(The) infinite family of symplectic tori in a fixed homology class

Abstract

Ronald Fintushel and Ronald J. Stern have proved the following theorem :

\begin{thm} Let $X$ be a simply connected symplectic 4-manifold which contains a c-embedded symplectic torus $T$. Then in each homology class $2m[T]$, $m\geq 2$, there is an infinite family of smoothly embedded symplectic tori, no two of which are smoothly isotopic. \end{thm} To say that a torus $T$ is $c-embedded$ means that $T$ is a smoothly embedded homologically essential torus of self-intersection zero which has a pair of simple curves which generate its first homology and which bound vanishing cycles (disks of self-intersection $-1$) in $X$. We will prove this theorem with different model from Birman and Menasco which R. Fintushel and R. J. Stern have used. We shall make proof of it by showing that if two smoothly embedded symplectic tori are smoothly isotopic then their Alexander polynomials are equal and finding an infinite family of braids whose Alexander polynomials are all distinct.