math.GT

For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$.

Given a canonically oriented Brieskorn sphere $Y=\Sigma(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $\Sigma(3,4,5),$ $\Sigma(2,5,7)$ and $\Sigma(2,3,6k+1)$ for $k\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.

Given a closed oriented surface $\Sigma$ of genus at least two, the Goldman trace map defines a function from the vector space generated by the free homotopy classes of oriented closed curves to the Poisson algebra of regular functions on the $G$-character variety where $G$ is a reductive (real or complex) linear Lie group. In this article, we prove that this map is never injective. For each $n$, we construct an explicit nonzero element of the vector space whose associated trace function vanishes on every homomorphism from $\pi_1(\Sigma)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $\pi_1(\Sigma)$, ensuring that no cancellation occurs at the level of free homotopy classes. This gives a uniform family of explicit kernel elements, proving Goldman's predicted non-injectivity of the trace map in arbitrary rank.

We construct an irreducible embedded projective plane in $S^4$. This gives a counterexample to the Kinoshita conjecture and answers Problem 4.37 of the K3 problem list. Moreover, we answer both Questions (i) and (ii) of Problem 4.37: (i) the connected sum $R\# R$ is a Klein bottle in $S^4$ with extremal normal Euler number that does not admit an unknotted projective plane summand, and (ii) we show that our projective plane $R$ is irreducible by showing that the peripheral map $\pi_1 (\partial (S^4\setminus\mathring{N}(R)))\to \pi_1 (S^4 \setminus \mathring{N}(R))$ has kernel of order $2$.