Immersions of manifolds with boundary (regular homotopy classes, h-principle) Have regular homotopy classes of immersions of manifolds with boundary been studied? i.e. immersions $(M, \partial M) \looparrowright (N, \partial N)$. The references I've looked at from this question state theorems for target manifolds without boundary.
Does the $h$-principle apply, or something like it?
Simple example to keep in mind: $M$ is $D^1$ with boundary $S^0$, $N$ is $D^1 \times \mathbb{R}$ with boundary $S^0 \times \mathbb{R}$, and let's say each boundary point of $M$ goes to a different component of $\partial N$. It seems to me at a glance that there is a regular homotopy class for each integer, corresponding to the number of counterclockwise turns made along the way from one component to the other. How is this proved? (and generalized?)
 A: There are two questions you could ask: about the space of immersions $Imm((M, \partial M), (N, \partial N))$ of $M$ in $N$ taking the boundray to the boundary, where the boundary is allowed to move, or you could fix an immersion $i : \partial M \looparrowright \partial N$ and consider the subspace $Imm((M, \partial M), (N, \partial N); i)$ of those immersions which restrict to $i$ on the boundary.
These fit into a fibration sequence
$$Imm((M, \partial M), (N, \partial N); i) \to Imm((M, \partial M), (N, \partial N)) \to Imm(\partial M, \partial N)$$
so the answers are closely related. The space $Imm(\partial M, \partial N)$ can certainly be studied by the Smale--Hirsch $h$-principle for immersions, but the usual proof of that theorem is an induction over handles, where handles are immersed with given bundary conditions: this proof applies equally well to $Imm((M, \partial M), (N, \partial N); i)$, too, and gives a bundle-theoretic description of this space. If you like, you can then combine these results to get a bundle-theoretic description of $Imm((M, \partial M), (N, \partial N))$. This will be: the space of bundle monomorphisms $\hat{f}: TM \to TN$, covering a map $f : (M, \partial M) \to (N, \partial N)$, which preserve the inward-pointing normal vector over the boundary.
