I have an answer in a different direction. It has been developed a theory (starting with Mañe, and then Bonatti-Diaz-Pujals-Ures) showing that robust dynamical behaviour (in the $C^1$-topology) implies the existence of $Df$-invariant geometric structures (hyperbolicity, partial hyperbolicity, dominated splitting...). In particular, one knows that a $C^1$-robustly transitive diffeomorphism is:

1) (Mañe) Anosov if $dim M=2$. By Franks' results, this implies that $M= T^2$ and $f$ is isotopic to a linear Anosov automorphism.

2) (Diaz-Pujals-Ures) Partially hyperbolic if $dim M=3$.

3) (Bonatti-Diaz-Pujals) Admits a dominated splitting if $dim M \geq 3$.

See the book by Bonatti-Diaz-Viana or the following survey (http://arxiv.org/abs/0912.2896 in french) for more information.

It seems natural to expect that the existence of such $Df$-invariant geometric structures impose some restrictions on the topology of the manifold which admits such diffeomorphisms. So, a relationship between $C^1$-robust dynamical behaviour and topological properties. It seems natural to study these questions in low-dimensions (such as in 1) the question is "solved" in dimension 2). In dimension 3, there are some guesses on which are the manifolds supporting (strong) partially hyperbolic systems. See the work of Bonatti-Wilkinson and Brin-Burago-Ivanov (they are in the author's websites). See also http://www.mat.puc-rio.br/edai/textos/potrie.pdf for some related discussion.

In particular, just to mention an open question which I find really interesting: Does there exists a robustly transitive diffeomorphism on $S^3$? Partial negative answers were given by Diaz-Pujals-Ures and Burago-Ivanov, but in all its generality it remains open.