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Mat. Sb. (N.S.), 1978, Volume 106(148), Number 4(8), Pages 566–588 (Mi msb2607)  

This article is cited in 13 scientific papers (total in 13 papers)

Intersections of loops in two-dimensional manifolds

V. G. Turaev

Abstract: Given an arbitrary smooth two-dimensional manifold $A$ with nonempty boundary and a point $a\in\partial A$, mappings $\mathbf Z[\pi_1(A,a)]\times\mathbf Z[\pi_1(A,a)]\to\mathbf Z[\pi_1(A,a)]$ and $\pi_1(A,a)\to\mathbf Z[\pi_1(A,a)]$. are constructed. In terms of them the author formulates and proves necessary and sufficient conditions for realizability of an element of the group $\pi_1(A,a)$ by a simple loop, conditions for the realizability of a few elements of $\pi_1(A,a)$ by nonintersecting loops and conditions for realizability of an automorphism of this group by a diffeomorphism $(A,a)\to(A,a)$.
Figures: 5.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1979, 35:2, 229–250

Bibliographic databases:

UDC: 513.83
MSC: Primary 57N05, 57M05, 57M25; Secondary 16A26
Received: 30.06.1977

Citation: V. G. Turaev, “Intersections of loops in two-dimensional manifolds”, Mat. Sb. (N.S.), 106(148):4(8) (1978), 566–588; Math. USSR-Sb., 35:2 (1979), 229–250

Citation in format AMSBIB
\by V.~G.~Turaev
\paper Intersections of loops in two-dimensional manifolds
\jour Mat. Sb. (N.S.)
\yr 1978
\vol 106(148)
\issue 4(8)
\pages 566--588
\jour Math. USSR-Sb.
\yr 1979
\vol 35
\issue 2
\pages 229--250

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    This publication is cited in the following articles:
    1. V. G. Turaev, “The fundamental groups of manifolds and Poincaré complexes”, Math. USSR-Sb., 38:2 (1981), 255–270  mathnet  crossref  mathscinet  zmath  isi
    2. V. G. Turaev, “Multiplace generalizations of the Seifert form of a classical knot”, Math. USSR-Sb., 44:3 (1983), 335–361  mathnet  crossref  mathscinet  zmath
    3. V. G. Turaev, O. Ya. Viro, “Intersection of loops in two-dimensional manifolds. II. Free loops”, Math. USSR-Sb., 49:2 (1984), 357–366  mathnet  crossref  mathscinet  zmath
    4. Turaev V., “Nilpotent Homotopy Types of Closed 3-Manifolds”, 1060, 1984, 355–366  mathscinet  zmath  isi
    5. K. A. Mitchell, J. P. Handley, J. B. Delos, S. K. Knudson, “Geometry and topology of escape. II. Homotopic lobe dynamics”, Chaos, 13:3 (2003), 892  crossref  mathscinet  zmath  adsnasa  isi
    6. Chernov V., Rudyak Y., “Toward a General Theory of Linking Invariants”, Geom. Topol., 9 (2005), 1881–1913  crossref  mathscinet  zmath  isi
    7. Thomas Church, Aaron Pixton, “Separating twists and the Magnus representation of the Torelli group”, Geom Dedicata, 2011  crossref
    8. Yurii Burman, Michael Polyak, “Whitney’s formulas for curves on surfaces”, Geom Dedicata, 151:1 (2011), 97  crossref
    9. A. G. Fedotov, “On the Realization of the Generalized Solenoid as a Hyperbolic Attractor of Sphere Diffeomorphisms”, Math. Notes, 94:5 (2013), 681–691  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. Massuyeau G., Oancea A., Salamon D.A., “Lefschetz Fibrations, Intersection Numbers, and Representations of the Framed Braid Group”, Bull. Math. Soc. Sci. Math. Roum., 56:4 (2013), 435–486  isi
    11. Cahn P. Chernov V., “Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra”, J. Lond. Math. Soc.-Second Ser., 87:3 (2013), 785–801  crossref  isi
    12. Cahn P., “A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface”, N. Y. J. Math., 19 (2013), 253–283  isi
    13. Massuyeau G., Turaev V., “Quasi-Poisson Structures on Representation Spaces of Surfaces”, Int. Math. Res. Notices, 2014, no. 1, 1–64  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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