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 ${\theta }^{\left(c\right)}$ defined by the first author satisfy ${\theta }^{\left(c\right)}\left(T_{a,b}\right)=(a-1)(b-1)/2$ for torus knots, whenever $c$ is a prime not dividing $ab$. Since ${\theta }^{\left(c\right)}$ 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 $Y=\Sigma (a_1,\cdots ,a_r)$ does not extend smoothly to any homology 4-ball bounding $Y$. In the case of a non-free cyclic group action of prime order, we prove that if the rank of $H{F}_{red}^{+}\left(Y\right)$ is greater than $p$ times the rank of $H{F}_{red}^{+}(Y/Z_p)$, then the $Z_p$-action on $Y$ does not extend smoothly to any homology 4-ball bounding $Y$. 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.