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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
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