3,091 research outputs found
Bounding the homological finiteness length
We give a criterion for bounding the homological finiteness length of certain
HF-groups. This is used in two distinct contexts. Firstly, the homological
finiteness length of a non-uniform lattice on a locally finite n-dimensional
contractible CW-complex is less than n. In dimension two it solves a conjecture
of Farb, Hruska and Thomas. As another corollary, we obtain an upper bound for
the homological finiteness length of arithmetic groups over function fields.
This gives an easier proof of a result of Bux and Wortman that solved a
long-standing conjecture. Secondly, the criterion is applied to integer
polynomial points of simple groups over number fields, obtaining bounds
established in earlier works of Bux, Mohammadi and Wortman, as well as new
bounds. Moreover, this verifes a conjecture of Mohammadi and Wortman.Comment: Revised versio
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
The Betti poset in monomial resolutions
Let be a finite partially ordered set with unique minimal element
. We study the Betti poset of , created by deleting elements for which the open interval is acyclic. Using basic
simplicial topology, we demonstrate an isomorphism in homology between open
intervals of the form and corresponding open intervals
in the Betti poset. Our motivating application is that the Betti poset of a
monomial ideal's lcm-lattice encodes both its -graded Betti
numbers and the structure of its minimal free resolution. In the case of rigid
monomial ideals, we use the data of the Betti poset to explicitly construct the
minimal free resolution. Subsequently, we introduce the notion of rigid
deformation, a generalization of Bayer, Peeva, and Sturmfels' generic
deformation
Homological Pisot Substitutions and Exact Regularity
We consider one-dimensional substitution tiling spaces where the dilatation
(stretching factor) is a degree d Pisot number, and where the first rational
Cech cohomology is d-dimensional. We construct examples of such "homological
Pisot" substitutions that do not have pure discrete spectra. These examples are
not unimodular, and we conjecture that the coincidence rank must always divide
a power of the norm of the dilatation. To support this conjecture, we show that
homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in
which the number of occurrences of a patch for a return length is governed
strictly by the length. The ERP puts strong constraints on the measure of any
cylinder set in the corresponding tiling space.Comment: 16 pages, LaTeX, no figure
A categorification of non-crossing partitions
We present a categorification of the non-crossing partitions given by
crystallographic Coxeter groups. This involves a category of certain bilinear
lattices, which are essentially determined by a symmetrisable generalised
Cartan matrix together with a particular choice of a Coxeter element. Examples
arise from Grothendieck groups of hereditary artin algebras.Comment: 34 pages. Substantially revised and final version, accepted for
publication in Journal of the European Mathematical Societ
On the deformation chirality of real cubic fourfolds
According to our previous results, the conjugacy class of the involution
induced by the complex conjugation in the homology of a real non-singular cubic
fourfold determines the fourfold up to projective equivalence and deformation.
Here, we show how to eliminate the projective equivalence and to obtain a pure
deformation classification, that is how to respond to the chirality question:
which cubics are not deformation equivalent to their image under a mirror
reflection. We provide an arithmetical criterion of chirality, in terms of the
eigen-sublattices of the complex conjugation involution in homology, and show
how this criterion can be effectively applied taking as examples -cubics
(that is those for which the real locus has the richest topology) and
-cubics (the next case with respect to complexity of the real locus). It
happens that there is one chiral class of -cubics and three chiral classes
of -cubics, contrary to two achiral classes of -cubics and three
achiral classes of -cubics.Comment: 25 pages, 8 figure
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