MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops based at a point in a contact manifold? Can that be made into a "Legendrian fundamental group" somehow?

I've heard that h-principles are somehow involved, but I'm not sure what the punchline is.

share|cite|improve this question
up vote 4 down vote accepted

In general, the (parametric) h-principle for Legendrian immersions implies that Legendrian immersions f:L->(M,\xi) are classified up to homotopy (through Legendrian immersions) by the following bundle-theoretic invariant: Choosing a compatible almost complex structure on \xi allows one to complexify the differential of f to an isomorphism d_C f: TL\otimes C -> f*\xi, and the relevant invariant is the homotopy class of this isomorphism of complex vector bundles (of course this is independent of the almost complex structure since the space of compatible almost complex structures is contractible).

The above holds in any contact manifold (M,\xi) of arbitrary dimension. Of course when M is 3-dimensional and L is S^1, f^*\xi is the unique complex line bundle over S^1, automorphisms of which are parametrized up to homotopy by pi_1(U(1))=Z. So (given that the h-principle also implies that any loop in a 3-manifold is homotopic to a Legendrian loop) it appears to always be the case that the "Legendrian fundamental group" surjects onto the standard fundamental group, with kernel Z.

When M=R^3 this invariant is equivalent to the rotation number that Steven mentioned. There's a proof of the relevant h-principle in the book by Eliashberg and Mishachev. The above discussion is partly based on Section 3.3 of arXiv:0210124 by Ekholm-Etnyre-Sullivan.

share|cite|improve this answer

I don't have a general answer, but for the standard tight contact structure \xi on R^3, see "A contact geometric proof of the Whitney-Graustein Theorem" by Geiges (arXiv:0801.0046). Proposition 4 says that regular Legendrian curves in (R^3, \xi) are classified up to homotopy through Legendrian curves by their rotation number, so I guess the "Legendrian fundamental group" should be Z in this case. The proof is less than a page long and very straightforward.

share|cite|improve this answer
Thanks, this paper looks very nice! – j.c. Oct 17 '09 at 23:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.