Geodesic
On Kirby calculus for null-homotopic framed links in 3–manifolds
Habiro, Kazuo1 ; Widmer, Tamara2
1Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
2Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland
Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 115-134
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

Kirby proved that two framed links in S3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke gave a necessary and sufficient condition for two framed links in a closed, oriented 3–manifold to be related by a finite sequence of these moves.

The purpose of this paper is twofold. We first give a generalization of Fenn and Rourke’s result to 3–manifolds with boundary. Then we apply this result to the case of framed links whose components are null-homotopic in the 3–manifold.

DOI : 10.2140/agt.2014.14.115
Classification : 57M25, 57M27
Keywords: $3$–manifold, framed link, surgery, Kirby calculus, null-homotopic link

Habiro, Kazuo  1   ; Widmer, Tamara  2

1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
2 Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland 
@article{10_2140_agt_2014_14_115,
     author = {Habiro, Kazuo and Widmer, Tamara},
     title = {On {Kirby} calculus for null-homotopic framed links in 3{\textendash}manifolds},
     journal = {Algebraic and Geometric Topology},
     pages = {115--134},
     year = {2014},
     volume = {14},
     number = {1},
     doi = {10.2140/agt.2014.14.115},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.115/}
}
TY  - JOUR
AU  - Habiro, Kazuo
AU  - Widmer, Tamara
TI  - On Kirby calculus for null-homotopic framed links in 3–manifolds
JO  - Algebraic and Geometric Topology
PY  - 2014
SP  - 115
EP  - 134
VL  - 14
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.115/
DO  - 10.2140/agt.2014.14.115
ID  - 10_2140_agt_2014_14_115
ER  - 
%0 Journal Article
%A Habiro, Kazuo
%A Widmer, Tamara
%T On Kirby calculus for null-homotopic framed links in 3–manifolds
%J Algebraic and Geometric Topology
%D 2014
%P 115-134
%V 14
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.115/
%R 10.2140/agt.2014.14.115
%F 10_2140_agt_2014_14_115
Habiro, Kazuo; Widmer, Tamara. On Kirby calculus for null-homotopic framed links in 3–manifolds. Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 115-134. doi: 10.2140/agt.2014.14.115

[1] A Adem, R J Milgram, Cohomology of finite groups, Grundl. Math. Wissen. 309, Springer (1994)

[2] J Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, IHÉS Publ. Math. 39 (1970) 5

[3] T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97

[4] C J Earle, J Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967) 557

[5] R Fenn, C Rourke, On Kirby's calculus of links, Topology 18 (1979) 1

[6] S Garoufalidis, A Kricker, A surgery view of boundary links, Math. Ann. 327 (2003) 103

[7] R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)

[8] K Habiro, Refined Kirby calculus for integral homology spheres, Geom. Topol. 10 (2006) 1285

[9] K Habiro, T Widmer, Kirby calculus for null-homologous framed links in $3$–manifolds, in preparation

[10] R Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978) 35

[11] J Roberts, Kirby calculus in manifolds with boundary, Turkish J. Math. 21 (1997) 111

Cité par Sources :