Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Average Betti numbers of induced subcomplexes in triangulations of
  manifolds   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.

8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th