A recent result in the theory of differentiable manifolds, is that it is possible to construct exotic differentiable structures on the 4-dimensional Euclidean space R4. A wide range of mathematical tools, from fields as diverse as topology, analysis and geometry, are used in this construction. The aim of this dissertation is to elucidate the topological background which provides the framework for the construction of the exotic R4’s. Also, the uniqueness of the differentiable structures on RN, for n>4 will be proved – providing some insight into the reason for the failure of uniqueness in R4. Microbundles are introduced and studied, and some results from engulfing theory are proved. Using semi-simplicial complexes, the classification of differentiable structures on manifolds can then be translated to a standard lifting problem in topology. This provides the background for the statement of the results from which an exotic R4 will be constructed.
|Subject||Mathematics, Mathematical statistics and Statistics|
|Subject 2||Mathematics, Mathematical statistics and Statistics|
|Degree Type||Masters degree|