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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Toward a general theory of linking invariants

Vladimir V Chernov and Yuli B Rudyak

Geometry & Topology 9 (2005) 1881–1913

arXiv: math.GT/0302295

Abstract

Let N1,N2,M be smooth manifolds with dimN1 + dimN2 + 1 = dimM and let ϕi, for i = 1,2, be smooth mappings of Ni to M where Imϕ1 Imϕ2 = . The classical linking number lk(ϕ1,ϕ2) is defined only when ϕ1[N1] = ϕ2[N2] = 0 H(M).

The affine linking invariant alk is a generalization of lk to the case where ϕ1[N1] or ϕ2[N2] are not zero-homologous. In [?] 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 1. In the case where ϕ1[N1] = ϕ2[N2] = 0 H(M), it is a splitting of the classical linking number into a collection of independent invariants.

To construct alk we introduce a new pairing μ on the bordism groups of spaces of mappings of N1 and N2 into M, not necessarily under the restriction dimN1 + dimN2 + 1 = dimM. For the zero-dimensional bordism groups, μ can be related to the Hatcher–Quinn invariant. In the case N1 = N2 = S1, 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
Mathematical Subject Classification 2000
Primary: 57R19
Secondary: 14M07, 53Z05, 55N22, 55N45, 57M27, 57R40, 57R45, 57R52
References
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Publication
Received: 30 January 2004
Revised: 20 September 2005
Accepted: 20 September 2005
Published: 6 October 2005
Proposed: Steve Ferry
Seconded: Ralph Cohen, Leonid Polterovich
Authors
Vladimir V Chernov
Department of Mathematics
6188 Bradley Hall
Dartmouth College
Hanover
New Hampshire 03755-3551
USA
Yuli B Rudyak
Department of Mathematics
University of Florida
358 Little Hall
Gainesville
Florida 32611-8105
USA