Volume 18, issue 3 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
Generating families and augmentations for Legendrian surfaces

Dan Rutherford and Michael G Sullivan

Algebraic & Geometric Topology 18 (2018) 1675–1731

We study augmentations of a Legendrian surface L in the 1–jet space, J1 M, of a surface M. We introduce two types of algebraic/combinatorial structures related to the front projection of L that we call chain homotopy diagrams (CHDs) and Morse complex 2–families (MC2Fs), and show that the existence of a ρ–graded CHD or a ρ–graded MC2F is equivalent to the existence of a ρ–graded augmentation of the Legendrian contact homology DGA to 2. A CHD is an assignment of chain complexes, chain maps, and homotopy operators to the 0–, 1–, and 2–cells of a compatible polygonal decomposition of the base projection of L with restrictions arising from the front projection of L. An MC2F consists of a collection of formal handleslide sets and chain complexes, subject to axioms based on the behavior of Morse complexes in 2–parameter families. We prove that if a Legendrian surface has a tame-at-infinity generating family, then it has a 0–graded MC2F and hence a 0–graded augmentation. In addition, continuation maps and a monodromy representation of π1(M) are associated to augmentations, and then used to provide more refined obstructions to the existence of generating families that (i) are linear at infinity or (ii) have trivial bundle domain. We apply our methods in several examples.

Legendrian surfaces, augmentations, generating families
Mathematical Subject Classification 2010
Primary: 53D42
Received: 5 April 2017
Revised: 17 December 2017
Accepted: 10 January 2018
Published: 3 April 2018
Dan Rutherford
Department of Mathematical Sciences
Ball State University
Muncie, IN
United States
Michael G Sullivan
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States