论文标题
Brieskorn球体，循环小组动作和Milnor猜想
Brieskorn spheres, cyclic group actions and the Milnor conjecture
论文作者
论文摘要
在本文中，我们进一步发展了两位作者的seiberg-witten-ploer共同体的理论，重点是Brieskorn同源性领域。我们获得以下应用程序。首先，我们证明了第一作者定义的结节一致性$θ^{（c）} $满足$θ^{（c）}（t_ {a，b}）=（a，b}）=（a-1）（a-1）（b-1）（b-1）/2 $对于torus打结，每当$ c $是$ c $的prime nove $ c $不划分$ ab ab $ ab $。由于$θ^{（c）} $是切片属的下限，因此可以使用ozsváth-szabó$τ$ invariant或rasmussen $ s $ invariant使用ozsváth-szabó$τ$ invariant的新证明。其次，我们证明，在Brieskorn同源性上进行的免费循环组动作$ 3 $ -SPHERE $ y =σ（a_1，\ dots，a_r）$不会平稳地扩展到任何可违约的平滑$ 4 $ manifold界限$ y $。这将是任意$ r $的一般性，即$ r = 3 $的Anvari-Hambleton的结果。第三，给定$ y =σ（a_1，\ dots，a_r）prime订单$ p $在$ y $上行动的有限子组，我们证明，如果$ hf_ {red}^+（red}^+（y）的等级大于$ p $ $ p $ y $ p $ y $ p $ $ hf_/p $ hf_ z $//z y/z^z^z^+a z^+red} $ y $上的$ \ mathbb {z} _p $ - $ y $上的Action不会平稳地扩展到任何合同的平滑$ 4 $ -Manifold界限$ y $。对于Brieskorn同源性领域，我们还证明了类似的非扩展结果。
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 $θ^{(c)}$ defined by the first author satisfy $θ^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever $c$ is a prime not dividing $ab$. Since $θ^{(c)}$ 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 $τ$-invariant or Rasmussen $s$-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology $3$-sphere $Y = Σ(a_1 , \dots , a_r)$ 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 $Y = Σ(a_1 , \dots , a_r)$ 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.
