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

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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.