gt.geometric topology - Homotopy classes of complex bundle maps and isotropic immersions into contact manifolds

jc, you'll have fun working out answers to examples of your first question. I'm only going to address the Legendrian case. If I didn't make mistakes, I'll conclude that Legendrian immersions of $S^n$ into $S^{2n+1}$ are all regular homotopic if $n$ is even, while if $n$ is odd they are classified by a rotation number. For the second question, I'll just suggest that you think about 1-forms valued in the orientation line bundle.

The h-principle you quoted (cf. Eliashberg-Mishachev's book) tells us that we need to understand homotopy classes $[f,l]$, where $f$ is a map $L to M$ and $l colon TLotimes mathbb{C}to f^* xi$ an injective bundle map. Understanding the set $[L,M]$ of homotopy classes of unbased maps is an obstruction theory problem, and in general pretty intractable, but $[S^k,M]$ is the set of $pi_1(M)$-orbits in $pi_k(M)$. The fibre of the forgetful map $[f,l]mapsto [f]$ is exactly the set of injective bundle maps $l colon TLotimes mathbb{C}to f^ast xi$ (this is an application of the covering homotopy property of bundles, bearing in mind that the injective bundle maps themselves form a fibre bundle).

Now fix $f$. In the Legendrian case, what we're looking is $pi_0$ of the space $I(f)$ of isomorphisms $TLotimes mathbb{C}to f^ast xi$. In terms of classifying maps, $pi_0 I(f)$ is the set of homotopy classes of homotopies from $t_{mathbb{C}}$, the composite of the tangent map $tcolon L to BO(n)$ with $BO(n)to BU(n)$, to $tilde{xi}f$. Necessary conditions for $I(f)neq emptyset$ are that, for all $j$, $p_j(TL) = pm f^ast c_{2j}(xi)$ (I forget what the sign should be) and $2 f^ast c_{2j+1}(xi)=0$. When $L=S^n$, you have to compare two elements of $pi_n BU(n)=pi_{n-1}U(n) = pi_{n-1}U(infty)$ which is $mathbb{Z}$ or $0$ according to whether $n$ is even or odd. Since the Pontryagin classes of $S^n$ are zero by the signature theorem, I think what you actually have to check is whether $f^* c_{n/2}(xi)=0$.

If one isomorphism exists, there will be more: we can compose with any complex automorphism of $T^ast Lotimes mathbb{C}$. Up to homotopy, automorphisms correspond to isomorphism classes of vector bundles over $S^1times L$ extending $T^ast Lotimes mathbb{C}to L$, and so one has to classify such extensions. This is another obstruction theory problem. When $L=S^n$, any two extensions agree over $S^1 vee L$, and over the last $(n+1)$-cell they differ by a map $S^{n+1} to BU(n)$.

So, when $L=S^n$, there's a freely transitive action of $pi_{n+1}BU(n)=pi_n U(n) = pi_n U(infty)$ on $I(f)$.

Examples: say $M=S^{2n+1}$, $xi$ any contact structure, $L=S^n$, $f$ any map (necessarily homotopically trivial). Then $f^ast xi$ is trivial, so $I(f)$ is non-empty. It is a singleton if $n$ is even, and is a freely transitive $mathbb{Z}$-set if $n$ is odd. When $n=1$, the $mathbb{Z}$ should be the classical rotation number.