Brieskorn spheres, cyclic group actions and the Milnor conjecture

Abstract

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 Ozsváth-Szabó $\tau$-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 $H{F}_{red}^{+}(Y)$ is greater than $p$ times the rank of $H{F}_{red}^{+}(Y/{\mathbb{Z}}_p)$, 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.

arXiv comment38 pages, accepted version. To appear in J. Topol

References in corpus (14)

• Knot Floer homology and the four-ball genus
• On the Floer homology of plumbed three-manifolds
• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
• Seiberg-Witten-Floer stable homotopy type of three-manifolds with b_1=0
• On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds
• HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
• HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence
• Seifert forms and concordance
• HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence
• Calculating Heegaard-Floer Homology by Counting Lattice Points in Tetrahedra
• More Brieskorn spheres bounding rational balls
• Cyclic Group Actions on Contractible 4-Manifolds
• Equivariant Seiberg-Witten-Floer cohomology
• A survey of the homology cobordism group