Brieskorn spheres, cyclic group actions and the Milnor conjecture (Q6564518)

The article further develops the Seiberg-Witten-Floer cohomology theory as first developed in [\textit{D. Baraglia} and \textit{P. Hekmati}, Algebr. Geom. Topol. 24, No. 1, 493--554 (2024; Zbl 1545.57013)] and applies it to Brieskorn homology spheres. Applications include a new proof of the Milnor conjecture and obstructions to extending group actions over a bounding 4-manifold. Let \(a_{1},\dots,a_{r}>1\) be pairwise coprime positive integers and let \(Y=K(a_{1},\dots,a_{r})\) be the Brieskorn homology sphere. \(Y\) is an integral homology 3-sphere and is also a Seifert 3-manifold, so there is an \(S^{1}\)-action on \(Y\). A \(\mathbb{Z}_{p}\)-action on \(Y\) is defined by restricting the circle action. The main results are obtained by considering the equivariant Seiberg-Witten-Floer cohomology of \(Y\) with respect to such \(\mathbb{Z}_{p}\)-actions. Let \(p>1\) be a prime and let \(Y=\Sigma_{p}(K)\) be the cyclic \(p\)-fold cover of \(S^{3}\) branched over a knot \(K\). The equivariant Seiberg-Witten-Floer cohomology of \(Y\) is applied to define knot concordance invariants \(\theta^{(p)}(K)\). The invariants are lower bounds for the slice genus \(g_{4}(K)\); indeed, \(g_{4}(K)\ge \theta^{(p)}(K)\) for all primes \(p\). The article considers the case where \(K=T_{a,b}\) is an \((a,b)\)-torus knot. Let \(c\) be a prime not dividing \(ab\), then \(\Sigma_{c}(T_{a,b})=K(a,b,c)\). The \(\mathbb{Z}_{c}\)-action resulting from the branched covering construction coincides with the restriction to \(\mathbb{Z}_{c}\) of the Seifert circle action.\N\NTheorem 1. Let \(a, b>1\) be coprimes and let \(c\) be a prime that does not divide the product \(ab\). Then \(\theta^{(c)}(T_{a,b})=\frac{1}{2}(a-1)(b-1)\).\N\NTherefore, \(g_{4}(K)\ge \frac{1}{2}(a-1)(b-1)\). The reverse is easy because the Milnor fiber of the singularity \(x^{a}=y^{b}\) has genus \(\frac{1}{2}(a-1)(b-1)\). So the Milnor conjecture \(g_{4}(K)= \frac{1}{2}(a-1)(b-1)\) follows as a consequence.\N\NThe equivariant delta invariants introduced in [loc. cit.] are an equivariant generalization of the Ozsváth-Szabó \(d\)-invariant and are equivariant homology cobordism invariants. Let \(Y\) be a rational homology sphere. Given a \(\mathbb{Z}_{p}\)-action on \(Y\) and a \(\mathbb{Z}_{p}\)-invariant \(spin^{c}\) structure \(\mathfrak{s}\) on \(Y\) the authors define the equivariant delta invariants of \((Y,\mathfrak{s})\) as the sequence of invariants \(\delta^{(p)}_{j}(Y,\mathfrak{s})\in\mathbb{Q}\) indexed by \(j\in\mathbb{N}\). When \(Y\) is an integral homology 3-sphere it has a unique \(spin^{c}\) structure which is automatically \(\mathbb{Z}_{p}\)-invariant. The important fact is that these invariants are invariant under equivariant homology cobordism. Therefore, they are obstructions to extending the \(\mathbb{Z}_{p}\)-action over an integral or rational homology 4-ball \(W\) such that \(\partial W= Y\). In [loc. cit.] the following result was proved;\N\NTheorem 2. Let \(Y\) be an integral homology 3-sphere on which \(\mathbb{Z}_{p}\) acts by orientation preserving diffeomorphisms. Suppose \(Y=\partial W\) where \(W\) is an integer homology 4-ball. If the \(\mathbb{Z}_{p}\)-action extends smoothly over \(W\), then \(\delta^{(p)}_{j}(Y)=\delta^{(p)}_{j}(-Y)=0\) for all \(j\).