I'd like some help with the following exercise, taken from Gromov's book Partial Differential Relations.
Let $M,N$ be stably parallelisable smooth manifolds with $\dim N>\dim M$ and let $f:M\rightarrow N$ be a smooth map. Find an immersion $M\looparrowright N$ in the homotopy class of $f$.
Here we recall that a manifold $V$ is stably parallelisable if $V\times\mathbb{R}$ is parallelisable. To solve the exercise you are, or course, supposed to use the
Smale-Hirsch Theorem: Immersions $V\looparrowright W$ satisfy the h-principle in the following two cases
i) $\dim W>\dim V$.
ii) $\dim W=\dim V$, and $V$ is an open manifold. $\square$
Now, since $M$, $N$ in the exercise are assumed to be stably parallelisable it is easy to cover $f\times 1:M\times\mathbb{R}\rightarrow N\times\mathbb{R}$ by an injective bundle map and apply the Smale-Hirsch statement to get that $M\times \mathbb{R}$ (and hence $M$) immerse in $N\times\mathbb{R}$ by a map which is homotopic to $f\times 1$. However I'm struggling to convince myself that I can project $N\times \mathbb{R}$ back to $N$ in such a way that the composite map remains an immersion.
Best Answer
The proof requires two steps, both steps being separate $h$-principles; one is Smale-Hirsch theorem as you said and the second is Gromov's $A$-directed embedding theorem.
By restricting your embedding $V \times \Bbb R \to W \times \Bbb R$ to $V \times 0$, you have an embedding $V \hookrightarrow W \times \Bbb R$ and as you concluded, you would like to position this embedded copy of $V$ such that restriction of the projection $W \times \Bbb R \to W$ gives an immersion $V \looparrowright W$.
Let $k = \dim V, l = \dim W$. Let $A \subset \mathrm{Grass}(k, W \times \Bbb R)$ be open subset of the Grassmannian of $k$-planes consisting of non-vertical planes, i.e., the planes which intersects the vertical lines $\{w\} \times \Bbb R$ transversely for all $w \in W$. You would like to homotope $V \hookrightarrow W \times \Bbb R$ to an embedding whose differential lands in $A$.
There is no obstruction at the level of differentials; given the tangent $k$-plane field along $V$ in $W \times \Bbb R$, one can always nudge the plane field to one which lands in $A$; this is because $k < l$, so there are at least two degrees of freedom for a ($k$-dim) tangent plane along $V$ in $W \times \Bbb R$ (which is $l+1$-dim) to move, since $(l+1) - k \geq 2$ so that we can simply rotate.
There is one more condition for the $h$-principle to go through; $A$ needs to be a complete subset of the Grassmannian, see Ch 4.6 in Eliashberg-Mishachev. This can be easily checked in the case of the above subset.
Given all of this, one can apply Gromov's theorem to homotope the embedding to be "$A$-directed", i.e., find a new embedding $V \hookrightarrow W \times \Bbb R$ whose differential lands in $A$. This concludes the second step in the proof of the exercise.
For a visually transparent proof of this corollary of the $A$-directed embedding theorem see Rourke-Sanderson, "The compression theorem I" (II and III are also beautiful papers which I highly recommend)