Geometry & Topology, Vol. 9 (2005) Paper no. 42, pages 1881--1913.

Toward a general theory of linking invariants

Vladimir V Chernov, Yuli B Rudyak

Abstract. Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2 disjoint. The classical linking number lk(phi_1,phi_2) is defined only when phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M).
The affine linking invariant alk is a generalization of lk to the case where phi_1*[N_1] or phi_2*[N_2] are not zero-homologous. In arXiv:math.GT/0207219 we constructed the first examples of affine linking invariants of nonzero-homologous spheres in the spherical tangent bundle of a manifold, and showed that alk is intimately related to the causality relation of wave fronts on manifolds. In this paper we develop the general theory.
The invariant alk appears to be a universal Vassiliev-Goussarov invariant of order < 2. In the case where phi_1*[N_1] and phi_2*[N_2] are 0 in homology it is a splitting of the classical linking number into a collection of independent invariants.
To construct alk we introduce a new pairing mu on the bordism groups of spaces of mappings of N_1 and N_2 into M, not necessarily under the restriction dim N_1 + dim N_2 +1 = dim M. For the zero-dimensional bordism groups, mu can be related to the Hatcher-Quinn invariant. In the case N_1=N_2=S^1, it is related to the Chas-Sullivan string homology super Lie bracket, and to the Goldman Lie bracket of free loops on surfaces.

Keywords. Linking invariants, winding numbers, Goldman bracket, wave fronts, causality, bordisms, intersections, isotopy, embeddings

AMS subject classification. Primary: 57R19. Secondary: 14M07, 53Z05, 55N22, 55N45, 57M27, 57R40, 57R45, 57R52.

E-print: arXiv:math.GT/0302295

DOI: 10.2140/gt.2005.9.1881

Submitted to GT on 30 January 2004. (Revised 20 September 2005.) Paper accepted 20 September 2005. Paper published 6 October 2005.

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Vladimir V Chernov, Yuli B Rudyak
Department of Mathematics, 6188 Bradley Hall
Dartmouth College, Hanover NH 03755-3551, USA
Department of Mathematics, University of Florida
358 Little Hall, Gainesville, FL 32611-8105, USA

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