Abstract
We introduce the class of dynamical 2-complexes. These complexes particularly allow the obtaining of a topological representation of any free group automorphism. A dynamical 2-complex can be roughly defined as a special polyhedron or standard 2-complex equipped with an orientation on its 1-cells satisfying two simple combinatorial properties. These orientations allow us to define non-singular semi-flows on the complex. The relationship with the free group automorphisms is done via a cohomological criterion to foliate the complex by compact graphs.
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References
Benedetti, R. and Petronio, C.: A finite graphic calculus for 3-manifolds, Manuscripta Math. 88(3) (1995), 291–310.
Benedetti, R. and Petronio, C.: Branched Standard Spines of 3-Manifolds, Lecture Notes in Math. 1653, Springer, New York, 1997.
Birman, J. S. and Williams, R. F.: Knotted periodic orbits in dynamical system I, Topology 22(1) (1983), 47–82.
Birman, J. S. and Williams, R. F.: Knotted periodic orbits in dynamical system II, Contemp.Math. 20 (1983), 1–60.
Casler, B. G.: An imbedding theorem for connected 3-manifolds with boundary, Proc. Amer. Math. Soc. 16 (1965), 559–566.
Christy, J.: Branched surfaces and attractors I: Dynamic branched surfaces, Trans. Amer. Math. Soc. 336 (1993), 759–784.
Christy, J.: Immersing branched surfaces in dimension three, Proc. Amer. Math. Soc. 115 (1992), 853–861.
Christy, J.: Standard spines and branched surfaces, MSRI Preprint (1993).
Floyd, W. and Oertel, U.: Incompressible branched surfaces via branched surfaces, Topology 23 (1983), 117–125.
Fried, D.: Geometry of cross-sections to flows, Topology 21 (1982), 353–371.
Gabai, D. and Oertel, U.: Essential laminations in 3-manifolds, Ann. Math. 130 (1989), 41–73.
Gautero, F.: CW-complexes dynamiques, doctoral thesis, University of Nice-Sophia Antipolis (1998).
Gautero, F.: Cross-sections to semi-flows, Preprint.
Gautero, F.: Foliations of 2-complexes, Preprint.
Ghrist, R. W., Holmes, P. J. and Sullivan, M. C.: Knots and Links in Three-Dimensional Flows, Lecture Notes in Math. 1654, Springer, New York, 1997.
Ikeda, H.: Acyclic fake surfaces, Topology 10 (1971), 9–36.
Matveev, S. V.: Special spines of piecewise linear manifolds, Math.USSR Sb. 21(2) (1973).
Munkres, J. R.: Elements of Algebraic Topology, Addison-Wesley, Menlo Park, CA, 1984.
Oertel, U.: Incompressible branched surfaces, Invent. Math. 76 (1984), 35–41.
Piergallini, R.: Standard moves for standard polyhedra and spines, Rend. Circ. Mat. Palermo (2) Suppl. 18 (1988), 391–414.
Thurston, W. P.: A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. (339) 59 (1986), 99–130.
Thurston, W. P.: In: S. Levy (ed.), Three-Dimensional Geometry and Topology volume 1, Princeton Math. Ser. 35, Princeton Univ. Press, 1997.
Tischler, D.: On fibering certain foliated manifolds over S1, Topology 9 (1970), 153–154.
Williams, R. F.: Expanding Attractors, Publ. Math. IHES 43 (1974), 169–203.
Williams, R. F.: The structure of Lorenz attractors, Publ. Math. IHES 50 (1979), 73–99.
Wright, P.: Formal 3-deformations of 2-polyhedra, Proc. Amer. Math. Soc. 37 (1973), 305–308.
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Gautero, F. Dynamical 2-Complexes. Geometriae Dedicata 88, 283–319 (2001). https://doi.org/10.1023/A:1013195402592
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DOI: https://doi.org/10.1023/A:1013195402592