Page 251 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 251
APPLIED COMBINATORIAL AND GEOMETRIC TOPOLOGY (MS-34)
greater dimensions, with motivations from combinatorics, topology, mathematical physics and
quantum gravity. However, it has been a challenge due to the intricate nature of combinatorics
and topology in dimensions 3 and greater. Here I will present results based on the model of
colored triangulations for PL-manifolds. They admit a representation as edge-colored graphs
which is amenable to combinatorial analysis. I will focus on a combinatorial generalization
of Euler’s relation for combinatorial maps, which is to bound the number of (d − 2)-simplices
linearly in the number of d-simplices, and identifying the triangulations which maximize the
bound for different colored building blocks. In 3d, I prove that those triangulations, for any
set of colored building blocks homeomorphic to the 3-ball, are in bijection with trees. In even
dimensions, we have proved that several universality classes can be achieved. Both those results
are in sharp contrast with the case of surfaces.
Kirby diagrams, edge-colored graphs and trisections of PL 4-manifolds
Maria Rita Casali, casali@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthor: Paola Cristofori
It is well-known that any framed link (L, c) uniquely represents the 3-manifold M 3(L, c) ob-
tained from S3 by Dehn surgery along (L, c), as well as the PL 4-manifold M 4(L, c) obtained
from D4 by adding 2-handles along (L, c). Moreover, if trivial dotted components are also al-
lowed (i.e. in case of a Kirby diagram (L(∗), c)), the associated PL 4-manifold M 4(L(∗), c) is
obtained from D4 by adding 1-handles along the dotted components and 2-handles along the
framed components.
In the present talk we present the relationship between framed links and/or Kirby diagrams
and the so called crystallization theory, which represents compact PL manifolds of arbitrary
dimension by regular edge-colored graphs: in particular, we describe how to construct a 5-
colored graph representing M 4(L(∗), c), directly “drawn over” a planar diagram of (L(∗), c)).
As a consequence, the combinatorial properties of Kirby diagrams yield upper bounds for
both the graph-defined invariants gem-complexity and generalized regular genus of the associ-
ated 4-manifold.
Further, the described relationship turns out to be strictly related to the possibility of study-
ing trisections of 4-manifolds via edge-colored graphs, also in the extended case of compact PL
4-manifolds with connected boundary.
Complexity of graph manifolds
Alessia Cattabriga, alessia.cattabriga@unibo.it
University of Bologna, Italy
Coauthor: Michele Mulazzani
The notion of complexity for compact 3-dimensional manifolds, was introduced by Matveev in
the nineties as a way to measure how complicated a manifold is; indeed, for closed orientable
irreducible manifolds, the complexity coincides with the minimum number of tetrahedra needed
to construct the manifold, with the only exceptions of the 3-sphere, the projective space and the
lens space L(3, 1) all having complexity zero. There exists a census of manifolds according
to increasing complexity that, for the orientable case, is Recognizer catalogue and includes all
249
greater dimensions, with motivations from combinatorics, topology, mathematical physics and
quantum gravity. However, it has been a challenge due to the intricate nature of combinatorics
and topology in dimensions 3 and greater. Here I will present results based on the model of
colored triangulations for PL-manifolds. They admit a representation as edge-colored graphs
which is amenable to combinatorial analysis. I will focus on a combinatorial generalization
of Euler’s relation for combinatorial maps, which is to bound the number of (d − 2)-simplices
linearly in the number of d-simplices, and identifying the triangulations which maximize the
bound for different colored building blocks. In 3d, I prove that those triangulations, for any
set of colored building blocks homeomorphic to the 3-ball, are in bijection with trees. In even
dimensions, we have proved that several universality classes can be achieved. Both those results
are in sharp contrast with the case of surfaces.
Kirby diagrams, edge-colored graphs and trisections of PL 4-manifolds
Maria Rita Casali, casali@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthor: Paola Cristofori
It is well-known that any framed link (L, c) uniquely represents the 3-manifold M 3(L, c) ob-
tained from S3 by Dehn surgery along (L, c), as well as the PL 4-manifold M 4(L, c) obtained
from D4 by adding 2-handles along (L, c). Moreover, if trivial dotted components are also al-
lowed (i.e. in case of a Kirby diagram (L(∗), c)), the associated PL 4-manifold M 4(L(∗), c) is
obtained from D4 by adding 1-handles along the dotted components and 2-handles along the
framed components.
In the present talk we present the relationship between framed links and/or Kirby diagrams
and the so called crystallization theory, which represents compact PL manifolds of arbitrary
dimension by regular edge-colored graphs: in particular, we describe how to construct a 5-
colored graph representing M 4(L(∗), c), directly “drawn over” a planar diagram of (L(∗), c)).
As a consequence, the combinatorial properties of Kirby diagrams yield upper bounds for
both the graph-defined invariants gem-complexity and generalized regular genus of the associ-
ated 4-manifold.
Further, the described relationship turns out to be strictly related to the possibility of study-
ing trisections of 4-manifolds via edge-colored graphs, also in the extended case of compact PL
4-manifolds with connected boundary.
Complexity of graph manifolds
Alessia Cattabriga, alessia.cattabriga@unibo.it
University of Bologna, Italy
Coauthor: Michele Mulazzani
The notion of complexity for compact 3-dimensional manifolds, was introduced by Matveev in
the nineties as a way to measure how complicated a manifold is; indeed, for closed orientable
irreducible manifolds, the complexity coincides with the minimum number of tetrahedra needed
to construct the manifold, with the only exceptions of the 3-sphere, the projective space and the
lens space L(3, 1) all having complexity zero. There exists a census of manifolds according
to increasing complexity that, for the orientable case, is Recognizer catalogue and includes all
249
