TITLE: Can surfaces be interestingly knotted in five-dimensional space?
QUESTION [22 upvotes]: It's possible this question is trivial, in which case it will be answered quickly.  In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves knots — one-dimensional compact manifolds mapped generically into three-dimensional compact manifolds — and it's natural to ask about "knots" in higher dimensions.  Of course, the space of generic maps of a one-dimensional compact manifold into a four-dimensional compact manifold is connected, so there is no interesting "knotting".  Instead, people usually think about "surface knots in 4d", which are usually defined as embedded compact 2-manifolds in a 4-manifold.
But surfaces can map into 4-space in much more interesting ways.  In particular, whereas a generic map from a 1-manifold to a 3-manifold is an embedding, two generic surfaces in 4-d can be "stuck" on each other: the generic behavior is to have point intersections.  So a richer theory than that of embedded surfaces in 4-space is one that allows for these point self-intersections — it would be the theory of connected components of the space of generic maps.
Still, though, thinking about these self-intersections is hard, and their existence is part of what makes 2-knot theory hard (for instance, it interferes with developing a good "Vassiliev" theory for 2-knots).  If you really want to reproduce the fact that generic maps have no self intersections, you should move the ambient space one dimension higher.
Hence my question:

Can compact 2-manifold embedded into a compact 5-manifold be interestingly "knotted"?  I.e. let $L$ be a compact 2-manifold and $M$ a compact 5-manifold; are there multiple connected components in the space of embeddings $L \hookrightarrow M$?

I expect the answer is "no", else I would have heard about it.  But my intuition is sufficiently poor that I thought it best to ask.

REPLY [6 votes]: I think the first reference in the literature for the result you want is Wen-Tsun Wu, On the isotopy of $C^r$-manifolds of dimension $n$ in Euclidean $(2n+1)$-space.  Sci. Record (N.S.) 2 1958 271--275. 
In it he proves any two $C^r$-embeddings $M^n \to \mathbb R^{2n+1}$ are isotopic, provided $n>1$.  $M^n$ is any compact $n$-manifold (the title does not mention he also assumed connected, but it is).  The techniques are now standard -- your two embeddings $f_0, f_1 : M \to \mathbb R^{2n+1}$ are homotopic $f_t : M \to \mathbb R^{2n+1}$ so you look at the "graph", $(x,t) \longmapsto (f_t(x), t)$ as a map $M \times [0,1] \to \mathbb R^{2n+!} \times [0,1]$ and then approximate by an immersion and consider appropriate usage of the Whitney trick.