Homology of balanced complexes via the Fourier transform

Let G_0,...,G_k be finite abelian groups and let G_0*...*G_k be the join of the 0-dimensional complexes G_i. We give a characterization of the integral k-coboundaries of subcomplexes of G_0*...*G_k in terms of the Fourier transform on the group G_0 \times ... \times G_k. This leads to an extension of a recent result of Musiker and Reiner on a topological interpretation of the cyclotomic polynomial.Comment: 8 page

Bounded quotients of the fundamental group of a random 2-complex

Let D denote the (n-1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of D obtained by starting with the full 1-skeleton of D and then adding each 2-simplex independently with probability p. For a fixed c>0 it is shown that if p=\frac{(6+7c) \log n}{n} then a.a.s. the fundamental group \pi(Y) does not have a nontrivial quotient of order at most n^c.Comment: 7 page

Graph codes and local systems

It is shown that the good expander codes introduced by Sipser and Spielman, can be realized as the first homology of a graph with respect to a certain twisted coefficient system.Comment: 4 page

Sum Complexes and Uncertainty Numbers

Let p be a prime and let A be a subset of F_p. For k<p let X_{A,k} be the (k-1)-dimensional complex on the vertex set F_p with a full (k-2)-skeleton whose (k-1)-faces are k-subsets S of F_p such that the sum of the elements of S belongs to A. The homology groups of X_{A,k} with field coefficients are determined. In particular it is shown that if |A| \leq k then H_{k-1}(X_{A,k};F_p)=0. This implies a homological characterization of uncertainty numbers of subsets of F_p.Comment: 16 page

An uncertainty inequality for finite abelian groups

Let G be a finite abelian group of order n. For a complex valued function f on G, let \fht denote the Fourier transform of f. The uncertainty inequality asserts that if f \neq 0 then |supp(f)| |supp(\fht)| \geq n. Answering a question of Terence Tao, the following improvement of the classical inequality is shown: Let d_1<d_2 be two consecutive divisors of n. If d_1 \leq k=|supp(f)| \leq d_2 then: |supp(\fht)| \geq \frac{n(d_1+d_2-k)}{d_1 d_2}Comment: 7 pages, no figure

Betti numbers of complexes with highly Connected links

Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes show that this bound is asymptotically sharp for all fixed j<k.Comment: 13 pages, 3 figure

Expansion of Building-Like Complexes

Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n \geq 1$, there exists a constant $\epsilon(n)>0$ such that for any $0 \leq k <n$ the $k$-th coboundary expansion constant of any $n$-dimensional spherical building is at least $\epsilon(n)$.Comment: 18 page

Maximal rank in matrix spaces via graph matchings

We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in terms of their maximal rank.Comment: 10 page

Homology of spaces of directed paths in Euclidean pattern spaces

The paper addresses certain topological aspects of Dijkstra's PV-model for parallel computations in concurrency theory. The main result is a computation of the homology of the trace space associated with PV-programs in which access and release of every resource happen without time delay.Comment: 21 pages, 4 figures. To be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springe

Sum complexes - a new family of hypertrees

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and \binom{n-1}{k} facets such that H_k(X;Q)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group Z_n. The sum complex X_A is the pure k-dimensional complex on the vertex set Z_n whose facets are subsets \sigma of Z_n such that |\sigma|=k+1 and \sum_{x \in \sigma}x \in A. It is shown that if n is prime then the complex X_A is a k-hypertree for every choice of A. On the other hand, for n prime X_A is k-collapsible iff A is an arithmetic progression in Z_n.Comment: 18 pages, 3 figure