Brieskorn spheres, cyclic group actions and the Milnor conjecture

In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants theta(c)$\theta <^>{(c)}$ defined by the first author satisfy theta(c)(Ta,b)=(a-1)(b-1)/2$\theta <^>{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever c$c$ is a prime not dividing ab$ab$. Since theta(c)$\theta <^>{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3-sphere Y=Sigma(a1,& ctdot;,ar)$Y = \Sigma (a_1, \dots, a_r)$ does not extend smoothly to any homology 4-ball bounding Y$Y$. In the case of a non-free cyclic group action of prime order, we prove that if the rank of HFred+(Y)$HF_{red}<^>+(Y)$ is greater than p$p$ times the rank of HFred+(Y/Zp)$HF_{red}<^>+(Y/\mathbb {Z}_p)$, then the Zp$\mathbb {Z}_p$-action on Y$Y$ does not extend smoothly to any homology 4-ball bounding Y$Y$. Third, we prove that for all but finitely many primes a similar non-extension result holds in the case that the bounding 4-manifold has positive-definite intersection form. Finally, we also prove non-extension results for equivariant connected sums of Brieskorn homology spheres.