PL equivariant surgery and invariant decompositions of 3-manifolds. (English) Zbl 0682.57005

Jaco, William; Rubinstein, J. Hyam

Adv. Math. 73, No. 2, 149-191 (1989).

Normal surface theory was introduced by W. Haken to show the existence of hierarchies for certain 3-manifolds M (“Haken-manifolds”). The authors study least weight surfaces, i.e. normal surfaces with minimal intersections with the 1-skeleton of a fixed triangulation of M. The main applications are constructive PL proofs of results which were previously obtained by Meeks-Yau via minimal surface theory. Specifically, let G be a group of simplicial homeomorphisms of M. The authors give PL proofs for (1) the Equivariant Sphere Theorem, (2) the existence of a G-invariant prime decomposition of M (if G is finite and M has no closed 1-handles), (3) the existence of a G-invariant characteristic submanifold V of a Haken-manifold M (if G is finite and M is not a torus bundle over \(S^ 1\) with V a neighborhood of the fiber).

Reviewer: W.Heil

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57S17	Finite transformation groups
57S15	Compact Lie groups of differentiable transformations

Keywords:

hierarchies; Haken-manifolds; least weight surfaces; normal surfaces; Equivariant Sphere Theorem; G-invariant prime decomposition; G-invariant characteristic submanifold

References:

[1]	{\scM. J. Dunwoody}, An equivariant sphere theorem, preprint. · Zbl 0592.57005
[2]	Edmonds, A, A topological proof of the equivariant Dehn’s lemma, Trans. amer. math. soc., 297, 605-615, (1986) · Zbl 0605.57004
[3]	Epstein, D.B.A, Projective planes in 3-manifolds, (), 469-484 · Zbl 0111.18801
[4]	Freedman, M; Hass, J; Scott, P, Least area incompressible surfaces in 3-manifolds, Invent. math., 71, 609-642, (1983) · Zbl 0482.53045
[5]	Gordon, C; Litherland, R, ()
[6]	Haken, W, Theorie der normal flachen, Acta math., 105, 245-375, (1961) · Zbl 0100.19402
[7]	Haken, W, Some results on surfaces in 3-manifolds, (), 34-98 · Zbl 0194.24902
[8]	Hempel, J, 3-manifolds, () · Zbl 0191.22203
[9]	Jaco, W, Lectures on three-manifold topology, (), Number 43 · Zbl 0433.57001
[10]	[J2]|{\scW. Jaco}, Normal surfaces and the projective solution space, Ensign. Math., to appear.
[11]	Jaco, W; Shalen, P.B, Seifert fibered spaces in 3-manifolds, Mem. amer. math. soc., 220, (1979) · Zbl 0471.57001
[12]	Johannson, K, Homotopy equivalences of 3-manifolds with boundary, () · Zbl 0542.57002
[13]	Knesser, H, Geschlossene flachen in dreidimensionalen mannigfaltigkeiten, Jahresbericht der deut. math. verein., 38, 248-260, (1929) · JFM 55.0311.03
[14]	Kim, P; Tollefson, J, PL involutions of fibered manifolds, Trans. amer. math. soc., 232, 221-237, (1977) · Zbl 0376.57021
[15]	Milnor, J, A unique factorization theorem for 3-manifolds, Amer. J. math., 84, 1-7, (1962) · Zbl 0108.36501
[16]	{\scW. H. Meeks and P. Scott}, Finite group actions on 3-manifolds, preprint. · Zbl 0626.57006
[17]	Meeks, W.H; Yau, S-T, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of math., 112, 441-485, (1980) · Zbl 0458.57007
[18]	Meeks, W.H; Yau, S-T, The equivariant Dehn’s lemma and loop theorem, Comment. math. helv., 56, 225-239, (1981) · Zbl 0469.57005
[19]	Meeks, W.H; Yau, S-T, The classical plateau problem and the topology of three dimensional manifolds, Topology, 21, 409-442, (1982) · Zbl 0489.57002
[20]	Meeks, W.H; Yau, S-T, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z., 179, 151-168, (1982) · Zbl 0479.49026
[21]	[M-Y5]|{\scW. H. Meeks and S-T. Yau}, “The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces, The Smith Conjecture” ({\scJ. Morgan and H. Bass}, Eds.), pp. 153-163, Series in Pure and Applied Mathematics Academic Press, New York.
[22]	Meeks, W.H; Simon, L; Yau, S-T, Embedded minimal surfaces, exotic spheres and manifolds with positive Ricci curvature, Ann. of math., 116, 621-659, (1982) · Zbl 0521.53007
[23]	Papakyriakopoulos, C, On Dehn’s lemma and the asphericity of knots, Ann. of math., 66, 1-26, (1957) · Zbl 0078.16402
[24]	Papakyriakopoulos, C, On solid tori, (), 281-299 · Zbl 0078.16305
[25]	Scott, P, There are no fake Seifert fibre spaces with infinite π1, Ann. of math., 117, 35-70, (1983) · Zbl 0516.57006
[26]	()
[27]	Stallings, J, On the loop theorem, Ann. of math., 72, 12-19, (1960) · Zbl 0094.36103
[28]	Stallings, J, Group theory and three-dimensional manifolds, () · Zbl 0122.27301
[29]	Waldhausen, F, On irreducible 3-manifolds which are sufficiently large, Ann. of math., 87, 56-88, (1968) · Zbl 0157.30603
[30]	Whitehead, J.H.C, On 2-spheres in 3-manifolds, Bull. amer. math. soc., 64, 161-166, (1958) · Zbl 0084.19103