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

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

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
\Bibitem{Tur78}
\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
\mathnet{http://mi.mathnet.ru/msb2607}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=507817}
\zmath{https://zbmath.org/?q=an:0422.57005|0384.57004}
\transl
\jour Math. USSR-Sb.
\yr 1979
\vol 35
\issue 2
\pages 229--250
\crossref{https://doi.org/10.1070/SM1979v035n02ABEH001471}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. V. G. Turaev, “Multiplace generalizations of the Seifert form of a classical knot”, Math. USSR-Sb., 44:3 (1983), 335–361
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
4. Turaev V., “Nilpotent Homotopy Types of Closed 3-Manifolds”, 1060, 1984, 355–366
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
6. Chernov V., Rudyak Y., “Toward a General Theory of Linking Invariants”, Geom. Topol., 9 (2005), 1881–1913
7. Thomas Church, Aaron Pixton, “Separating twists and the Magnus representation of the Torelli group”, Geom Dedicata, 2011
8. Yurii Burman, Michael Polyak, “Whitney’s formulas for curves on surfaces”, Geom Dedicata, 151:1 (2011), 97
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
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
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
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
13. Massuyeau G., Turaev V., “Quasi-Poisson Structures on Representation Spaces of Surfaces”, Int. Math. Res. Notices, 2014, no. 1, 1–64
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