We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.

This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.

We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems.

We construct geometric models for K-homology with coefficients based on the theory of Z/k-manifolds. To do so, we generalize the operations and relations Baum and Douglas put on spinc-manifolds to spinc Z/kZ-manifolds. We then de fine a model for K-homology with coefficients in Z/k using cycles of the form ((Q,P), (E,F), f) where (Q, P) is a spinc Z/k-manifold, (E, F) is a Z/k-vector bundle over (Q, P) and f is a continuous map from (Q, P) into the space whose K-homology we are modelling. Using results of Rosenberg and Schochet, we then construct an analytic model for K-homology with coefficients in Z/k and a natural map from our geometric model to this analytic model. We show that this map is an isomorphism in the case of finite CW-complexes. Finally, using direct limits, we produced geometric models for K-homology with coefficients in any countable abelian group.

Thesis (Ph. D.)--School of Economics and Political Science, London.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20882号 / 理博第4334号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 上 正明, 准教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We interpret the generalised gauge symmetry introduced in string theory and M-Theory as a special case of Grothendieck's stability equivalence relation in the definition of the 0th K-group and we calculate the Euler number of the elliptic de Rham complex twisted by a flat connection. Then using Polyakov's classical equivalence of flat bundles with non-linear sigma models we define a new topological invariant for foliations using techniques from noncommutative geometry, in particular the Connes' pairing between K-Theory and cyclic cohomology. This new invariant classifies foliations up to Morita equivalence.

L'objet de cette thèse est de démontrer une formule reliant les métriques de Ray-Singer hypoelliptique et de Milnor sur le déterminant de la cohomologie d'une variété riemannienne compacte par une déformation à la Witten du laplacien hypoelliptique en théorie de de Rham. / The purpose of this thesis is to prove a formula relating the hypoelliptic Ray-Singermetric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.

John Lott berechnete eine Signatur mit ganzzahligen Werten für den Orbitraum einer kompakten, orientierbaren (4k + 1)-Mannigfaltigkeit mit einer halbfreien S1-Wirkung. Diese Signatur ist eine Homotopieinvariante für den Orbitraum. Allerdings konstruierte er keinen Operator vom Dirac-Typ, der die Signatur als Index besitzt. In dieser Arbeit konstruieren wir einen solchen Operator auf dem Orbitraum der S1-Wirkung, einem Thom-Mather stratifizierten Raum mit einem singulären Stratum von positiver Dimension, und wir zeigen, dass der Operator im wesentlichen eindeutig bestimmt ist. Ferner zeigen wir, dass sein Index mit Lotts Signatur übereinstimmt, zumindest wenn der stratifizierte Raum die sogenannte Witt-Bedingung erfüllt. Wirnennendiesen Operator den induzierten Dirac-Schrödinger Operator. Unsere Konstruktionsstrategie ist es, einen geeigneten S1-invarianten transversal elliptischen Operator erster Ordnung auf den S1-invarianten Differentialformen zu definieren, der den gesuchten Operator auf den Differentialformen des Orbitraums induziert. Die Witt-Bedingung, eine topologische Bedingung, welche in diesem Fall von der Kodimension der betrachteten Punktmenge abhängt, lässt verschiedene analytische Schlussfolgerungen zu. Insbesondere ist, wenn die Bedingung nicht erfüllt ist, der Hodge-de Rham Operator auf dem Quotientenraum nicht notwendigerweise essentiell selbstadjungiert und die Wahl einer Randbedingung ist daher notwendig. Diese Wahlfreiheit erscheint unnatürlich in Anbetracht der Tatsache, dass Lotts Signatur unabhängig von der Witt-Bedingung wohldefiniert ist. Der Dirac-Schrödinger Operator, der in dieser Arbeit konstruiert wird, unterschei- det sich vom Hodge-de Rham Operator durch einen Term nullter Ordnung, welcher sicherstellt, dass der Operator wesentlich selbstadjungiert ist. Außerdem antikommutiert dieser Term nullter Ordnung mit der Signatur-Involution, wodurch der gesamte Operator zerfällt und so der Index berechnet werden kann, auch wenn die Witt-Bedingung nicht erfüllt ist. / John Lott has computed an integer-valued signature for the orbit space of a compact orientable (4k + 1) manifold with a semi-free S1-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on the orbit space, a Thom-Mather stratified space with one singular stratum of positive dimension, and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so called Witt condition. We call this operator the induced Dirac-Schrödinger operator. The strategy of the construction is to “push down” an appropriate S1-invariant first order transversally elliptic operator to the quotient space. The Witt condition, a topological condition which in this case depends on the codi- mension of the fixed point set, has various analytic consequences. In particular, when not satisfied, the Hodge-de Rham operator on the quotient space does not need to be essentially self-adjoint and therefore a choice of boundary conditions is required. This choice freedom is not natural in view of the fact that Lott’s signature is well defined independently of the Witt condition. The Dirac-Schrödinger operator constructed in this Thesis differs from the Hodge-de Rham operator by a zero order term which ensures it to be essentially self-adjoint. Moreover, this zero order term anti-commutes with the chirality involution allowing the whole operator to split so that the index can be computed even if the Witt condition is not satisfied.

