View source for Linking coefficient

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An integer or fraction associated with two disjoint cycles $z^{k-1}$ and $z^{n-k}$ in an $n$-dimensional manifold $M$ whose homology classes are members of the torsion subgroups of the integral homologies $H_{k-1}(M,\mathbf Z)$ and $H_{n-k}(M,\mathbf Z)$, respectively. The simplest example is the linking coefficient of two non-intersecting closed rectifiable curves $L_1,L_2$ in $\mathbf R^3$, given by the so-called Gauss integral:

$$I=\frac{1}{4\pi}\int\limits_{L_1}\int\limits_{L_2}\frac{(x_1-x_2)\,dx_1\,dx_2}{|x_1-x_2|^3}$$

(here $x_1$ and $x_2$ are the radius vectors of $L_1$ and $L_2$).

The concept of the linking coefficient generalizes to the case of closed oriented manifolds $M^{k-1}$ and $M^{n-k}$ in $\mathbf R^n$: the linking coefficient is equal to the degree of the mapping (cf. [[Degree of a mapping|Degree of a mapping]]) $\chi$ of the oriented direct product $M^{k-1}\times M^{n-k}$ into the sphere $S^{n-1}\subset\mathbf R^n$, where $\chi(x,y)$, $x\in M^{n-k}$, is the point at which $S^{n-1}$ is cut by a ray through the origin parallel to the vector $(x,y)$. The linking coefficient is equal to the [[Intersection index (in homology)|intersection index (in homology)]] of any $k$-chain $C^k$ such that $\partial C^k=\alpha z^{k-1}$ with the cycle $z^{n-k}$, divided by $\alpha$. This number is independent of the choice of $C^k$. If the roles of the cycles $z^{k-1}$ and $z^{n-k}$ are interchanged, the linking coefficient is multiplied (in the orientable case) by $(-1)^{k(n-k)}$. If either of the cycles is replaced by a homological cycle in the complement to the other cycle, the linking coefficient remains the same. This is the basis for the linking interpretation of [[Alexander duality|Alexander duality]]. If one of the cycles is replaced by a homological cycle, the linking coefficient varies by an integer, thus defining a pairing of torsion subgroups in $H_{k-1}(M,\mathbf Z)$ and $H_{n-k}(M,\mathbf Z)$ with values in the quotient group $\mathbf Q/\mathbf Z$, where $\mathbf Q$ denotes the set of rational numbers. This pairing establishes a [[Pontryagin duality|Pontryagin duality]] between them. In particular, considering the torsion subgroup of $H_m(M,\mathbf Z)$ in the case $n=2m+1$, it defines a non-degenerate quadratic form of self-linkings with values in $\mathbf Q/\mathbf Z$, which is a homotopy invariant of the manifold. For example, this was what led to the first observations of asymmetric manifolds, which were particular lens manifolds (cf. [[Lens space|Lens space]]).

Linking coefficients are also considered in other coefficient domains; for example, if a group $\pi$ is acting freely on the manifold, the homology groups are group modules and the linking coefficient is defined in a suitably localized group ring.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Seifert,   W. Threlfall,   "Lehrbuch der Topologie" , Chelsea, reprint  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Smooth manifolds and their applications in homotopy theory"  ''Transl. Amer. Math. Soc.'' , '''11'''  (1959)  pp. 1–114  ''Trudy Mat. Inst. Steklov.'' , '''45'''  (1955)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , Springer  (1978)</TD></TR></table>