Homotopy invariants of links
In this paper the author introduces a concordance invariant of links, \(\theta_ k\), which is defined whenever all the Milnor \({\bar \mu}\) invariants of length \(\leq k\) vanish. Equivalently, \(\theta_ k\) is defined whenever the lower central series quotients of the fundamental group up to order k are indistinguishable from those of a trivial link. Various facts about \(\theta_ k\) make it possible to give a precise formula for the number of independent Milnor numbers of length \(k+1.\) The details are as follows: Let L be a link with m components in \(S^ 3\). Let \(\pi\) denote the fundamental group of the complement, F denote the free group on m letters, and \(\pi /\pi^{(k)}\), \(F/F^{(k)}\) denote the lower central series quotients for \(k=1,2,...,\omega\). Let \(\{\bigvee^{m}_{i=1}S^ 1\subset S^ 3-L\},\) denoted by \(\tau\), be a choice of meridians for L. This induces a homomorphism \(F\to \pi\). If all Milnor numbers of length \(\leq k\) vanish, this induces an isomorphism \(F/F^{(k)}\cong \pi /\pi^{(k)}\). Let K be the Eilenberg-MacLane space \(K(F/F^{(k)},1)\) obtained from \(\bigvee^{m}_{i=1}S^ 1\) by adding cells of dimension \(\geq 2\) and let \(K_ k=K/\bigvee^{m}_{i=1}S^ 1\). Extension over meridianal disks gives a canonical map \(\rho\) : \(S^ 3\to K_ k\). Define \(\theta_ k(L,\tau)=[\rho]\in \pi_ 3(K_ k)\). The following properties of \(\theta_ k\) hold: Theorem: 1. \(\theta_ k(L,\tau)=0\) if L is a boundary link. 2. If \(k>\ell\) there is a homomorphism \(\psi_{k,\ell}=\pi_ 3(K_{\ell})\to \pi_ 3(K_{\ell})\) such that \(\psi_{k,\ell}(\theta_ k(L,\tau))=\theta_{\ell}(L,\tau)\). 3. If \(\tau_ 1\), \(\tau_ 2\) are two choices of meridians for L then there is an automorphism \(\alpha\) of \(\pi_ 3(K_ k)\) such that \(\alpha (\tau_ 1)=\tau_ 2\). 4. \(\theta_ k(L,\tau)\) is additive under connected sum of based links. 5. If \(x\in \pi_ 3(K_ k)\) then there is a link L such that \(\theta_ k(L,\tau)\) is defined and equals x. If \(L_ 1\) and \(L_ 2\) are two links with equal \(\theta_ k\) then they are concordant in a special way which respects their lower central series quotients. Now consider what happens if we consider the same link but change the choice of \(\tau\) of the meridians. Theorem: Let \(L\subset S^ 3\) be a link for which all \({\bar \mu}\)-invariants of length \(\leq k\) are zero. Let \(\tau_ 1\), \(\tau_ 2\) be two choices of meridians. Then \(\theta_ k(L,\tau_ 1)\equiv \theta_ k(L,\tau_ 2)\) modulo image \(\{\psi_{k+1,k}: \pi_ 3(K_{k+1})\to \pi_ 3(K_ k)\}.\) In view of the above we can define an invariant \({\bar \theta}{}_ k(L)\in \pi_ 3(K_ k)/image \psi_{k+1,k}=C_ k\) which is independent of the choice of meridians. We have the following Theorem: Let L and \(L'\) be two m-component links with vanishing \({\bar \mu}\)- invariants of length \(\leq k\). Then L and \(L'\) have the same \({\bar \mu}\)- invariants of length \(k+1\) if and only if \({\bar \theta}{}_ k(L)={\bar \theta}_ k(L').\) The rank of the group \(C_ k\) above can be calculated and this gives a formula for the number of independent \({\bar \mu}\)-invariants. Let \(N_ k=1/k\sum_{d| k}\mu (d) m^{k/d}\) where \(\mu\) (d) is the Möbius function. Then \(N_ k\) is the number of basic commutators of length k in the free group on m letters. Theorem: \(C_ k\) is free Abelian of rank \(mN_ k-N_{k+1}\).
lower central series quotients of the fundamental group
scientific article; zbMATH DE number 4094402
scientific article