Toward a general theory of linking invariants

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.