Consider the smooth 3-manifolds, $M$, which can be 'sliced', i.e. $$M=\bigcup_{t\in I}\Sigma_t$$ where $t$ is a real parameter, $I$ is some interval of the real line, and $\Sigma_t$ is a smooth 2-manifold $\forall t$.
The topology of the 2-manifolds is given entirely by the genus, $g$. Here it is a function of the parameter $t$: $g\equiv g(t)$. It seems to me that the topology of such an $M$ is entirely specified by the function $g(t)$. Is this correct and is it useful?
A typical instance of $g(t)$ might be:
I should add that at the instances where the genus changes, I expect $\Sigma_t$ to actually be a singular manifold.
First of all, the assertion "The topology of the 2-manifolds is given entirely by the genus, $g$" is false. It is true under some extra assumptions, namely, that the manifold is connected, compact and, say, orientable (or nonorientable). Thus, let me assume that each $\Sigma_t$ is connected, compact and orientable.
Next, you wrote the ordinary union, $\bigcup_{t\in I}\Sigma_t$, but you surely meant a disjoint union, $$ \coprod_{t\in I} \Sigma_t, $$ i.e. you forgot to assume that for different values of $t$ the surfaces are disjoint.
However, even with this assumption the conjecture that the topology of $M$ is uniquely determined by the function $t\mapsto g(t)$, where $g(t)$ is the genus of $\Sigma_t$, is false even if you additionally assume that the 3-dimensional manifolds are compact, orientable and connected. Examples are given by various 3-dimensional manifolds fibered over the circle. Up to a homeomorphism, there are infinitely many 3-dimensional manifolds which can be sliced as $$ \coprod_{t\in I} \Sigma_t, g(t)=1 ~~~\forall t. $$ These manifolds are torus bundles over the circle. The interval $I$ in this case is $[0,1)$ and, moreover, in a suitable sense, $\Sigma_t$ depends smoothly on $t$. On the other hand, if you do not assume that surfaces are orientable, there are examples of connected sliced manifolds where the topology of the slices depends on $t$ (switches from orientable to nonorientable and the genus changes as well).