Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\Delta$ be a simplical complex.

Call $\Delta$ pure if all the maximal faces have the same dimension.

Call $\Delta$ Eulerian if it is pure and $\chi (lk (F))= \chi (S^{dim (lk(F))})$ for any $F \in \Delta$.

Question 1: From what I understand characterizing the $f$-vectors of pure simplicial complexes is hopeless at the moment. What if however I give you an $f$ (or $h$)-vector? What are some necessary conditions that I can check for it to be the $f$-vector of a pure simplicial complex.

Say I give you something like this: $(1,19,99,276, 504, 630, 546, 324, 126, 28)$. One can check the Kruskal-Katona bounds (which hold in this case) and even produce the shifted complex, however this is not generally pure even if the $f$-vector is pure. Are there any similar constructions of these type?

Questions 2: What can one say about the $h$-vectors of Eulerian complexes other that they are symmetric?

Thank you!

share|improve this question
1  
I don't understand your example $f$-vector. Are there $19$ edges and only one vertex? –  Vidit Nanda Sep 25 '13 at 1:58
    
The first entry is usually denoted by $f_{-1}$ and is always $1$ since it counts the empty set in the simplicial complex. So in this case the complex would have $19$ vertices and $99$ edges. –  Alexandru Papiu Sep 25 '13 at 4:58

2 Answers 2

A related question is to obtain information on f-vectors of pure multicomplexes (or order ideals of monomials). Any restriction on them would also be a restriction for simplicial complexes. For information on pure multicomplexes see http://atlas.mat.ub.edu/RGiAG/memo647.pdf. Chapter 8 of this monograph is on simplicial complexes. There it is explained that a complete characterization is probably hopeless since it would include a characterization of the orders of finite projective planes.

share|improve this answer

As Gil remarks in his comment, Corollary 1 of the paper which I mentioned does not in fact imply the upper bound conjecture except when one additionally assumes isolated singularities. Still, I hope that this paper and its references may be of some use to the OP.


At least partial answers to both your questions may be found in the following paper:

Patricia Hersh and Isabella Novik, A short simplicial $h$-vector and the upper bound theorem, Discrete & Computational Geometry, 28 (3): 283-289 (2002).

In particular, the main theorem implies (via Corollary 1) that if the dimension $d$ is odd, then the first $d-1$ entries of your $f$-vector will be bounded above by those of the cyclic polytope $C(n,d)$. For additional details, look for the "upper bound conjecture" as well as the references in this paper.

I don't think much is known when $d$ is even.

share|improve this answer
    
Vidit, the paper by Hersh and Novik deals with Eulerian complexes with isolated singularities, which is a very special case. –  Gil Kalai Sep 27 '13 at 3:20
    
@GilKalai Thank you for noticing that I had overestimated the consequences of that result in Hersh-Novik. I will modify my answer. –  Vidit Nanda Sep 27 '13 at 4:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.