\N\NThe sequence of invariants \(\{\delta^{(p)}_{j}(Y)\}_{j\ge 0}\) is decreasing and eventually constant. By setting \(\delta^{(p)}_{\infty}(Y)=\lim_{j\to\infty}\delta^{(p)}_{j}(Y)\), the authors prove the following result for a Brieskorn homology 3-sphere \(Y=\Sigma(a_{1},\dots,a_{r})\):\N\N(1) \(\delta^{(p)}_{j}(Y)=\delta^{(p)}_{\infty}(Y)\) for all \(j>0\).\N\N(2) \(\delta(Y)\le \delta^{(p)}_{\infty}(Y)\le -\lambda(Y)\).\N\N(3) \(\lambda(Y)\le \delta^{(p)}_{j}(-Y)\le \delta(Y)\) for all \(j\ge 0\), where \(\lambda(Y)\) is the Casson invariant of \(Y\) and \(\delta(Y)=\frac{d(Y)}{2}\) is half the Ozsváth-Szabó \(d\)-invariant. If \(Y\) is a Brieskorn homology 3-sphere bounding a contractible 4-manifold, then \(\delta(Y)=0\), \(\delta^{(p)}_{j}(Y)=\delta^{(p)}_{\infty}(Y)\) and\N\[\N\delta^{(p)}_{\infty}(-Y)\le \delta^{(p)}_{j}(-Y)\le 0\N\]\Nfor all \(j\ge 0\). Thus \(\delta^{(p)}_{j}(\pm Y)=0\) for all \(j\ge 0\) if and only if \(\delta^{(p)}_{\infty}(\pm Y)=0\). When the \(\mathbb{Z}_{p}\)-action on \(Y=K(a_{1},\dots,a_{r})\) is considered, the article proves the following result;\N\NTheorem 3. Let \(Y=K(a_{1},\dots,a_{r})\) be a Brieskorn homology 3-sphere and let \(p\) be a prime not dividing \(a_{1},\dots,a_{r}\). Set \(Y_{0}=Y/\mathbb{Z}_{p}\). Then for any \(spin^{c}\)-structure \(\mathfrak{s}_{0}\) on \(Y_{0}\) we have\N\[\N\delta^{(p)}_{\infty}(Y)-\delta(Y)=rk\big(HF^{+}_{red}(Y)\big)-rk(HF^{+}_{red}\big(Y_{0},\mathfrak{s}_{0})\big).\N\]\NTheorem 4. We have \(rk\big(HF^{+}_{red}(Y)\big)>rk(HF^{+}_{red}\big(Y_{0},\mathfrak{s}_{0})\big)\) except in the following cases:\N\N(1) \(Y=\Sigma(2,3,5)\) and \(p\) is an arbitrary prime.\N\N(2) \(Y=\Sigma(2,3,11)\) and \(p=5\).\N\NIn Case (1) we have \(rk\big(HF^{+}_{red}(Y)\big)=rk(HF^{+}_{red}\big(Y_{0},\mathfrak{s}_{0})\big)=0\) and in Case (2) we have \N\N\(rk\big(HF^{+}_{red}(Y)\big)=rk(HF^{+}_{red}\big(Y_{0},\mathfrak{s}_{0})\big)=1\).\N\NFrom the combination of these results, the authors prove the next corollaries:\N\NCorollary 5. Under the hypothesis of Theorem 5, we have \(\delta^{(p)}_{\infty}(Y)>\delta(Y)\) except in the cases:\N\N(1) \(Y=\Sigma(2,3,5)\) and \(p\) is an arbitrary prime.\N\N(2) \(Y=\Sigma(2,3,11)\) and \(p=5\).\N\NIn both cases we have \(\delta^{(p)}_{\infty}(Y)=\delta(Y)=1\).\N\NCorollary 6. Let \(Y=K(a_{1},\dots,a_{r})\) be a Brieskorn homology 3-sphere and let \(m>1\) be an integer not dividing \(a_{1},\dots,a_{r}\). Suppose \(Y=\partial W\) where \(W\) is a smooth rational homology 4-ball and \(m\) does not divide \(\vert H^{2}(W;\mathbb{Z})\vert\). Then the \(\mathbb{Z}_{m}\)-action on \(Y\) does not extend smoothly to \(W\).\N\NThe \(\delta\)-invariants are also useful to obstruct the extension of the \(\mathbb{Z}_{p}\)-action over a positive definite 4-manifold bounding \(Y\). The article proves the following result:\N\NTheorem 7. Let \(Y=K(a_{1},\dots,a_{r})\) be a Brieskorn homology 3-sphere and let \(p\) be an arbitrary prime. If \(\delta^{(p)}_{\infty}(Y)>0\) or \(\delta^{(p)}_{\infty}(-Y)>0\), then the \(\mathbb{Z}_{p}\)-action on \(Y\) does not extend smoothly to any smooth, compact, oriented, 4-manifold \(W\) with positive definite intersection form and with \(b_{1}(W)=0\) bounding \(Y\).