  The linking number $lk$ is defined if link components are zero homologous.
Our affine linking invariant $alk$ generalizes $lk$ to the case of linked
submanifolds with arbitrary homology classes. We apply $alk$ to the study of
causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a
globally hyperbolic spacetime $(X^{m+1}, g)$. The spherical cotangent bundle
$ST^*M$ is identified with the space $N$ of all null geodesics in $(X,g).$
Hence the set of null geodesics passing through a point $x\in X$ gives an
embedded $(m-1)$-sphere $S_x$ in $N=ST^*M$ called the sky of $x.$ Low observed
that if the link $(S_x, S_y)$ is nontrivial, then $x,y\in X$ are causally
related. This motivated the problem (communicated by Penrose) on the Arnold's
1998 problem list to apply link theory to the study of causality. The spheres
$S_x$ are isotopic to fibers of $(ST^*M)^{2m-1}\to M^m.$ They are nonzero
homologous and $lk(S_x,S_y)$ is undefined when $M$ is closed, while $alk(S_x,
S_y)$ is well defined. Moreover, $alk(S_x, S_y)\in Z$ if $M$ is not an
odd-dimensional rational homology sphere. We give a formula for the increment
of $\alk$ under passages through Arnold dangerous tangencies. If $(X,g)$ is
such that $alk$ takes values in $\Z$ and $g$ is conformal to $g'$ having all
the timelike sectional curvatures nonnegative, then $x, y\in X$ are causally
related if and only if $alk(S_x,S_y)\neq 0$. We show that $x,y$ in
nonrefocussing $(X, g)$ are causally unrelated iff $(S_x, S_y)$ can be deformed
to a pair of $S^{m-1}$-fibers of $ST^*M\to M$ by an isotopy through skies. Low
showed that if $(\ss, g)$ is refocussing, then $M$ is compact. We show that the
universal cover of $M$ is also compact.
