Homology and embedded surfaces of genus $g$

Question

Suppose to have a closed manifold $M$ of dimension $n\geq 3$ and suppose that there is an element in Homology $[S]\in H_2(M,\mathbb{Z})$ that is represented by a surface $S$ homeomorphic to the 2-torus (if you want it induces the support of a simplicial complex homeomorphic to the torus). $$S \simeq \mathbb{T}^2.$$ Then we have also the multiples $k[S] $ where $k\in \mathbb{Z}$, I wonder if we can think of this multiples as surfaces of genus $k$ embedded into M. Or more formally $k[S]$ is represented by a $k$-genus surface.

Answer 1

This is not true in general. For example consider $\mathbb CP^2$. Lets $a$ be a generator of $H_2$. Then $3a$ represent by a torus (think that 3 sphere or complex line intersect each other tranversally pairwise, so it is a torus). But $6a$ cannot be represent by a genus 2 surface. Now $6a$ can represent by $\sigma = \{[x:y:z:]\in \mathbb CP^2| x^6+y^6+z^6=0\}$ . Adjunction formula tells us that $2g(\sigma)-2=[\sigma]^2-c_1(\mathbb CP^2)[\sigma]=d^2-3d$ (where $d$ is the degree and $c_1$ denote the first Chern class). So $g(\sigma)=10$ ( since $d=6$). In-fact this is the minimum genus. For more details, see chapter 2 of Kirby Calculus-Gompf & Stipsicz.

One case when your question has a positive answer- If $S$ has a trivial normal bundle inside $M$. Then you can take parallel $k$ copies of $S$ and join them by annulus. And that gives you a genus $k$ surface representation.