A manifold calculus approach to link maps and the linking number

Munson, Brian A¹

¹Department of Mathematics, Wellesley College, Wellesley, MA 02481, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 2323-2353

Résumé

We study the space of link maps Link(P1,…,Pk;N), the space of smooth maps P1 ⊔⋯ ⊔ Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation resembles recent work of the author on embeddings, and is used to show that the Borromean rings are linked.

Détail

DOI : 10.2140/agt.2008.8.2323

Keywords: calculus of functors, link map, linking number

Bibliographie

[1] V V Chernov, Y B Rudyak, Toward a general theory of linking invariants, Geom. Topol. 9 (2005) 1881

[2] T G Goodwillie, Calculus. I. The first derivative of pseudoisotopy theory, $K$–Theory 4 (1990) 1

[3] T G Goodwillie, J R Klein, Excision estimates for spaces of Poincaré embeddings, In preparation

[4] T G Goodwillie, J R Klein, M S Weiss, Spaces of smooth embeddings, disjunction and surgery, from: "Surveys on surgery theory, Vol. 2", Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 221

[5] T G Goodwillie, J R Klein, M S Weiss, A Haefliger style description of the embedding calculus tower, Topology 42 (2003) 509

[6] T G Goodwillie, B Munson, Connectivity estimates for spaces of link maps, In preparation

[7] T G Goodwillie, M Weiss, Embeddings from the point of view of immersion theory: Part II, Geom. Topol. 3 (1999) 103

[8] N Habegger, U Kaiser, Link homotopy in the $2$–metastable range, Topology 37 (1998) 75

[9] N Habegger, X S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389

[10] A Hatcher, F Quinn, Bordism invariants of intersections of submanifolds, Trans. Amer. Math. Soc. 200 (1974) 327

[11] M W Hirsch, Differential topology, Graduate Texts in Math. 33, Springer (1976)

[12] J Klein, B.Williams, Homotopical intersection theory I

[13] U Koschorke, A generalization of Milnor's $\mu$–invariants to higher-dimensional link maps, Topology 36 (1997) 301

[14] U Koschorke, Link maps in arbitrary manifolds and their homotopy invariants, J. Knot Theory Ramifications 12 (2003) 79

[15] U Koschorke, Link homotopy in $S^n\times \mathbb R^{m-n}$ and higher order $\mu$–invariants, J. Knot Theory Ramifications 13 (2004) 917

[16] U Koschorke, Linking and coincidence invariants, Fund. Math. 184 (2004) 187

[17] U Koschorke, D Rolfsen, Higher-dimensional link operations and stable homotopy, Pacific J. Math. 139 (1989) 87

[18] J P Levine, An approach to homotopy classification of links, Trans. Amer. Math. Soc. 306 (1988) 361

[19] W S Massey, D Rolfsen, Homotopy classification of higher-dimensional links, Indiana Univ. Math. J. 34 (1985) 375

[20] J Milnor, Link groups, Ann. of Math. $(2)$ 59 (1954) 177

[21] J Milnor, Topology from the differentiable viewpoint, University Press of Virginia (1965)

[22] B A Munson, Embeddings in the $3/4$ range, Topology 44 (2005) 1133

[23] R Schneiderman, P Teichner, Higher order intersection numbers of $2$–spheres in $4$–manifolds, Algebr. Geom. Topol. 1 (2001) 1

[24] G P Scott, Homotopy links, Abh. Math. Sem. Univ. Hamburg 32 (1968) 186

[25] A Skopenkov, On the generalized Massey–Rolfsen invariant for link maps, Fund. Math. 165 (2000) 1

[26] M Weiss, Embeddings from the point of view of immersion theory: Part I, Geom. Topol. 3 (1999) 67