SGGTC this Friday: Daniel Álvarez-Gavela

[Oleg Lazarev]

Dear all, 

Daniel Álvarez-Gavela from Princeton/IAS will give a talk this Friday at 1:00 in Math 417.  We will meet at 11:30 in the lobby before his talk and go out for lunch.  

Best, 

Oleg

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Daniel Álvarez-Gavela: K-theoretic invariants for Legendrians via parametrized Morse theory 

Abstract: Joint work with K. Igusa. It was remarked by Eliashberg and Gromov that any Legendrian link in the standard contact R^3 which is generated by a 1 parametric family of functions corresponding to a pseudo-isotopy with nontrivial Wh_2 invariant is not Legendrian isotopic to an unlink of standard Legendrian unknots. Similarly, one can show that any Legendrian link in R^5 which is generated by a 2 parametric family of functions corresponding to a loop of pseudo-isotopies with nontrivial Wh_3 invariant is not Legendrian isotopic to an unlink of standard Legendrian unknots. In a similar spirit, we study a family of Legendrian submanifolds in the 1-jet space of any oriented surface which carry nontrivial and distinct K_3 invariants. In our setting the product K_1 x K_2 \to K_3 together with a standardization of the K_2 factor allows us to understand these examples more simply in terms of a K_1 type invariant. We call this invariant the Legendrian Turaev torsion. As an application, we show that in the 1-jet space of any oriented surface there exist pairs of Legendrian links which (a) are formally isotopic (b) cannot be distinguished by any natural Legendrian invariant (c) yet are not Legendrian isotopic. These examples appeared in a different guise in work of K. Igusa and J. Klein on the higher Reidemeister torsion of circle bundles, where in particular an explicit picture for the exotic element of K_3(Z)=Z/48 was given.