Toward a general theory of linking invariants

Chernov, Vladimir V; Rudyak, Yuli B

doi:10.2140/gt.2005.9.1881

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Toward a general theory of linking invariants

Vladimir V Chernov and Yuli B Rudyak

Geometry & Topology 9 (2005) 1881–1913

DOI: 10.2140/gt.2005.9.1881

arXiv: math.GT/0302295

Abstract

Let $N_{1}, N_{2}, M$ be smooth manifolds with $dim N_{1} + dim N_{2} + 1 = dim M$ and let $ϕ_{i}$ , for $i = 1, 2$ , be smooth mappings of $N_{i}$ to $M$ where $Im ϕ_{1} \cap Im ϕ_{2} = \emptyset$ . The classical linking number $lk (ϕ_{1}, ϕ_{2})$ is defined only when $ϕ_{1 *} [N_{1}] = ϕ_{2 *} [N_{2}] = 0 \in H_{*} (M)$ .

The affine linking invariant $alk$ is a generalization of $lk$ to the case where $ϕ_{1 *} [N_{1}]$ or $ϕ_{2 *} [N_{2}]$ 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 $\leq 1$ . In the case where $ϕ_{1 *} [N_{1}] = ϕ_{2 *} [N_{2}] = 0 \in H_{*} (M)$ , 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 $N_{1}$ 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, $μ$ 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

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

Volume 9, issue 4 (2005)