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Preprint

10 August 2022

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 the following applications. First, we show that the knot concordance invariants \(\theta^{(c)}\) defined by the first author satisfy \(\theta^{(c)}(T_{a,b}) = (a-1)(b-1)/2\) for torus knots, whenever \(c\) is a prime not dividing \(ab\). Since \(\theta^{(c)}\) is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture of a similar flavour to the proofs using the Ozsv\'ath-Szab\'o \(\tau\)-invariant or Rasmussen \(s\)-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology \(3\)-sphere \(Y = \Sigma(a_1 , \dots , a_r)\) does not extend smoothly to any contractible smooth \(4\)-manifold bounding \(Y\). This generalises to arbitrary \(r\) the result of Anvari-Hambleton in the case \(r=3\). Third, given a finite subgroup of the Seifert circle action on \(Y = \Sigma(a_1 , \dots , a_r)\) of prime order \(p\) acting non-freely on \(Y\), we prove that if the rank of \(HF_{red}^+(Y)\) is greater than \(p\) times the rank of \(HF_{red}^+(Y/\mathbb{Z}_p)\), then the \(\mathbb{Z}_p\)-action on \(Y\) does not extend smoothly to any contractible smooth \(4\)-manifold bounding \(Y\). We also prove a similar non-extension result for equivariant connected sums of Brieskorn homology spheres.