9,711 research outputs found
Finite random coverings of one-complexes and the Euler characteristic
This article presents an algebraic topology perspective on the problem of
finding a complete coverage probability of a one dimensional domain by a
random covering, and develops techniques applicable to the problem beyond the
one dimensional case. In particular we obtain a general formula for the chance
that a collection of finitely many compact connected random sets placed on
has a union equal to . The result is derived under certain topological
assumptions on the shape of the covering sets (the covering ought to be {\em
good}, which holds if the diameter of the covering elements does not exceed a
certain size), but no a priori requirements on their distribution. An upper
bound for the coverage probability is also obtained as a consequence of the
concentration inequality. The techniques rely on a formulation of the coverage
criteria in terms of the Euler characteristic of the nerve complex associated
to the random covering.Comment: 25 pages,2 figures; final published versio
Enumerating Colorings, Tensions and Flows in Cell Complexes
We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex , generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in for
some ) or integral (with values in ). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series
Euler Characteristics of Categories and Homotopy Colimits
In a previous article, we introduced notions of finiteness obstruction, Euler
characteristic, and L^2-Euler characteristic for wide classes of categories. In
this sequel, we prove the compatibility of those notions with homotopy colimits
of I-indexed categories where I is any small category admitting a finite
I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit
Formula include formulas for products, homotopy pushouts, homotopy orbits, and
transport groupoids. We also apply our formulas to Haefliger complexes of
groups, which extend Bass--Serre graphs of groups to higher dimensions. In
particular, we obtain necessary conditions for developability of a finite
complex of groups from an action of a finite group on a finite category without
loops.Comment: 44 pages. This final version will appear in Documenta Mathematica.
Remark 8.23 has been improved, discussion of Grothendieck construction has
been slightly expanded at the beginning of Section 3, and a few other minor
improvements have been incoporate
Free resolutions via Gr\"obner bases
For associative algebras in many different categories, it is possible to
develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining
relations for an algebra of such a category provides a "monomial replacement"
of this algebra. The main goal of this article is to demonstrate how this
machinery can be used for the purposes of homological algebra. More precisely,
our approach goes in three steps. First, we define a combinatorial resolution
for the monomial replacement of an object. Second, we extract from those
resolutions explicit representatives for homological classes. Finally, we
explain how to "deform" the differential to handle the general case. For
associative algebras, we recover a well known construction due to Anick. The
other case we discuss in detail is that of operads, where we discover
resolutions that haven't been known previously. We present various
applications, including a proofs of Hoffbeck's PBW criterion, a proof of
Koszulness for a class of operads coming from commutative algebras, and a
homology computation for the operads of Batalin--Vilkovisky algebras and of
Rota--Baxter algebras.Comment: 34 pages, 4 figures. v2: added references to the work of
Drummond-Cole and Vallette. v3: added an explicit description of homology
classes in the monomial case and more examples, re-structured the exposition
to achieve more clarity. v4: changed the presentation of the main
construction to make it clearer, added another example (a computation of the
bar homology of Rota--Baxter algebras
Quillen homology for operads via Gr\"obner bases
The main goal of this paper is to present a way to compute Quillen homology
of operads. The key idea is to use the notion of a shuffle operad we introduced
earlier; this allows to compute, for a symmetric operad, the homology classes
and the shape of the differential in its minimal model, although does not give
an insight on the symmetric groups action on the homology. Our approach goes in
several steps. First, we regard our symmetric operad as a shuffle operad, which
allows to compute its Gr\"obner basis. Next, we define a combinatorial
resolution for the "monomial replacement" of each shuffle operad (provided by
the Gr\"obner bases theory). Finally, we explain how to "deform" the
differential to handle every operad with a Gr\"obner basis, and find explicit
representatives of Quillen homology classes for a large class of operads. We
also present various applications, including a new proof of Hoffbeck's PBW
criterion, a proof of Koszulness for a class of operads coming from commutative
algebras, and a homology computation for the operads of Batalin-Vilkovisky
algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint
arXiv:0912.4895. Final version, to appear in Documenta Mat
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