Cohomological rigidity of real Bott manifolds

Kamishima, Yoshinobu¹ ; Masuda, Mikiya²

¹Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan
²Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan

Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2479-2502

Résumé

A real Bott manifold is the total space of an iterated ℝℙ1–bundle over a point, where each ℝℙ1–bundle is the projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with ℤ∕2–coefficients are isomorphic.

A real Bott manifold is a real toric manifold and admits a flat Riemannian metric invariant under the natural action of an elementary abelian 2–group. We also prove that the converse is true, namely a real toric manifold which admits a flat Riemannian metric invariant under the action of an elementary abelian 2–group is a real Bott manifold.

Détail

DOI : 10.2140/agt.2009.9.2479

Keywords: real toric manifold, real Bott tower, flat Riemannian manifold

Bibliographie

[1] A Borel, F Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958) 458

[2] V M Buchstaber, Circle actions on toric manifolds and their applications, MIMS EPrint: 2006.12

[3] V M Buchstaber, T E Panov, Torus actions and their applications in topology and combinatorics, 24, Amer. Math. Soc. (2002)

[4] S Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008) 2391

[5] S Choi, M Masuda, D Y Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. (to appear)

[6] S Choi, M Masuda, D Y Suh, Topological classification of generalized Bott towers, Trans. Amer. Math. Soc. (2010) 1097

[7] M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417

[8] A Garrison, R Scott, Small covers of the dodecahedron and the 120-cell, Proc. Amer. Math. Soc. 131 (2003) 963

[9] M Grossberg, Y Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994) 23

[10] G Hochschild, J P Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953) 110

[11] P Kleinschmidt, A classification of toric varieties with few generators, Aequationes Math. 35 (1988) 254

[12] S Mac Lane, Homology, 114, Academic Press (1963)

[13] M Masuda, Equivariant cohomology distinguishes toric manifolds, Adv. Math. 218 (2008) 2005

[14] M Masuda, Classification of real Bott manifolds

[15] M Masuda, Cohomological non-rigidity of generalized real Bott manifolds of height 2, Proc. of the Steklov Institute dedicated to the 100th Anniversary of L S Pontryagin (to appear)

[16] M Masuda, T E Panov, Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb. 199 (2008) 95

[17] M Masuda, D Y Suh, Classification problems of toric manifolds via topology, from: "Toric topology", Contemp. Math. 460, Amer. Math. Soc. (2008) 273

[18] G D Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. (1968) 53

[19] A Nazra, Real Bott tower, Masters thesis, Tokyo Metropolitan University (2008) (A summary can be found in Diffeomorphism type of real Bott towers, Geometry of transformation groups and related topics, RIMS Kôkyûroku No. 1612, Kyoto University, (2008) 165–176)

[20] T Oda, Convex bodies and algebraic geometry, 3:15, Springer (1988)

[21] S Payne, Topology of real toric surfaces

[22] H Sato, Smooth toric Fano five-folds of index two, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006) 106

[23] V Uma, On the fundamental group of real toric varieties, Proc. Indian Acad. Sci. Math. Sci. 114 (2004) 15

[24] J A Wolf, Spaces of constant curvature, Publish or Perish (1984)