Computing spectral sequences

In this paper, a set of programs enhancing the Kenzo system is presented. Kenzo is a Common Lisp program designed for computing in Algebraic Topology, in particular it allows the user to calculate homology and homotopy groups of complicated spaces. The new programs presented here entirely compute Serre and Eilenberg-Moore spectral sequences, in particular the groups and differential maps for arbitrary r. They also determine when the spectral sequence has converged and describe the filtration of the target homology groups induced by the spectral sequence

Integration of the kenzo system within sagemath for new algebraic topology computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before

A Bennequin-type inequality and combinatorial bounds

In this paper we provide a new Bennequin-type inequality for the Rasmussen- Beliakova-Wehrli invariant, featuring the numerical transverse braid invariants (the c-invariants) introduced by the author. From the Bennequin type-inequality, and a combinatorial bound on the value of the c-invariants, we deduce a new computable bound on the Rasmussen invariant.Comment: 24 pages, 5 figures and 2 tables. Minor revisions, title change

Constructive category theory and applications to equivariant sheaves

In this thesis we create a purely categorical framework for cohomology computations of G-equivariant coherent sheaves on projective space for a finite group G. For this, we develop three different sub-frameworks: First, we construct a skeletal tensor category SRep(G) equivalent to the representation category Rep(G) of G. Second, we design, in the context of an arbitrary abelian category, an algorithm for computing spectral sequences which is suitable for a direct computer implementation, i.e., it only uses categorical constructions provided by the axioms of an abelian category. Last, we describe how to internalize the exterior algebra E and its modules in a tensor category. Combining our three sub-frameworks yields an algorithm for computing spectral sequences within the category of E-modules internal to SRep(G). Thanks to an equivariant version of the famous BGG-correspondence, we can use such an algorithm for computing cohomology groups of G-equivariant sheaves on projective space. Furthermore, this algorithm allows us to compute a new invariant called spectral cohomology table which in this thesis is proven to be stronger than the classical cohomology table. Since our framework can be described in purely categorical language, a software project in GAP facilitating the implementation of abstract categories and categorical algorithms was born during the writing of this thesis: Cap (Categories, Algorithms, Programming). The categorical framework along with all algorithms presented in this thesis is implemented in Cap.In dieser Arbeit geben wir der Kohomologieberechnung G-äquivarianter Garben auf dem projektiven Raum für endliche Gruppen G einen konstruktiven kategoriellen Rahmen. Dazu gehen wir in drei Schritten vor: Wir konstruieren zuerst eine skeletale Tensorkategorie SRep(G), welche äquivalent zur Darstellungskategorie Rep(G) von G ist. Danach entwerfen wir einen ausschließlich auf den Axiomen einer abelschen Kategorie beruhenden Algorithmus zur Berechnung von Spektralsequenzen. Im Anschluss behandeln wir die äußere Algebra E und ihre Moduln intern in einer Tensorkategorie. Die Kombination dieser drei Schritte ergibt einen Algorithmus zur Berechnung von Spektralsequenzen innerhalb der Kategorie von E-Moduln intern in SRep(G). Dank der berühmten BGG-Korrespondenz kann dieser Algorithmus zur Bestimmung von Kohomologiegruppen G-äquivarianter Garben auf dem projektiven Raum genutzt werden. Darüber hinaus ermöglicht er die Berechnung von Spektral-Kohomologietabellen - eine neue Invariante, welche stärker ist als klassische Kohomologietabellen, wie in dieser Arbeit gezeigt wird. Durch die konstruktive Anwendung rein kategorieller Konzepte entstand während des Verfassens dieser Arbeit ein Software-Projekt in GAP zur Vereinfachung der Implementation abstrakter Kategorien und kategorieller Algorithmen: Cap (Categories, Algorithms, Programming). Alle Ergebnisse und Algorithmen dieser Arbeit wurden in Cap realisiert und implementiert

A geometric decomposition of spaces into cells of different types II: Homology theory

We develop the homology theory of CW (A)-complexes, generalizing the classical cellular homology theory for CW-complexes. A CW (A)-complex is a topological space which is built up out of cells of a certain core A.Fil: Minian, Elias Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Ottina, Enzo Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Universidad Nacional de Cuyo. Facultad de Ciencias Exactas y Naturales; Argentin