Orientation-reversing involutions on handlebodies

Authors:
John Kalliongis and Darryl McCullough

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

MSC (1991):
Primary 57M60; Secondary 57S25

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

MathSciNet review:
1329535

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Abstract | References | Similar Articles | Additional Information

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 -fibers of some -bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.

**1.**F. Bonahon, Cobordisme des difféomorphisms des surfaces, C. R. Acad. Sci. Paris Sér. A**290**(1987), 765--767. MR**81d:57028****2.**P.E. Conner and F. Raymond,*Actions of compact Lie groups on aspherical manifolds*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens Ga., 1969), Markham, Chicago, 1970, pp. 227-264. MR**42:6839****3.**M. Davis and J. Morgan,*Finite group actions on homotopy spheres*, The Smith Conjecture (J. Morgan and H. Bass, eds.), Academic Press, Orlando, 1984, pp. 181-226. MR**86i:57002****4.**P.K. Kim and J. Tollefson,*Splitting the PL involutions of nonprime 3-manifolds*, Michigan Math. J.**27**(1980), 259-274. MR**81m:57007****5.**P.K. Kim and J. Tollefson,*PL involutions of fibered 3-manifolds*, Trans. Amer. Math. Soc.**232**(1974), 221--237. MR**56:13223****6.**D. McCullough and A. Miller,*Homeomorphisms of -manifolds with compressible boundary*, Mem. Amer. Math. Soc.**344**(1986), 1--100. MR**87i:57013****7.**D. McCullough, A. Miller, and B. Zimmermann,*Group actions on handlebodies*, Proc. London Math. Soc.**59**(3) (1989)), 373--416. MR**90h:57014****8.**W. Meeks and P. Scott,*Finite group actions on -manifolds*, Invent. Math.**86**(1986), 287--346. MR**88b:57039****9.**W. Meeks and S-T. Yau,*The equivariant Dehn's lemma and loop theorem*, Comment. Math. Helv.**56**(1981), 225--239. MR**83b:57006****10.**R. Nelson,*Some fiber-preserving involutions of orientable 3-dimensional handlebodies*, Houston J. Math.**9**(1983), 255--269. MR**84k:57029****11.**R. Nelson,*A unique decomposition of involutions of handlebodies*, Proc. Amer. Math. Soc.**93**(1985), 358--362. MR**86g:57031****12.**M. Stephanus,*Orientation reversing involutions of a handlebody*, Dissertation at St. Louis University, 1995.**13.**W. Thurston,*The geometry and topology of 3-manifolds*, mimeographed notes, Princeton University.

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Additional Information

**John Kalliongis**

Affiliation:
Department of Mathematics, St. Louis University, St. Louis, Missouri 63103

Email:
kalliongisje@sluvca.slu.edu

**Darryl McCullough**

Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019

Email:
dmccullough@uoknor.edu

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

Keywords:
3-manifold,
orbifold,
handlebody,
orientation-reversing,
group action,
involution,
$I$-bundle,
fiber-preserving,
compression body,
classification

Received by editor(s):
June 29, 1994

Received by editor(s) in revised form:
May 4, 1995

Article copyright:
© Copyright 1996
American Mathematical Society