On manifolds with nonhomogeneous factors

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces

Topological Manifolds

Topological Manifolds are abstract spaces that locally resemble Euclidean space. For example, consider a round globe and a flat map. The map is a 2-dimensional representation of a 3-dimensional space. Given any point on the globe we can find a corresponding position on the map, and vice versa. This correspondence is called a chart. With a sufficient number of charts, we can describe the whole space. Such a collection of charts is called an Atlas. It is possible to construct different Atlases for the same space, allowing us to move from one chart, to the space, to another chart. This process is called a transition map. The areas of focus for this project include several examples of manifolds such as curves, n-spheres, and the torus. We explore and illustrate different approaches to charts on these manifolds, the properties of a manifold, examples of spaces that fail to meet these requirements, and the derivation of transition maps

The triangulation of manifolds

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.Comment: 26 pages, 2 figures. version 2: spellings corrected, analytic speculations in 4.8.2 sharpene

Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.Comment: 26 pages, accepted in "Journal of Topology and Analysis

The Implicit Function Theorem for continuous functions

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps.Comment: 9 pages, no figure