Geometry & Topology 9 (2005) 1881–1913

DOI: 10.2140/gt.2005.9.1881

Abstract

Let N₁, N₂, M be smooth manifolds with dim N₁ +dim N₂ +1= dim M and let φ_i, for i=1,2, be smooth mappings of N_i to M where Imφ₁∩Imφ₂=Ø. The classical linking number lk(φ₁,φ₂) is defined only when φ_1*[N₁]=φ_2*[N₂]=0 in H_*(M).

The affine linking invariant alk is a generalization of lk to the case where φ_1*[N₁] or φ_2*[N₂] 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 ≤ 1. In the case where φ_1* [N₁]=φ_2*[N₂]=0 in 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 dim N₁+dim N₂+1=dim M. 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

Primary: 57R19

Secondary: 14M07, 53Z05, 55N22, 55N45, 57M27, 57R40, 57R45, 57R52

References

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