Algebraic & Geometric Topology

On diffeomorphisms over surfaces trivially embedded in the 4–sphere

Susumu Hirose

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A surface in the 4–sphere is trivially embedded, if it bounds a 3–dimensional handle body in the 4–sphere. For a surface trivially embedded in the 4–sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.

Article information

Algebr. Geom. Topol., Volume 2, Number 2 (2002), 791-824.

Received: 6 March 2002
Accepted: 4 September 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57N05: Topology of $E^2$ , 2-manifolds 20F38: Other groups related to topology or analysis

knotted surface mapping class group spin mapping class group


Hirose, Susumu. On diffeomorphisms over surfaces trivially embedded in the 4–sphere. Algebr. Geom. Topol. 2 (2002), no. 2, 791--824. doi:10.2140/agt.2002.2.791.

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  • J S Birman, H Hilden, On the mapping class group of closed surface as covering spaces, from: “Advances in the theory of Riemann surfaces" Ann. of Math. Studies 66(1971) 81–115
  • J Dieudonné, La gémmetrie des groupes classiques, (3-rd edn.), Ergebnisse der Math. u.i. Grundz. 5, Springer, 1971
  • J L Harer, Stability of the homology of the moduli spaces of Riemann surfaces with spin structure, Math. Ann. 287(1990) 323–334
  • J L Harer, The rational Picard group of the moduli space of Riemann surfaces with spin structure, Contemp. Math. 150(1993) 107–136
  • S Hirose, On diffeomorphisms over $T^2$-knot, Proc. of A.M.S. 119(1993) 1009–1018
  • Z Iwase, Dehn surgery along a torus $T^2$-knot. II, Japan. J. Math. 16(1990) 171–196
  • D Johnson, The structure of the Torelli Group I: A finite set of generators for ${\cal I}$, Annals of Math. 118(1983) 423–442
  • D Johnson, The structure of the Torelli Group III: The abelianization of ${\cal I}$, Topology 24(1985) 127–144
  • W Magnus, A Karras, D Solitar, Combinatorial Group Theory, Dover 1975
  • J M Montesinos, On twins in the four-sphere I, Quart. J. Math. Oxford (2) 34(1983) 171–199