Related to the previous post, I have been thinking about when, given submanifolds $A, B \subset M$, one can have a Legendrian isotopy from $S^*_A M$ to $S^*_B M$. The simplest kind of such isotopy is one which begins and ends with Reeb flow, and in the interim, remains in the complement of both.
For this, we may as well cut out neighborhoods of $A$ and $B$ and consider the resulting manifold with boundary. Its boundaries are the cosphere bundles over $A$ and $B$; we are asking for a Legendrian isotopy an outward-oriented copy of one to an inward-oriented copy of the other. Remarkably, this puts extremely strong constraints on $M$. For the nonce, let us forget that the boundaries came from cosphere bundles.
Theorem. Let $M$ be a manifold with boundary $\partial M = M_{in} \cup M_{out}$. If there’s a Legendrian isotopy from the positive conormal of $M_{in}$ to the negative conormal of $M_{out}$, then $M$ is a $h$-cobordism.
Proof. More precisely, let’s write $i: C_{in} \to M$ for a collar neighborhood of $M_{in}$ and $o: C_{out} \to M$ for a collar neighborhood of $C_{out}$. We are assuming a Legendrian isotopy from the positive conormal of the inner boundary of $C_{in}$ (aka $ss^\infty i_* \mathbb{Z}$) to the negative conormal of the inner boundary of $C_{out}$ (aka $ss^\infty o_! \mathbb{Z}$).
Let $\Phi$ denote the sheaf quantization of this isotopy. Then $ss^\infty(\Phi(i_*\mathbb{Z})) = ss^\infty(o_! \mathbb{Z})$. Moreover, the stalk of $\Phi(i_*\mathbb{Z})$ at the outer boundary $C_{in}$ is just $\mathbb{Z}$, since the isotopy stays away from here. The resulting sheaf is a rank one local system on the complement of the interior of $C_{out}$.
Sheaf quantization acts as the identity on the category of local systems; in particular, preserving $\mathbb{Z}_M$. Thus it preserves $Hom(\mathbb{Z}_M, \cdot)$; thus there’s a nontrivial map $\mathbb{Z}_M \to \Phi(i_*\mathbb{Z})$. Thus $\Phi i_*\mathbb{Z}$ is just the constant local system on the complement of the interior of $C_{out}$, and the above map induces
$$H^*(M, \mathbb{Z}) = Hom(\mathbb{Z}_M, \mathbb{Z}_M) = Hom(\mathbb{Z}_M, \Phi i_* \mathbb{Z}_M) = Hom(\mathbb{Z}_M, i_* \mathbb{Z}) = H^*(M_{in}, \mathbb{Z}_M)$$
where the second to last step is due to the fact that sheaf quantization is an equivalence of categories.
Thus the inclusion $M_{in} \to M$ is a $\mathbb{Z}$-homology equivalence. Playing the same game as above with nontrivial local systems (as in Guillermou) shows that moreover pullback along this inclusion induces an isomorphism of categories of local systems, hence is a homotopy equivalence. $\blacksquare$
E.g. invoking Smale’s h-cobordism theorem, I learn that any manifold both of whose ends are simply connected of dimension $\ge 5$, which are moreover Legendrian isotopic, must be a cylinder.
But really one wants to upgrade the conclusion of the above theorem to the statement that $M$ is an $s$-cobordism. Maybe techniques like Abouzaid-Kragh used in here work for this.
[Indeed, this statement can possibly be deduced from some version of what is currently known about the nearby Lagrangian conjecture: take the movie of the isotopy and turn it into an exact (?) Lagrangian. One should worry a bit about this being the noncompact setting, about Maslov classes, and also about the fact that at least if I allow the isotopy to close back up, e.g. the isotopy from the cocircle over the north pole on the sphere to the cocircle over south pole on the sphere, followed by the trivial isotopy back to the north pole, then the movie certainly cannot be made exact — the movie’s a klein bottle; this would violate known statements re. NLC. Possibly one is saved in this setting by the fact that the isotopy doesn’t close.]
Remark. At least the case of sphere boundaries isn’t really saying anything interesting. Indeed, suppose given such an isotopy in the conormal bundle, one has a degree one map relative the boundaries from a cylinder to the base; by capping off, one has a degree 1 map from a sphere to the capped off base, hence the capped of base is a sphere, hence once you remove the caps it’s a cylinder. Maybe this isn’t saying anything interesting in general. I.e.,
Q: suppose one has a cobordism in which the two ends are homotopic. Must it be an h-cobordism?
Hi Vivek,
Here are some related questions about Legendrian
1. How to test if two Legendrians are isotopic in $S^*M$? Or in the complement of some other singular Legendrian $L$? Is there any sheafy way to do so?
2. If one view a constructible sheaf $F \in Sh(M, L^\infty)$ as a functor from $Open(M)$ to chain complexes, by $U \mapsto F(U)$, can one replace the category of open sets by some other category of test objects? Here the category of open sets are viewed as a full subcategory of $Sh(M)$ with objects of costandard objects on these open sets. I am interested in the vague question as ‘How many test objects do you need to fully know a constructible sheaf’.