#### Volume 9, issue 4 (2005)

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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 ${N}_{1},{N}_{2},M$ be smooth manifolds with $dim{N}_{1}+dim{N}_{2}+1=dimM$ and let ${\varphi }_{i}$, for $i=1,2$, be smooth mappings of ${N}_{i}$ to $M$ where $Im{\varphi }_{1}\cap Im{\varphi }_{2}=\varnothing$. The classical linking number $lk\left({\varphi }_{1},{\varphi }_{2}\right)$ is defined only when ${\varphi }_{1\ast }\left[{N}_{1}\right]={\varphi }_{2\ast }\left[{N}_{2}\right]=0\in {H}_{\ast }\left(M\right)$.

The affine linking invariant $alk$ is a generalization of $lk$ to the case where ${\varphi }_{1\ast }\left[{N}_{1}\right]$ or ${\varphi }_{2\ast }\left[{N}_{2}\right]$ 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 $\le 1$. In the case where ${\varphi }_{1\ast }\left[{N}_{1}\right]={\varphi }_{2\ast }\left[{N}_{2}\right]=0\in {H}_{\ast }\left(M\right)$, 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 $dim{N}_{1}+dim{N}_{2}+1=dimM$. 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
##### Mathematical Subject Classification 2000
Primary: 57R19
Secondary: 14M07, 53Z05, 55N22, 55N45, 57M27, 57R40, 57R45, 57R52