#### 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.

Notes on file formats
Vladimir V Chernov, Yuli B Rudyak

Department of Mathematics, 6188 Bradley Hall

Dartmouth College,
Hanover NH 03755-3551, USA

and

Department of Mathematics,
University of Florida

358 Little Hall, Gainesville, FL 32611-8105, USA

Email: Vladimir.Chernov@dartmouth.edu, rudyak@math.ufl.edu

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**