Geodesic
Average Betti numbers of induced subcomplexes in triangulations of manifolds
Giulia Codenotti1 ; Jonathan Spreer ; Francisco Santos
1Free University of Berlin
The electronic journal of combinatorics, Tome 27 (2020) no. 3
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We study a variation of Bagchi and Datta's $\sigma$-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 $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ 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.
DOI : 10.37236/8564
Classification : 05E45, 13F55, 57M15
Mots-clés : Bagchi and Datta's \(\sigma \)-vector, Betti numbers

Giulia Codenotti  1   ; Jonathan Spreer    ; Francisco Santos 

1 Free University of Berlin 
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     title = {Average {Betti} numbers of induced subcomplexes in triangulations of manifolds},
     journal = {The electronic journal of combinatorics},
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Giulia Codenotti; Jonathan Spreer; Francisco Santos. Average Betti numbers of induced subcomplexes in triangulations of manifolds. The electronic journal of combinatorics, Tome 27 (2020) no. 3. doi: 10.37236/8564

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