Orientador: Ketty Abaroa de Rezende / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T08:12:09Z (GMT). No. of bitstreams: 1 MontufarLopez_HernanRoberto_D.pdf: 8015438 bytes, checksum: 1175f8d0f78fe476b09178b6e50f10ec (MD5) Previous issue date: 2010 / Resumo: Em [6] são apresentadas condições necessárias e suficientes para a estabilidade estrutural e o teorema de densidade para campos de vetores em 2-variedades com singularidades simples dos seguintes tipos: cone, guarda-chuva de Whitney, ponto duplo e ponto triplo. Nesta tese, estudamos os fluxos induzidos por estes campos de vetores, que denominamos fluxos Gutierrez-Sotomayor, do ponto de vista topológico utilizando a teoria de Conley. Apresentamos uma fórmula dinâmico-topológica que relaciona o índice de Conley de uma variedade com singularidades simples M que possui uma estratificação que a decompõe numa união disjunta da sua parte regular e da sua parte singular. Usando essa estratificação mostramos que se a singularidade está na parte singular S de M o seu índice pode ser calculado tanto com respeito a M como com respeito a S. Definimos uma função de Lyapunov, neste contexto, e mostramos sua existência para fluxos sem órbitas periódicas e sem ciclos singulares. Em seguida, por uma análise da seqüência de homologia longa exata de um par índice determinamos propriedades que um grafo de Lyapunov deve satisfazer para estar associado a um fluxo. Também abordamos a questão da realização de grafos de Lyapunov abstratos. Para isto, primeiramente apresentamos a igualdade de Poincaré-Hopf, para o caso bidimensional, que caracteriza a relação entre o primeiro número de Betti das 1-variedades ramificadas que são fronteiras de um bloco isolante com seu número de componentes de fronteira e o índice numérico de Conley. Em seguida, mostramos que dados números inteiros positivos que satisfaçam a condição de Poincaré-Hopf sempre é possível construir um bloco isolante que satisfaz estes dados dinâmicos e homológicos / Abstract: In [6] a characterization and genericity theorem for C1-structurally stable vector fields tangent to a 2-dimensional compact subset M of Rk are established. Also in [6], new types of structurally stable singularities and periodic orbits are presented. In this thesis we study the continuous flows associated to these vector fields, which we refer to as the Gutierrez-Sotomayor flows on manifolds M with simple singularities using Conley Index Theory. We consider a stratification of M which decomposes it into a union of its regular and singular strata. We prove certain Euler type formulas which relate topology of M and dynamics on the singular strata. We show the existence of a Lyapunov function for Gutierrez-Sotomayor flows without periodic orbits and singular cycles in this context. Using long exact sequence analysis of index pairs we determine necessary and sufficient conditions for a Gutierrez-Sotomayor flow to be defined on an isolating block. We organize this combinatorially with the aid of Lyapunov graphs and using a Poincar'e-Hopf equality we give necessary conditions for a Lyapunov graph to be associated to a Gutierrez-Sotomayor flow and we also prove these conditions are sufficient / Doutorado / Geometria e Topologia / Doutor em Matemática