# Homology and embedded surfaces of genus $g$

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.

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.