4,131 research outputs found
Extending the notion of AT-Model for integer homology computation
When the ground ring is a field, the notion of algebraic topological model (AT-model) is a useful tool for computing (co)homology, representative (co)cycles of (co)homology generators and the cup product on cohomology of nD digital images as well as for controlling topological information when the image suffers local changes [6,7,9]. In this paper, we formalize the notion of λ-AT-model (λ being an integer) which extends the one of AT-model and allows the computation of homological information in the integer domain without computing the Smith Normal Form of the boundary matrices. We present an algorithm for computing such a model, obtaining Betti numbers, the prime numbers p involved in the invariant factors (corresponding to the torsion subgroup of the homology), the amount of invariant factors that are a power of p and a set of representative cycles of the generators of homology mod p, for such p
A Tool for Integer Homology Computation: Lambda-At Model
In this paper, we formalize the notion of lambda-AT-model (where is
a non-null integer) for a given chain complex, which allows the computation of
homological information in the integer domain avoiding using the Smith Normal
Form of the boundary matrices. We present an algorithm for computing such a
model, obtaining Betti numbers, the prime numbers p involved in the invariant
factors of the torsion subgroup of homology, the amount of invariant factors
that are a power of p and a set of representative cycles of generators of
homology mod p, for each p. Moreover, we establish the minimum valid lambda for
such a construction, what cuts down the computational costs related to the
torsion subgroup. The tools described here are useful to determine topological
information of nD structured objects such as simplicial, cubical or simploidal
complexes and are applicable to extract such an information from digital
pictures.Comment: Journal Image and Vision Computing, Volume 27 Issue 7, June, 200
Constructive Algebraic Topology
The classical ``computation'' methods in Algebraic Topology most often work
by means of highly infinite objects and in fact +are_not+ constructive. Typical
examples are shown to describe the nature of the problem. The Rubio-Sergeraert
solution for Constructive Algebraic Topology is recalled. This is not only a
theoretical solution: the concrete computer program +Kenzo+ has been written
down which precisely follows this method. This program has been used in various
cases, opening new research subjects and producing in several cases significant
results unreachable by hand. In particular the Kenzo program can compute the
first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of
Tenerife, September 199
Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups
We extend each higher Johnson homomorphism to a crossed homomorphism from the
automorphism group of a finite-rank free group to a finite-rank abelian group.
We also extend each Morita homomorphism to a crossed homomorphism from the
mapping class group of once-bounded surface to a finite-rank abelian group.
This improves on the author's previous results [Algebr. Geom. Topol. 7
(2007):1297-1326]. To prove the first result, we express the higher Johnson
homomorphisms as coboundary maps in group cohomology. Our methods involve the
topology of nilpotent homogeneous spaces and lattices in nilpotent Lie
algebras. In particular, we develop a notion of the "polynomial straightening"
of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page
Fluxes in F-theory Compactifications on Genus-One Fibrations
We initiate the construction of gauge fluxes in F-theory compactifications on
genus-one fibrations which only have a multi-section as opposed to a section.
F-theory on such spaces gives rise to discrete gauge symmetries in the
effective action. We generalize the transversality conditions on gauge fluxes
known for elliptic fibrations by taking into account the properties of the
available multi-section. We test these general conditions by constructing all
vertical gauge fluxes in a bisection model with gauge group SU(5) x Z2. The
non-abelian anomalies are shown to vanish. These flux solutions are dynamically
related to fluxes on a fibration with gauge group SU(5) x U(1) by a conifold
transition. Considerations of flux quantization reveal an arithmetic constraint
on certain intersection numbers on the base which must necessarily be satisfied
in a smooth geometry. Combined with the proposed transversality conditions on
the fluxes these conditions are shown to imply cancellation of the discrete Z2
gauge anomalies as required by general consistency considerations.Comment: 30 pages; v2: typos correcte
Chain Homotopies for Object Topological Representations
This paper presents a set of tools to compute topological information of
simplicial complexes, tools that are applicable to extract topological
information from digital pictures. A simplicial complex is encoded in a
(non-unique) algebraic-topological format called AM-model. An AM-model for a
given object K is determined by a concrete chain homotopy and it provides, in
particular, integer (co)homology generators of K and representative (co)cycles
of these generators. An algorithm for computing an AM-model and the
cohomological invariant HB1 (derived from the rank of the cohomology ring) with
integer coefficients for a finite simplicial complex in any dimension is
designed here. A concept of generators which are "nicely" representative cycles
is also presented. Moreover, we extend the definition of AM-models to 3D binary
digital images and we design algorithms to update the AM-model information
after voxel set operations (union, intersection, difference and inverse)
Realizing modules over the homology of a DGA
Let A be a DGA over a field and X a module over H_*(A). Fix an
-structure on H_*(A) making it quasi-isomorphic to A. We construct an
equivalence of categories between A_{n+1}-module structures on X and length n
Postnikov systems in the derived category of A-modules based on the bar
resolution of X. This implies that quasi-isomorphism classes of A_n-structures
on X are in bijective correspondence with weak equivalence classes of
rigidifications of the first n terms of the bar resolution of X to a complex of
A-modules. The above equivalences of categories are compatible for different
values of n. This implies that two obstruction theories for realizing X as the
homology of an A-module coincide.Comment: 24 page
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