* Universidad Nacional Autónoma de Mexico, México

Instituto de Matemáticas UNAM and CIMAT Circuito Interior S/N, Ciudad Universitaria, 04510 C.P. 3600 México D.F., México

e-mail: ficomx@yahoo.com.mx

Departamento de Matemáticas CUCEI-Universidad de Guadalajara and CIMAT A.C. Callejón Jalisco S/N Valenciana, 36240 A.P. 402 Guanajuanto, México

Abstract. In this note we prove that, if N₃ = P#P#P, where P := RP², then the canonical homomorphism from Diff(N₃) onto the homeotopy group Mod(N3) has a section. To do this we first prove that Mod(N₃) = GL(2; Z).

Keywords and phrases. Homeotopy group, non-orientable surface.

2000 Mathematics Subject Classification. Primary: 57M60. Secondary: 20F38.

Resumen. En esta nota probamos que, si N₃ = P#P#P, donde P := RP², entonces el homomorfismo canónico de Diff(N₃) sobre el grupo de homeotopía Mod(N₃) tiene una sección. Para hacer esto, primero probamos que Mod(N₃) = GL(2; Z).

FULL TEXT IN PDF

[1] S. Akbulut & H. King, Submanifolds and the homology of non singular algebraic varieties, Amer. J. Math., 107 (1985), 45-83.         [ Links ]

[2] J. S. Birman & D. R. J. Chillingworth, On the homeotopy group of a nonorientable surface, Proc. Camb. Phil. Soc., 71 (1972), 437-448.         [ Links ]

[3] J. S. Birman & M. H. Hilden, Lifting and projecting homeomorphisms, Arch. Math., 23 (1972), 428-434.         [ Links ]

[4] D. B. A. Epstein, Curves on 2¡manifolds and isotopies, Acta Math., 115 (1966), 83-107.         [ Links ]

[5] N. V. Ivanov, Mapping Class Groups, Handbook of Geometric Topology, Elsevier Science, N.H. (2002), 523-633.         [ Links ]

[6] W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc., 59 (1963), 307-317.         [ Links ]

[7] J. M. Márquez, On the trigenus of surface bundles over S¹, Aportaciones Matemáticas, 35 (2005), 201-215.         [ Links ]

[8] G. Mikhalkin, Blowup equivalence of smooth closed manifolds, Topology, 36 (1997), 287-299.         [ Links ]

[9] Sh. Morita, Characteristic classes of surface bundles, Bull. Amer. Math. Soc., 11 (1984) 2, 386-388.         [ Links ]

[10] Sh. Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987) 3, 552-577.         [ Links ]

[11] D. Rolfsen, Knots and links, Math. Lectures Series. 7. Berkeley, Ca. Publish Perish, Inc. 1976.         [ Links ]

[12] H. Torriani, Subgroups of the Klein bottle group and the mapping class group of the Klein Bottle, Rend. Mat. Appl., 7 (1987) 7, 215-222.         [ Links ]

(Recibido en mayo de 2006. Aceptado en julio de 2006)