Plan's theorem for links
Consider the n-fold cyclic branched covering \(K_ n\) of a link K in the 3-sphere \(S^ 3\) or, more generally, in an integral homology 3-sphere. When K is a knot, namely when K is connected, a theorem of Antonio Plans asserts that the first homology group \(H_ 1(K_ n)\) of \(K_ n\) is a direct double, namely splits as a direct sum \(H\oplus H\). This theorem fails when K is not connected, and the authors consider what remains of this result in this case. They show that, if n is odd, the torsion-free quotient and the p-torsion subgroups of \(H_ 1(K_ n)\) are direct doubles for all primes p not dividing n. For n even, there is a natural map \(\pi: K_ n\to K_ 2\), inducing a homomorphism \(\pi_*: H_ 1(K_ n)\to H_ 1(K_ n)\). The authors then show that the torsion-free quotient and the p-torsion groups of the kernel of \(\pi_*\) are direct doubles for all primes not dividing n/2.
n-fold cyclic branched covering of a link
scientific article; zbMATH DE number 4147102