Algebraic & Geometric Topology

On diffeomorphisms over nonorientable surfaces standardly embedded in the $4$–sphere

Susumu Hirose

Abstract

For a nonorientable closed surface standardly embedded in the $4$–sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou–Marin quadratic form of this embedded surface.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 1 (2012), 109-130.

Dates
Received: 11 September 2011
Revised: 24 October 2011
Accepted: 6 November 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715334

Digital Object Identifier
doi:10.2140/agt.2012.12.109

Mathematical Reviews number (MathSciNet)
MR2889548

Zentralblatt MATH identifier
1244.57043

Citation

Hirose, Susumu. On diffeomorphisms over nonorientable surfaces standardly embedded in the $4$–sphere. Algebr. Geom. Topol. 12 (2012), no. 1, 109--130. doi:10.2140/agt.2012.12.109. https://projecteuclid.org/euclid.agt/1513715334

References

• J S Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–238
• D R J Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 65 (1969) 409–430
• L Guillou, A Marin, Une extension d'un théorème de Rohlin sur la signature, C. R. Acad. Sci. Paris Sér. A 285 (1977) A95–A98
• S Hirose, On diffeomorphisms over surfaces trivially embedded in the $4$–sphere, Algebr. Geom. Topol. 2 (2002) 791–824
• S Hirose, Surfaces in the complex projective plane and their mapping class groups, Algebr. Geom. Topol. 5 (2005) 577–613
• S Hirose, A Yasuhara, Surfaces in $4$–manifolds and their mapping class groups, Topology 47 (2008) 41–50
• D Johnson, The structure of the Torelli group I: A finite set of generators for ${\mathscr I}$, Ann. of Math. 118 (1983) 423–442
• W B R Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963) 307–317
• W B R Lickorish, On the homeomorphisms of a non-orientable surface, Proc. Cambridge Philos. Soc. 61 (1965) 61–64
• Y Matsumoto, An elementary proof of Rochlin's signature theorem and its extension by Guillou and Marin, from: “À la recherche de la topologie perdue”, (L Guillou, A Marin, editors), Progr. Math. 62, Birkhäuser, Boston, MA (1986) 119–139
• J M Montesinos, On twins in the four-sphere I, Quart. J. Math. Oxford Ser. 34 (1983) 171–199
• T Nowik, Immersions of non-orientable surfaces, Topology Appl. 154 (2007) 1881–1893
• B Szepietowski, Crosscap slides and the level $2$ mapping class group of a nonorientable surface
• B Szepietowski, A finite generating set for the level $2$ mapping class group of a nonorientable surface