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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?)

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    $\begingroup$ Quick thought, you can get rid of the boundary by going to the compact double, $M \sqcup_{\partial M} (-M)$, that is $M$ glued to an orientation reflected copy of itself along $\partial M$, and same with $N$. I suspect that this can be made to play nice with both immersions and homotopy classes, though I wouldn't guess exactly how. $\endgroup$
    – Igor Khavkine
    Commented Oct 28, 2017 at 8:55
  • $\begingroup$ Very interesting idea @IgorKhavkine! I'll think about that. In the example I gave, it feels like your construction takes regular homotopy classes injectively to regular homotopy classes of the circle in the cylinder. (to the even integers?) Thanks! $\endgroup$
    – Haggai Nuchi
    Commented Oct 28, 2017 at 9:27

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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.

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  • $\begingroup$ Thank you! I'm interested in $Imm((M, \partial M), (N, \partial N))$. Do you happen to know a reference for this stuff, or is it simple enough (once you get comfortable) that no one bothers to write it down? The last bit, in particular, of bundle monomorphisms preserving inward-pointing normals sounds especially like what I'm looking for. $\endgroup$
    – Haggai Nuchi
    Commented Oct 28, 2017 at 9:36
  • $\begingroup$ @HaggaiNuchi I agree with Oscar that this can be proved using a modification of the standard techniques, but if you want to see it in some detail and in more generality (Smale-Hirsch for immersions of stratified spaces in other stratified spaces), take a look at Section 6.3 here which is a joint work with Mahan Mj. $\endgroup$
    – Balarka Sen
    Commented May 29, 2022 at 20:34
  • $\begingroup$ Thanks @BalarkaSen! I'm embarrassed to say that I only barely remember posting this question and don't at all remember why I wanted to know the answer. But past-me would have very much appreciated your response! $\endgroup$
    – Haggai Nuchi
    Commented May 30, 2022 at 1:53

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