Average Betti numbers of induced subcomplexes in triangulations of manifolds

We study a variation of Bagchi and Datta’s σ-vector of a simplicial complex C, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of C. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations o...

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Authors: Codenotti, Giulia, Spreer, Jonathan, Santos, Francisco|||0000-0003-2120-9068
Format: article
Publication Date:2020
Country:España
Institution:Universidad de Cantabria (UC)
Repository:UCrea Repositorio Abierto de la Universidad de Cantabria
Language:English
OAI Identifier:oai:repositorio.unican.es:10902/20620
Online Access:http://hdl.handle.net/10902/20620
Access Level:Open access
Keyword:Triangulations of manifolds
σ-vector
µ-vector
τ -vector
Graded Betti numbers
Stacked and neighborly spheres
Billera-Lee polytopes
Simplicial complexes
Perfect elimination order
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spelling Average Betti numbers of induced subcomplexes in triangulations of manifoldsCodenotti, GiuliaSpreer, JonathanSantos, Francisco|||0000-0003-2120-9068Triangulations of manifoldsσ-vectorµ-vectorτ -vectorGraded Betti numbersStacked and neighborly spheresBillera-Lee polytopesSimplicial complexesPerfect elimination orderWe study a variation of Bagchi and Datta’s σ-vector of a simplicial complex C, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of C. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of C. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given f-vector. For the first entry of σ, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on σ can be used to obtain lower bounds on the f-vector of triangulated 4-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.Santos is also supported by grants MTM2014-54207-P and MTM2017-83750-P of the Spanish Ministry of Science.Electronic Journal of CombinatoricsUniversidad de Cantabria20202020-08-21journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articlehttp://hdl.handle.net/10902/20620The electronic journal of combinatorics 27(3) (2020)reponame:UCrea Repositorio Abierto de la Universidad de Cantabriainstname:Universidad de Cantabria (UC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nd/4.0/info:eu-repo/semantics/openAccessoai:repositorio.unican.es:10902/206202026-06-02T12:39:31Z
dc.title.none.fl_str_mv Average Betti numbers of induced subcomplexes in triangulations of manifolds
title Average Betti numbers of induced subcomplexes in triangulations of manifolds
spellingShingle Average Betti numbers of induced subcomplexes in triangulations of manifolds
Codenotti, Giulia
Triangulations of manifolds
σ-vector
µ-vector
τ -vector
Graded Betti numbers
Stacked and neighborly spheres
Billera-Lee polytopes
Simplicial complexes
Perfect elimination order
title_short Average Betti numbers of induced subcomplexes in triangulations of manifolds
title_full Average Betti numbers of induced subcomplexes in triangulations of manifolds
title_fullStr Average Betti numbers of induced subcomplexes in triangulations of manifolds
title_full_unstemmed Average Betti numbers of induced subcomplexes in triangulations of manifolds
title_sort Average Betti numbers of induced subcomplexes in triangulations of manifolds
dc.creator.none.fl_str_mv Codenotti, Giulia
Spreer, Jonathan
Santos, Francisco|||0000-0003-2120-9068
author Codenotti, Giulia
author_facet Codenotti, Giulia
Spreer, Jonathan
Santos, Francisco|||0000-0003-2120-9068
author_role author
author2 Spreer, Jonathan
Santos, Francisco|||0000-0003-2120-9068
author2_role author
author
dc.contributor.none.fl_str_mv Universidad de Cantabria
dc.subject.none.fl_str_mv Triangulations of manifolds
σ-vector
µ-vector
τ -vector
Graded Betti numbers
Stacked and neighborly spheres
Billera-Lee polytopes
Simplicial complexes
Perfect elimination order
topic Triangulations of manifolds
σ-vector
µ-vector
τ -vector
Graded Betti numbers
Stacked and neighborly spheres
Billera-Lee polytopes
Simplicial complexes
Perfect elimination order
description We study a variation of Bagchi and Datta’s σ-vector of a simplicial complex C, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of C. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of C. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given f-vector. For the first entry of σ, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on σ can be used to obtain lower bounds on the f-vector of triangulated 4-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.
publishDate 2020
dc.date.none.fl_str_mv 2020
2020-08-21
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10902/20620
url http://hdl.handle.net/10902/20620
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nd/4.0/
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Electronic Journal of Combinatorics
publisher.none.fl_str_mv Electronic Journal of Combinatorics
dc.source.none.fl_str_mv The electronic journal of combinatorics 27(3) (2020)
reponame:UCrea Repositorio Abierto de la Universidad de Cantabria
instname:Universidad de Cantabria (UC)
instname_str Universidad de Cantabria (UC)
reponame_str UCrea Repositorio Abierto de la Universidad de Cantabria
collection UCrea Repositorio Abierto de la Universidad de Cantabria
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869404183453499392
score 15.300719

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