Gem-induced trisections of compact PL 4-manifolds
-
Maria Rita Casali
and
Paola Cristofori
Abstract
The idea of studying trisections of closed smooth 4-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 standard simply-connected 4-manifolds, via the so-called simple crystallizations. In the present paper we propose generalizations of these ideas by taking into consideration a possible extension of trisections to compact PL 4-manifolds with connected boundary, which is related to Birman’s special Heegaard sewing, and by analyzing gem-induced trisections, i.e. trisections that can be induced not only by simple crystallizations, but also by any 5-colored graph encoding a PL 4-manifold with empty or connected boundary. This last notion gives rise to that of G-trisection genus, as an analogue, in this context, of the well-known trisection genus. We give conditions on a 5-colored graph ensuring one of its gem-induced trisections – if any – to realize the G-trisection genus, and prove how to determine it directly from the graph. As a consequence, we detect a class of closed simply-connected 4-manifolds, comprehending all standard ones, for which both G-trisection genus and trisection genus coincide with the second Betti number and also with half the value of the graph-defined PL invariant regular genus. Moreover, the existence of gem-induced trisections and an estimation of the G-trisection genus via surgery description is obtained, for each compact PL 4-manifold admitting a handle decomposition lacking in 3-handles.
Funding statement: This work was supported by GNSAGA of INDAM and by the University of Modena and Reggio Emilia, project: “Discrete Methods in Combinatorial Geometry and Geometric Topology”.
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Articles in the same Issue
- Frontmatter
- On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem
- Non-commutative Khinchine-type inequality for dependent random variables and overview of its applications in data science
- Connections and genuinely ramified maps of curves
- Caustics of pseudo-spherical surfaces in the Euclidean 3-space
- On liftings of modular forms and Weil representations
- Approximation via statistical measurable convergence with respect to power series for double sequences
- Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems
- Gem-induced trisections of compact PL 4-manifolds
- Categories of modules, comodules and contramodules over representations
- On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions
- On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
- Congruence subgroups and crystallographic quotients of small Coxeter groups
- Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2
- Hardy and BMO spaces on Weyl chambers
- De Branges–Rovnyak spaces and local Dirichlet spaces of higher order
Articles in the same Issue
- Frontmatter
- On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem
- Non-commutative Khinchine-type inequality for dependent random variables and overview of its applications in data science
- Connections and genuinely ramified maps of curves
- Caustics of pseudo-spherical surfaces in the Euclidean 3-space
- On liftings of modular forms and Weil representations
- Approximation via statistical measurable convergence with respect to power series for double sequences
- Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems
- Gem-induced trisections of compact PL 4-manifolds
- Categories of modules, comodules and contramodules over representations
- On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions
- On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
- Congruence subgroups and crystallographic quotients of small Coxeter groups
- Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2
- Hardy and BMO spaces on Weyl chambers
- De Branges–Rovnyak spaces and local Dirichlet spaces of higher order
