Orientation reversal of manifolds

Müllner, Daniel¹

¹Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 82, 53115 Bonn, Germany

Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2361-2390

Résumé

We call a closed, connected, orientable manifold in one of the categories TOP, PL or DIFF chiral if it does not admit an orientation-reversing automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly chiral if it does not admit a self-map of degree − 1. We prove that there are strongly chiral, smooth manifolds in every oriented bordism class in every dimension ≥ 3. We also produce simply-connected, strongly chiral manifolds in every dimension ≥ 7. For every k ≥ 1, we exhibit lens spaces with an orientation-reversing self-diffeomorphism of order 2k but no self-map of degree − 1 of smaller order.

Détail

DOI : 10.2140/agt.2009.9.2361

Keywords: orientation, reversal, oriented, manifold, chiral, chirality, amphicheiral, amphicheirality, achiral, degree

Bibliographie

[1] D Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965) 365

[2] M Belolipetsky, A Lubotzky, Finite groups and hyperbolic manifolds, Invent. Math. 162 (2005) 459

[3] K S Brown, Cohomology of groups, 87, Springer (1982)

[4] P E Conner, Differentiable periodic maps, 738, Springer (1979)

[5] M H Freedman, F Quinn, Topology of 4–manifolds, 39, Princeton Univ. Press (1990)

[6] P A Griffiths, J W Morgan, Rational homotopy theory and differential forms, 16, Birkhäuser (1981)

[7] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[8] J C Hausmann, Geometric Hopfian and non-Hopfian situations, from: "Geometry and topology (Athens, Ga., 1985)" (editors C McCrory, T Shifrin), Lecture Notes in Pure and Appl. Math. 105, Dekker (1987) 157

[9] M W Hirsch, Differential topology, 33, Springer (1994)

[10] H Hopf, Zur Algebra der Abbildungen von Mannigfaltigkeiten, J. Reine Angew. Math. 163 (1930) 71

[11] B Huppert, Endliche Gruppen. I, 134, Springer (1967)

[12] D W Kahn, Induced maps for Postnikov systems, Trans. Amer. Math. Soc. 107 (1963) 432

[13] K Kawakubo, Orientation reversing involution, J. Math. Kyoto Univ. 16 (1976) 113

[14] M A Kervaire, J W Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963) 504

[15] R Kirby, Problems in low dimensional manifold theory, from: "Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., 1976), Part 2" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 273

[16] S O Kochman, Bordism, stable homotopy and Adams spectral sequences, 7, Amer. Math. Soc. (1996)

[17] D Kotschick, Orientation-reversing homeomorphisms in surface geography, Math. Ann. 292 (1992) 375

[18] M Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999) 707

[19] M Kreck, Simply connected asymmetric manifolds, J. Topol. 2 (2009) 249

[20] W Lück, A basic introduction to surgery theory, from: "Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001)" (editors F T Farrell, L Göttsche, W Lück), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys. (2002) 1

[21] D M Meyer, L Smith, Poincaré duality algebras, Macaulay’s dual systems, and Steenrod operations, 167, Cambridge Univ. Press (2005)

[22] J W Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358

[23] J W Milnor, D Husemoller, Symmetric bilinear forms, 73, Springer (1973)

[24] J W Milnor, J D Stasheff, Characteristic classes, 76, Princeton Univ. Press (1974)

[25] J Morgan, G Tian, Ricci flow and the Poincaré conjecture, 3, Amer. Math. Soc. (2007)

[26] D Müllner, Orientation reversal of manifolds, PhD Thesis, Universität Bonn, Bonner Math. Schriften 392 (2009)

[27] S P Novikov, Topological invariance of rational Pontrjagin classes, Dokl. Akad. Nauk SSSR 163 (1965) 298

[28] P Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953) 458

[29] Ş Papadima, Classification of Poincaré duality algebras over the rationals, Geom. Dedicata 17 (1984) 199

[30] V Puppe, Do manifolds have little symmetry?, J. Fixed Point Theory Appl. 2 (2007) 85

[31] H L Rosenzweig, Bordism of involutions on manifolds, Illinois J. Math. 16 (1972) 1

[32] M Rueff, Beiträge zur Untersuchung der Abbildungen von Mannigfaltigkeiten, Compositio Math. 6 (1939) 161

[33] N Saveliev, Lectures on the topology of 3–manifolds. An introduction to the Casson invariant, , Walter de Gruyter Co. (1999)

[34] N Saveliev, Invariants for homology 3–spheres., 140 (I), Springer (2002)

[35] Z Sela, Endomorphisms of hyperbolic groups. I. The Hopf property, Topology 38 (1999) 301

[36] L Siebenmann, On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology 3–spheres, from: "Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen)" (editors U Koschorke, W D Neumann), Lecture Notes in Math. 788, Springer (1980) 172

[37] C T C Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960) 292

[38] C T C Wall, Resolutions for extensions of groups, Proc. Cambridge Philos. Soc. 57 (1961) 251

[39] C T C Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962) 421

[40] J H C Whitehead, Combinatorial homotopy. I, Bull. Amer. Math. Soc. 55 (1949) 213

[41] A V Zhubr, Closed simply connected six-dimensional manifolds: proofs of classification theorems, Algebra i Analiz 12 (2000) 126