Let $E$ be the space of codimension zero embeddings of a manifold N (PL or Diff, whichever you prefer) into a manifold V that are homotopy equivalences. Let $I$ be the analogous space for immersions.
I have read that if $N$ has a handle decomposition with handles of index less than $d$ where $dim(N)=dim(V)=d+k+1$, then the inclusion $E \rightarrow I$ is (k-d)-connected, citing "general position of handles". Is anyone able to write out a more detailed argument?
This comes from pg. 146 of "SPACES OF PL MANIFOLDS AND CATEGORIES OF SIMPLE MAPS", by Waldhausen, Rognes, and Jahren. I'll add that these manifolds are definitely not simply connected.
Here is my best approximation to an argument:
This inclusion being (k-d)-connected is equivalent to asking that every bundle of immersions $N \rightarrow V$ over spheres of dimension less than or equal to $k-d+1$ can be fiberwise homotoped (through immersions) to an embedding. Now I imagine that we proceed by induction on a handle decomposition of $N$ and prove there are essentially (k-d+1) directions I can homotope my handles so that the restriction to them is an embedding. Then I use the amount of directions to argue I can give a cohesive homotopy along every fiber of the sphere.
Could someone formalize this or just give a complete proof in another manner?