arXiv subjects

Jonathan D. Williams

Publications and source records attributed to Jonathan D. Williams.

4 recordsLinked to original sources

arXiv2019

Existence of 2-parameter crossings, with applications

A Morse 2-function is a generic smooth map from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move these arcs around in B by a homotopy and to give a library of examples when M is a closed 4-manifold. The last two sections give applications to the theory of crown diagrams of smooth 4-manifolds.

math.GT↗

arXiv2018

Uniqueness of surface diagrams of smooth 4-manifolds

In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Each surface diagram comes from a certain type of map from the 4-manifold to the two-sphere. The aim of this paper is to give a uniqueness theorem stating that surface diagrams coming from maps within a fixed homotopy class are unique up to four moves: stabilization, handleslide, multislide, and shift.

math.GT↗

arXiv2013

Holomorphic polygons and smooth 4-manifold invariants

Any smooth, closed oriented 4-manifold has a surface diagram of arbitrarily high genus g>2 that specifies it up to diffeomorphism. The goal of this paper is to prove the following statement: For any smooth, closed oriented 4-manifold M, there is a sequence of weak A-infinity algebras indexed by g, and the homotopy equivalence class of each entry of this sequence is a diffeomorphism invariant of M.

math.SG↗

arXiv2009

The h-principle for broken Lefschetz fibrations

It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken fibration in a given homotopy class of smooth maps. One notable application is that adding an additional "projection" move generates all broken fibrations, regardless of homotopy class. The paper ends with further applications and open problems.

math.GT↗