32 research outputs found
The Trisection Genus of Standard Simply Connected PL 4-Manifolds
Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. In this note we show that the K3 surface has trisection genus 22. This implies that the trisection genus of all standard simply connected PL 4-manifolds is known. We show that the trisection genus of each of these manifolds is realised by a trisection that is supported by a singular triangulation. Moreover, we explicitly give the building blocks to construct these triangulations
Gem-induced trisections of compact PL -manifolds
The idea of studying trisections of closed smooth -manifolds via
(singular) triangulations, endowed with a suitable vertex-labelling by three
colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by
Spreer and Tillmann to colored triangulations associated to the so called
simple crystallizations of standard simply-connected -manifolds. The present
paper performs a generalization of these ideas along two different directions:
first, we take in consideration also compact PL -manifolds with connected
boundary, introducing a possible extension of trisections to the boundary case;
then, we analyze the trisections induced not only by simple crystallizations,
but by any 5-colored graph encoding a simply-connected -manifold. This
extended notion is referred to as gem-induced trisection, and gives rise to the
G-trisection genus, generalizing the well-known trisection genus. Both in the
closed and boundary case, we give conditions on a 5-colored graph which ensure
one of its gem-induced trisections - if any - to realize the G-trisection
genus, and prove how to determine it directly from the graph itself. Moreover,
the existence of gem-induced trisections and an estimation of the G-trisection
genus via surgery description is obtained, for each compact simply-connected PL
4-manifold admitting a handle decomposition lacking in 1-handles and 3-handles.
As a consequence, we prove that the G-trisection genus equals for all
-bundles of , and hence it is not finite-to-one.Comment: 25 pages, 14 figures. Updated to most recent versio
Bridge trisections in rational surfaces
We study smooth isotopy classes of complex curves in complex surfaces from
the perspective of the theory of bridge trisections, with a special focus on
curves in and . We are
especially interested in bridge trisections and trisections that are as simple
as possible, which we call "efficient". We show that any curve in
or admits an efficient
bridge trisection. Because bridge trisections and trisections are nicely
related via branched covering operations, we are able to give many examples of
complex surfaces that admit efficient trisections. Among these are
hypersurfaces in , the elliptic surfaces , the Horikawa
surfaces , and complete intersections of hypersurfaces in
. As a corollary, we observe that, in many cases, manifolds that
are homeomorphic but not diffeomorphic have the same trisection genus, which is
consistent with the conjecture that trisection genus is additive under
connected sum. We give many trisection diagrams to illustrate our examples.Comment: 46 pages, 28 color figure
Computing Matveev's complexity via crystallization theory: the boundary case
The notion of Gem-Matveev complexity has been introduced within
crystallization theory, as a combinatorial method to estimate Matveev's
complexity of closed 3-manifolds; it yielded upper bounds for interesting
classes of such manifolds. In this paper we extend the definition to the case
of non-empty boundary and prove that for each compact irreducible and
boundary-irreducible 3-manifold it coincides with the modified Heegaard
complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via
Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all
Seifert 3-manifolds with base and two exceptional fibers and,
therefore, for all torus knot complements.Comment: 27 pages, 14 figure