I am currently studying a little bit of general position and transversality theory and I was told that it is necessary to consider at least a picewise linear embedding of two submanifolds to be able to state a General Position Theorem for Embeddings such as this one found in: Rourke, Sanderson, Introduction to piecewise linear topology:
Theorem Let $ Q^q, P_0\subset P^p$ be closed subpolyhedra of the unbounded manifold $M^m$ with $cl(P-P_0)$ compact. Let $\epsilon>0$ be given. Then there is an $\epsilon$-isotopy of $M$ with compact support, fixed on $P_0$ and finishing with $h:M\rightarrow M$ such that $$dim(h(P-P_0)\cap Q) \leq p+q-m$$ Addendum If $p+q=m$ then we can also arrange that $h(P-P_0)$ meets Q transversely.
Now what could happen if $P$ or $Q$ weren't even polyhedra? What kind of ugly thing could happen with a topological embedding so that no general position is possible? If possible I would need some references.
Thank's a lot!
First of all, you should at least restrict to the category of topological manifolds (and topological submanifolds). The there is indeed a notion of transversality and a general position theorem, but you can easily spend a year trying to understand the precise statement (e.g. what is a normal microbundle?). The result is mostly due to Rob Kirby and Larry Siebenmann (under some dimension assumption) and to Frank Quinn in full generality. See this paper and references therein. (Just keep in mind that the book reference [4] there is basically unreadable unless your speciality is higher-dimensional topology.)