三分量链环的三重环绕数、模糊Hopf不变量和积分公式
Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links
摘要：
约翰·米尔诺（John Milnor）在1954年发表的高级论文中将3维球体中的三分量链归类为链同伦。一个完整的不变量集由组成部分的成对环绕数p、q和r以及另一个整数μ的剩余类给出，该整数μ是标题中的“三重环数“，它是以p、q和r的最大公因数为模定义的。对于每一个这样的链环L，我们将一个从3-环面到2-球面的几何自然特征映射g_L以这样的方式联系起来，使得L的链环同伦成为g_L的同伦。映射的3-torus的2球被归类为同伦的列夫庞特里亚金在1941年。一个完整的不变量集由它们对二维坐标子环面的限制度p、q和r以及另一个整数nu的剩余类给出，nu是一个“模糊的Hopf不变量”，它是p、q和r的最大公因数的两倍的模。证明了L的分支的成对环绕数p，q和r等于其特征映射g_L限制在二维子环面上的次数，并且证明了L的2倍Milnor μ-不变量等于g_L的Pontryagin μ-不变量.当p、q和r都为零时，mu-和nu-不变量是普通整数。在这种情况下，我们使用J.H. C.怀特黑德的霍普夫不变量的积分公式，适用于3-环面到2-球面的映射，连同3-环面上的标量拉普拉斯算子的基本解的公式作为三个变量的傅里叶级数，提供了nu的显式积分公式，从而为mu。
英文摘要：
Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r. To each such link L we associate a geometrically natural characteristic map g_L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g_L. Maps of the 3-torus to the 2-sphere were classified up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer nu, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r. We show that the pairwise linking numbers p, q and r of the components of L are equal to the degrees of its characteristic map g_L restricted to the 2-dimensional subtori, and that twice Milnor's mu-invariant for L is equal to Pontryagin's nu-invariant for g_L. When p, q and r are all zero, the mu- and nu-invariants are ordinary integers. In this case we use J. H. C. Whitehead's integral formula for the Hopf invariant, adapted to maps of the 3-torus to the 2-sphere, together with a formula for the fundamental solution of the scalar Laplacian on the 3-torus as a Fourier series in three variables, to provide an explicit integral formula for nu, and hence for mu.
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