Orientation-reversing involutions on handlebodies

Kalliongis, John; McCullough, Darryl

doi:10.1090/S0002-9947-96-01515-2

Orientation-reversing involutions on handlebodies
HTML articles powered by AMS MathViewer

by John Kalliongis and Darryl McCullough

Trans. Amer. Math. Soc. 348 (1996), 1739-1755

DOI: https://doi.org/10.1090/S0002-9947-96-01515-2

PDF | Request permission

Abstract:

The observation that the quotient orbifold of an orientation- reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the $I$-fibers of some $I$-bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.

References

Francis Bonahon, Cobordisme des difféomorphismes de surfaces, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 16, A765–A767 (French, with English summary). MR 577154
P. E. Conner and Frank Raymond, Actions of compact Lie groups on aspherical manifolds, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 227–264. MR 0271958
John W. Morgan and Hyman Bass (eds.), The Smith conjecture, Pure and Applied Mathematics, vol. 112, Academic Press, Inc., Orlando, FL, 1984. Papers presented at the symposium held at Columbia University, New York, 1979. MR 758459
Paik Kee Kim and Jeffrey L. Tollefson, Splitting the PL involutions of nonprime $3$-manifolds, Michigan Math. J. 27 (1980), no. 3, 259–274. MR 584691
Paik Kee Kim and Jeffrey L. Tollefson, PL involutions of fibered $3$-manifolds, Trans. Amer. Math. Soc. 232 (1977), 221–237. MR 454981, DOI 10.1090/S0002-9947-1977-0454981-2
Darryl McCullough and Andy Miller, Homeomorphisms of $3$-manifolds with compressible boundary, Mem. Amer. Math. Soc. 61 (1986), no. 344, xii+100. MR 840832, DOI 10.1090/memo/0344
Darryl McCullough, Andy Miller, and Bruno Zimmermann, Group actions on handlebodies, Proc. London Math. Soc. (3) 59 (1989), no. 2, 373–416. MR 1004434, DOI 10.1112/plms/s3-59.2.373
William H. Meeks III and Peter Scott, Finite group actions on $3$-manifolds, Invent. Math. 86 (1986), no. 2, 287–346. MR 856847, DOI 10.1007/BF01389073
William H. Meeks III and Shing Tung Yau, The equivariant Dehn’s lemma and loop theorem, Comment. Math. Helv. 56 (1981), no. 2, 225–239. MR 630952, DOI 10.1007/BF02566211
Roger B. Nelson, Some fiber preserving involutions of orientable $3$-dimensional handlebodies, Houston J. Math. 9 (1983), no. 2, 255–269. MR 703274
Roger B. Nelson, A unique decomposition of involutions of handlebodies, Proc. Amer. Math. Soc. 93 (1985), no. 2, 358–362. MR 770554, DOI 10.1090/S0002-9939-1985-0770554-6
M. Stephanus, Orientation reversing involutions of a handlebody, Dissertation at St. Louis University, 1995.
W. Thurston, The geometry and topology of 3-manifolds, mimeographed notes, Princeton University.