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- W2022888947 abstract "Given a closed, irreducible, orientable 3-manifold M, let x denote the Thurston norm on H2 (M; R) . Suppose g, h, and f are three homology classes of H2(M; Z) carried by a single face of the x-unit sphere in H2(M; R) . In this paper it is shown that there exists a taut, oriented branched surface carrying representatives of g and h and a semi-taut oriented branched surface carrying representatives of all three homology classes. Throughout, M will be a closed, irreducible, orientable 3-manifold. In [T] Thurston introduced a semi-norm x on H2(M; R) . For g E H2(M; Z), x(g) is defined to be the infimum, taken over all embedded surfaces G representing g, of x-(G) = Ei IX(Gi) I where the sum is over all components Gi of G with X(Gi) < 0. Then x is extended to all of H2(M; R) using homogeneity and continuity. The x-unit ball x Z). All homology classes represented by surfaces which are not tori and are carried by a given semi-taut oriented branched surface are carried by a single face of the x-unit sphere adx. The question arises as to how the faces of d5x relate to taut (or semi-taut) oriented branched surfaces and has been considered in [M], [0], and [S]. An incompressible surface S in a closed irreducible 3-manifold M is an embedded surface without any 2-sphere components satisfying the condition that if a simple closed curve in S bounds a disc in M then it bounds a disc in S. An oriented surface S is said to be X_-minimizing if it has no 2-sphere components, [S] ? 0, and x([S]) = -f(S) where x is Thurston's norm on H2(M, AM), [S] is the homology class represented by S, and x(S) is the Euler characteristic. A X--minimizing surface may contain compressible or homologically trivial torus components. In the literature, a norm-minimizing surface usually refers to an incompressible, X--minizing surface. Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 57N1 0; Secondary 57M99. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + S.25 per page" @default.
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- W2022888947 date "1994-02-01" @default.
- W2022888947 modified "2023-09-25" @default.
- W2022888947 title "Branched surfaces and Thurston’s norm on homology" @default.
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- W2022888947 doi "https://doi.org/10.1090/s0002-9939-1994-1246537-4" @default.
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