# Parameterized Morse Theory in Low-Dimensional and Symplectic Topology (14w5119)

Arriving in Banff, Alberta Sunday, March 23 and departing Friday March 28, 2014

## Organizers

(University of Georgia)

(University of Massachusetts)

## Objectives

The goal of this workshop is to bring together researchers from different fields in low-dimensional and symplectic topology, who might not communicate frequently with each other due to the nature of their problems, but who are using potentially similar tools in parameterized Morse theory. Some such fields have been described above in the {\em short overview} section.To involve younger researchers in these goals, we plan to invite graduate students and new postdocs, in addition to the list of more senior participants given below.When people working in one field gain a clearer picture of how parameterized Morse theory is being used in other fields, they will come away with new techniques that they can use in their own fields as well as new ideas for how their own techniques may contribute to the other fields. In addition, there are possible foundational problems in parameterized Morse theory the solutions of which are critical to all of the programs mentioned above. We anticipate participants from diverse backgrounds collaborating intensively to work toward such solutions. One concrete objective, therefore, is to foster new inter-disciplinary collaborations which should lead to significant scientific results and publications. Below are some examples of how such interactions could play out.Recently, Henry and Rutherford construct a differential graded algebra using handle-slides from the Cerf theory of the $N$-functions arising in Legendrian generating families and speculate that it recovers the full Chekanov-Eliashberg DGA \cite{HR}. This DGA has proven to be useful, computable and an intrinsically interesting invariant, as it is a combinatorial reformulation of the Level 1" relative symplectic field theory defined by Eliashberg, Hofer and Givental using pseudo-holomoprhic curves \cite{EGH}. The boundary map in the Henry-Rutherford DGA counts disks that appear to be swept out by gradient flow lines in a $1$--parameter family of Morse functions; this idea could transfer to the setting of Morse $2$--functions on $4$--manifolds, in which a DGA is generated by crossings and handle-slides in the critical value graphic, and the boundary map counts compact surfaces swept out by families of gradient flow lines, appropriately defined. This might have the potential to produce invariants of homology (homotopy?) $4$--spheres, in which case gauge theory fails, and ideally these invariants would clearly measure obstructions to constructing a diffeomorphism with $S^4$. This could be a step towards resolving one of the most significant open problems in topology today, the smooth $4$--dimensional Poincar\'{e} conjecture.Gay and Kirby \cite{GKMorse2F} develop techniques to eliminate certain classes of singularities in Morse $2$--functions and homotopies between Morse $2$--functions, thus simplifying the collection of building blocks and moves needed when decomposing manifolds using Morse $2$--functions. The results are general but the techniques are, in some sense, even more general and could be useful in the $n$--category and TQFT context when, for example, finding generators and relations for certain topological $n$--categories.Sabloff and Traynor have proved Poincar\'e duality for generating homology, and also proved that Lagrandian cobordisms induce a TQFT-like'' structure on the theory, conjecturally equivalent to the symplectic field theory version \cite{SJ2}. Thus there is an important project, to prove this conjectural equivalence, as well as significant connections with the general theory of TQFT's.In another direction, a $k$-family of isotopic Legendrian submanifolds produces a bi-indexed $(k,N)$-parameter family of functions. Sabloff and Sullivan use these $k$-families of generating homologies to say things about the higher homotopy groups of the space of Legendrian submanifolds \cite{SS}. They apply Hutchings' framework of families of Morse functions \cite{Hu}, which in principle, applies to any type of Floer theory modeled on Morse theory. The framework uses continuation maps; however, it can also be defined using the explicit singularities of Cerf theory. A natural conjecture is that these two methods produce the same set of invariants. Both \cite{Hu} and \cite{SS} are framed just for $k$-family isotopies. In Morse theory examples, such as generating families, these ideas should naturally extend to the TQFT set-up of \cite{SP}.Given a smooth $4$--manifold $X$, consider the minimum, over all generic homotopies $f_t : X \to \mathbb{R}$ between Morse functions $f_0$ and $f_1$ with the property that $f_0 = -f_1$, of the number of births and deaths of cancelling pairs of critical points in the homotopy. This is easy to define, obviously an invariant, and hard to compute. However, the techniques used by Johnson \cite{JJ} to bound the number of stabilizations needed to make two Heegaard splittings the same could generalize precisely to this setting, giving computable bounds and thus, potentially, new ways to distinguish homeomorphic but non-diffeomorphic $4$--manifolds. \begin{thebibliography}{99}\bibitem{DR} Fuchs, Dmitry; Rutherford, Dan. \newblock{Generating families and Legendrian contact homology in the standard contact space.} \newblock{{\em J. Topol.} 4 (2011), no. 1, 190-226.}\bibitem{EGH} Eliashberg, Y.; Givental, A.; Hofer, H. \newblock{Introduction to symplectic field theory.} \newblock{GAFA 2000 (Tel Aviv, 1999). {\em Geom. Funct. Anal.} 2000, Special Volume, Part II, 560--673.}\bibitem{GKMorse2F} Gay, David T.; Kirby, Robion. \newblock {Indefinite Morse 2-functions; broken fibrations and generalizations}. \newblock {arXiv:1102.0750}\bibitem{GKreconstruct} Gay, David T.; Kirby, Robion. \newblock {Reconstructing 4-manifolds from Morse 2-functions}. \newblock In {\em Proceedings of the {F}reedman {F}est, {G}eometry \& {T}opology {M}onographs (18) 2012}.\bibitem{HR} Henry, B. Michael; Rutherford, Dan. \newblock{A combinatorial DGA for Legendrian knots from generating families.} \newblock{arXiv:1106.3357}\bibitem{Hu} Hutchings, Michael. \newblock{Floer homology of families. I.} \newblock{{\em Algebr. Geom. Topol.} 8 (2008), no. 1, 435-492.}\bibitem{JJ} Johnson, Jesse. \newblock Bounding the stable genera of heegaard splittings from below. \newblock {\em J. Topol.}, 3:668--690, 2010. \bibitem{NT} Ng, Lenhard; Traynor, Lisa. \newblock{Legendrian solid-torus links.} \newblock{{\em J. Symplectic Geom.} 2 (2004), no. 3, 411--443.}\bibitem{RS} Rubinstein, Hyam; Scharlemann, Martin. \newblock Comparing {H}eegaard splittings of non-{H}aken 3-manifolds. \newblock {\em Topology}, 35:1005--1026, 1996.\bibitem{SJ1} Sabloff, Joshua M.; Traynor, Lisa. \newblock{Obstructions to the existence and squeezing of Lagrangian cobordisms.} \newblock{{\em J. Topol. Anal.} 2 (2010), no. 2, 203--232.}\bibitem{SJ2} Sabloff, Joshua M.; Traynor, Lisa. \newblock{Obstructions to Lagrangian Cobordisms between Legendrian Submanifolds.} \newblock{arXiv:1109.5660} \bibitem{SS} Sabloff, Joshua M.; Sullivan, Michael. \newblock{Families of Generating Families for Legendrian Submanifolds.} \newblock{In preparation.}\bibitem{SP} Schommer-Pries, Christopher J. \newblock {The Classification of Two-Dimensional Extended Topological Field Theories}. \newblock {arXiv:1112.1000}\bibitem{Vi} Viterbo, Claude. \newblock{Generating functions, symplectic geometry, and applications.} \newblock{{\em Proceedings of the International Congress of Mathematicians}, Vol. 1, 2 (Z\"{u}rich, 1994), 537--547, Birkh\"{a}user, Basel, 1995.}\end{thebibliography}