Page 389 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 389
TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
is called triangulation. The name comes from the fact that when space X is a two-dimensional
surface, triangulation means representing it as the union of adjacent triangles with edges meet-
ing at the vertices. In the case of higher dimensions, the basic cells are the ”i” - dimensional
simplices. One of the natural questions is to find a triangulation with the minimum number of
vertices, respectively of all simplexes (or estimate these numbers). This lecture will be devoted
to this problem. We will present a new method based on the notion of covering type estimating
from below the number of vertices by the weighted length of the elements in the cohomology
ring H∗(X), or the weighted Lusternik-Schirelmann category theory. As a consequence, we
got not only a unified method of proof of estimates of the number of vertices of the minimal
triangulations derived originally by ad hock combinatorial methods, but also sharper estimates,
or estimates for the families of manifolds not studied earlier.
References
[1] Duan, W. Marzantowicz, and X. Zhao, On the number of simplices required to
triangulate a Lie group, Topology & Applications, Volume 293, 15 April 2021,
https://doi.org/10.1016/j.topol.2020.107559
[2] D. Govc, W. Marzantowicz and P. Pavešic´, Estimates of covering type and the number of
vertices of minimal triangulations, Discr. Comp. Geom. (2019), 63 (2020), no. 1, 31–48.
[3] D. Govc, W. Marzantowicz and P. Pavešic´, How many simplices are needed to triangulate
a Grassmannian?, TMNA, Volume 56, No. 2, 2020, 501- 518.
[4] M. Karoubi, C. Weibel, On the covering type of a space, L’Enseignement Math. 62 (2016),
457–474.
On the set of periods for the Morse-Smale diffeomorphisms
Adrian Myszkowski, adrian.myszkowski@pg.edu.pl
Gdansk University of Technology, Poland
Coauthors: Grzegorz Graff, Małgorzata Lebiedz´
We apply the representation of Lefschetz numbers of iterates in the form of so-called periodic
expansion to determine the minimal set of Lefschetz periods MPerL(f ). Applying this approach
we present an algorithmic method for finding minimal Lefschetz periods for Morse-Smale dif-
feomorphisms on Mg and Ng, respectively an orientable and non-orientable compact surface
without boundary of genus g. Llibre and Sirvent calculated MPerL(f ) for several genuses
g 9 of Mg and Ng. Our approach makes it possible to find easily MPerL(f ) for much bigger
values of g.
Research supported by the National Science Centre, Poland within the grant Sheng 1 UMO-
2018/30/Q/ST1/00228.
387
is called triangulation. The name comes from the fact that when space X is a two-dimensional
surface, triangulation means representing it as the union of adjacent triangles with edges meet-
ing at the vertices. In the case of higher dimensions, the basic cells are the ”i” - dimensional
simplices. One of the natural questions is to find a triangulation with the minimum number of
vertices, respectively of all simplexes (or estimate these numbers). This lecture will be devoted
to this problem. We will present a new method based on the notion of covering type estimating
from below the number of vertices by the weighted length of the elements in the cohomology
ring H∗(X), or the weighted Lusternik-Schirelmann category theory. As a consequence, we
got not only a unified method of proof of estimates of the number of vertices of the minimal
triangulations derived originally by ad hock combinatorial methods, but also sharper estimates,
or estimates for the families of manifolds not studied earlier.
References
[1] Duan, W. Marzantowicz, and X. Zhao, On the number of simplices required to
triangulate a Lie group, Topology & Applications, Volume 293, 15 April 2021,
https://doi.org/10.1016/j.topol.2020.107559
[2] D. Govc, W. Marzantowicz and P. Pavešic´, Estimates of covering type and the number of
vertices of minimal triangulations, Discr. Comp. Geom. (2019), 63 (2020), no. 1, 31–48.
[3] D. Govc, W. Marzantowicz and P. Pavešic´, How many simplices are needed to triangulate
a Grassmannian?, TMNA, Volume 56, No. 2, 2020, 501- 518.
[4] M. Karoubi, C. Weibel, On the covering type of a space, L’Enseignement Math. 62 (2016),
457–474.
On the set of periods for the Morse-Smale diffeomorphisms
Adrian Myszkowski, adrian.myszkowski@pg.edu.pl
Gdansk University of Technology, Poland
Coauthors: Grzegorz Graff, Małgorzata Lebiedz´
We apply the representation of Lefschetz numbers of iterates in the form of so-called periodic
expansion to determine the minimal set of Lefschetz periods MPerL(f ). Applying this approach
we present an algorithmic method for finding minimal Lefschetz periods for Morse-Smale dif-
feomorphisms on Mg and Ng, respectively an orientable and non-orientable compact surface
without boundary of genus g. Llibre and Sirvent calculated MPerL(f ) for several genuses
g 9 of Mg and Ng. Our approach makes it possible to find easily MPerL(f ) for much bigger
values of g.
Research supported by the National Science Centre, Poland within the grant Sheng 1 UMO-
2018/30/Q/ST1/00228.
387
