×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. J. Dunwoody; M. J. Dunwoody · 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, (Proc. London Math. Soc., 11 (1961)), 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., (Bass, H.; Morgan, J., The Smith Conjecture (1984), Academic Press: Academic Press New York) · Zbl 0599.57001
[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, (Studies in Modern Topology (1968), Math. Assoc. of Amer., Prentice-Hall: Math. Assoc. of Amer., Prentice-Hall Englewood Cliffs, NJ), 34-98 · Zbl 0194.24902
[8] Hempel, J., 3-manifolds, (Ann. of Math. Studies, Vol. 86 (1976), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ) · Zbl 0191.22203
[9] Jaco, W., Lectures on Three-Manifold Topology, (CBMS Lecture Series (1980), Amer. Math. Soc: Amer. Math. Soc Providence, RI), Number 43 · Zbl 0433.57001
[10] [J2]|W. JacoEnsign. Math.; [J2]|W. JacoEnsign. Math.
[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, (Lecture Notes in Mathematics, 761 (1979), Springer: Springer Berlin) · Zbl 0542.57002
[13] Knesser, H., Geschlossene Flachen in dreidimensionalen Mannigfaltigkeiten, Jahresbericht der Deut. Math. Verein., 38, 248-260 (1929)
[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] W. H. Meeks and P. Scott; W. H. Meeks and P. Scott · 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]|W. H. Meeks and S-T. YauJ. Morgan and H. Bass; [M-Y5]|W. H. Meeks and S-T. YauJ. Morgan and H. Bass
[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, (Proc. London Math. Soc., 7 (1957)), 281-299 · Zbl 0078.16305
[25] Scott, P., There are no fake Seifert fibre spaces with infinite \(π_1\), Ann. of Math., 117, 35-70 (1983) · Zbl 0516.57006
[26] (Morgan, J.; Bass, H., The Smith Conjecture. The Smith Conjecture, Series in Pure and Applied Mathematics (1984), Academic Press: Academic Press New York) · Zbl 0599.57001
[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, (Yale Math. Monographs, 4 (1971), Yale Univ. Press: Yale Univ. Press New Haven, CN) · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.