In this thesis, we study the group of base-point preserving homotopy classes of homotopy self-equivalences of a four-manifold. Based on the approach of Hambleton and Kreck, an explicit description of this group is obtained when the fundamental group of the manifold is either a free group or a two-dimensional Poincare duality group. As a byproduct, a classification of such four-manifolds up to s-cobordism is obtained by using the modified surgery theory of Kreck. / Thesis / Doctor of Philosophy (PhD)

Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator. / QC 20101108

There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernel of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Variants of the conjugate gradient algorithm along with different preconditioners are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. In addition, a parallel version of Craig's method with incomplete LU factorization preconditioning is implemented on a shared memory parallel computer with various levels and degrees of parallelism (e.g., linear algebra, function and Jacobian matrix evaluation, etc.). An in-depth study is presented for each of these levels with respect to the speedup in execution time obtained with the parallelism, the time spent implementing the parallel code and the extra memory allocated by the parallel algorithm. / Master of Science

Este trabalho apresenta a teoria de homotopia simples, desenvolvida por J. H. C. Whitehead, com o objetivo de obter um método para classificar espaços com o mesmo tipo de homotopia. Com esta motivação, Whitehead introduz o conceito de equivalência de homotopia simples entre complexos simpliciais, que posteriormente é generalizado para complexos CW, espaços criados pelo próprio Whitehead. Um resultado imediato desta teoria é que quando dois espaços têm o mesmo tipo de homotopia simples, eles têm o mesmo tipo de homotopia. A recíproca desta afirmação é então conjecturada. Mostraremos que trata-se de uma conjectura falsa, contudo a investigação de sua confirmação produz um material que toma rumo próprio. Nosso enfoque são os aspectos algébricos envolvidos nesta investigação / This work presents the simple-homotopy theory, developed by J. H. C. Whitehead, with the goal to get an method to classify spaces with the same homotopy type. So, with this motivation, Whitehead introduced the concept of simple-homotopy equivalence between simplicial complexes, that later was generalized for CW complexes, spaces created by himself. An immediate result of this theory is that, if two spaces have the same simple-homotopy type, they have the same homotopy type. Then, the reciprocal statement is conjectured. We will show that the conjecture is not true, but the research about its truthfulness produces a material that takes its own way. Our approach are the algebraic aspects involved in this research

In this thesis we investigate the role of odd Poisson brackets in related areas of supergeometry. In particular we study three different cases of their appearance: Courant algebroids and their homotopy analogues, weak Poisson structures and their relation to foliated manifolds, and the structure of odd Poisson manifolds and their modular class. In chapter 2 we introduce the notion of a homotopy Courant algebroid, a subclass of which is suggested to stand as the double objects to L-bialgebroids. We provide explicit formula for the higher homotopy Dorfman brackets introduced in this case, and the higher relations between these and the anchor maps. The homotopy Loday structure is investigated, and we begin a discussion of what other constructions in the theory of Courant algebroids can be carried out in this homotopy setting. Chapter 3 is devoted to lifting a weak Poisson structure corresponding to a local foliation of a submanifold to a weak Koszul bracket, and interpreting the results in terms of the cohomology of an associated differential. This bracket is shown to produce a bracket on co-exact differential forms. In chapter 5 studies classes of second order differential operators acting on semidensities on an arbitrary supermanifold. In particular, when the supermanifold is odd Poisson, we given an explicit description of the modular class of the odd Poisson manifold, and provide the first non-trivial examples of such a class. We also introduce the potential field of a general odd Laplacian, and discuss its relation to the geometry of the odd Poisson manifold and its status as a connection-like object.

Este trabalho apresenta a teoria de homotopia simples, desenvolvida por J. H. C. Whitehead, com o objetivo de obter um método para classificar espaços com o mesmo tipo de homotopia. Com esta motivação, Whitehead introduz o conceito de equivalência de homotopia simples entre complexos simpliciais, que posteriormente é generalizado para complexos CW, espaços criados pelo próprio Whitehead. Um resultado imediato desta teoria é que quando dois espaços têm o mesmo tipo de homotopia simples, eles têm o mesmo tipo de homotopia. A recíproca desta afirmação é então conjecturada. Mostraremos que trata-se de uma conjectura falsa, contudo a investigação de sua confirmação produz um material que toma rumo próprio. Nosso enfoque são os aspectos algébricos envolvidos nesta investigação / This work presents the simple-homotopy theory, developed by J. H. C. Whitehead, with the goal to get an method to classify spaces with the same homotopy type. So, with this motivation, Whitehead introduced the concept of simple-homotopy equivalence between simplicial complexes, that later was generalized for CW complexes, spaces created by himself. An immediate result of this theory is that, if two spaces have the same simple-homotopy type, they have the same homotopy type. Then, the reciprocal statement is conjectured. We will show that the conjecture is not true, but the research about its truthfulness produces a material that takes its own way. Our approach are the algebraic aspects involved in this research

In this thesis we use the theory of algebraic operads to define a complete invariant of real and rational homotopy classes of maps of topological spaces and manifolds. More precisely let f,g : M -&gt; N be two smooth maps between manifolds M and N. To construct the invariant, we define a homotopy Lie structure on the space of linear maps between the homology of M and the homotopy groups of N, and a map mc from the set of based maps from M to N, to the set of Maurer-Cartan elements in the convolution algebra between the homology and homotopy. Then we show that the maps f and g are real (rational) homotopic if and only if mc(f) is gauge equivalent to mc(g), in this homotopy Lie convolution algebra. In the last part we show that in the real case, the map mc can be computed by integrating certain differential forms over certain subspaces of M. We also give a method to determine in certain cases, if the Maurer-Cartan elements mc(f) and mc(g) are gauge equivalent or not. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript.</p>

This licentiate thesis consists of two papers about topics related to representation stability for different automorphisms groups of topological spaces and manifolds. In Paper I, we study the rational homology groups of \textit{Torelli groups} of smooth, compact and orientable surfaces. The Torelli group of a smooth surface is the group of isotopy classes of orientation preserving diffeomorphisms that act trivially on the first homology group of the surface. In the paper, we study a certain class of stable homology classes, i.e. classes that exist for sufficiently large genus, and explicitly describe the image of these classes under a higher degree version of the \textit{Johnson homomorphism}, as a representation of the symplectic group. This gives a lower bound on the dimension of the stable homology of the group, as well as providing some further evidence that these homology groups satisfy representation stability for symplectic groups, in the sense of Church and Farb. In Paper II, we study pointed homotopy automorphisms of iterated wedge sums of spaces as well as boundary relative homotopy automorphisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these, for simply connected CW-complexes and closed manifolds respectively,  satisfy representation stability for symmetric groups, in the sense of Church and Farb.

We build a model structure from the simple point of departure of a structured interval in a monoidal category — more generally, a structured cylinder and a structured co-cylinder in a category.

In this work, we prove that the set of link-homotopy classes of generalized string links over a closed, connected and orientable surface M of genus g &ge; 1 form a group, denoted by Bn(M) and we find a presentation for it. Moreover, we prove that its normal subgroup PBnn(M), namely, the homotopy string links over M, is bi-orderable. These results extend results proved by Juan GonzalezMeneses in [GM], [GM2] and Ekaterina Yurasovskaya in [Y], respectively. Also, we obtain an exact sequence for link-homotopy braid groups, which is an extension of [Go, Theorem 1]. / Sem resumo