Irreducible 3-manifold, needs of smooth sphere. I'm trying to get acquainted with 3-manifold so bear with me if the question is rather basic. My definiton of Irreducible 3-manifold is the following:
$\textbf{Definition:}$ A 3-manifold $M$ is irreducible if any 2-sphere $S^{2}\subset M$ bounds a ball $B^{3}\subset M$.
Now what I don't really understand is the needs to have a smooth structure on the sphere, for what I understand it is usefull because it gives me the "possibility" of considering a tubular neighborhood but why is it needed in the definiton?
Related to this is the Alexander horn sphere an example of "what could go wrong"?
Last question and I'm going in deep water so forgive me for any error: can't I always give a smooth structure to a 3-manifold as Top and Smooth are equivalent cathegory in dimension 3? Is the problem that a smooth structure on a topological sphere might not be THE smooth structure of a smooth sphere?
Moreover is anyone of you knows reference for books or notes on somewhat related topic I would love it. Hatcher has some 3-manifold notes but they seem to start "too fast" for me, nothing more introductory?
Thanks in advance.
 A: What you need here is not a "smooth structure on the sphere." The correct setting is the following:
Suppose that $M$ is a smooth 3-dimensional manifold and $S\subset M$ is a smooth submanifold diffeomorphic to $S^2$.
Alternatively, you can work with a triangulated 3-manifold and a simplicial subcomplex $S\subset M$ which is homeomorphic to $S^2$.
Or, you can work purely topologically and assume that $M$ is a topological 3-dimensional manifold and $S\subset M$ is a locally flat surface homeomorphic to $S^2$. Here a surface $S\subset M$ is called locally flat if $\forall x\in S$ there exists a neighborhood $U$ of $x$ in $M$ and a homeomorphism $U\to R^3$ sending $U\cap S$ to a linear hyperplane in $R^3$.
Which of these you find more convenient depends on your preferences and knowledge of 3-dimensional topology. Given any of these settings you can continue and define $M$ to be irreducible if every sphere in $M$, satisfying one of the above conditions, bounds a 3-ball in $M$.
To see why a condition on $S$ is needed, let's make up the following fake definition:
Definition. We say that a topological 3-dimensional manifold $M$ is ireducible (note the misspelling!) if every topological submanifold $S\subset M$ homeomorphic to $S^2$ has the property that one of the components of $M\setminus S$ is homeomorphic to $R^3$.
However, there are wild spheres $S\subset S^3$ such that neither component of $S^3\setminus S$ is homeomorphic to $R^3$. From this, one can conclude (with a bit of thought) that there are no ireducible 3-dimensional manifolds, making the above definition useless.
