997) 561

[44] G A Swarup, On a theorem of C B Thomas, J. London Math. Soc. 8 (1974) 13

[45] C B Thomas, The oriented homotopy type of compact $3$–manifolds, Proc. London Math. Soc. 19 (1969) 31

[46] V G Turaev, Towards the topological classification of geometric $3$–manifolds, from: "Topology and geometry — Rohlin Seminar" (editor O Y Viro), Lecture Notes in Math. 1346, Springer (1988) 291

[47] V Turaev, A combinatorial formulation for the Seiberg–Witten invariants of $3$–manifolds, Math. Res. Lett. 5 (1998) 583

[48] T Yoshida, Simply connected smooth $4$–manifolds which admit nontrivial smooth $S^{1}$ actions, Math. J. Okayama Univ. 20 (1978) 25