Hayes, El
(2023)
Smooth Structures on Spheres.
Masters thesis, National University of Ireland Maynooth.
Abstract
We provide an account of Milnor's construction of an exotic 7-sphere and the subsequent
rapid development of differential topology used to produce and classify exotic manifolds.
We begin by giving some necessary background, assuming only previous knowledge of real
analysis and linear algebra. Smooth manifolds, vector bundles, and fibre bundles are introduced,
along with some operations on manifolds not usually seen in a first course, before
giving a review of all necessary algebraic topology. We pay particular attention to the homology
of smooth manifolds, as this will form the basis for the constructions of the following
section. We then introduce characteristic classes, which are one of the main ingredients
in constructing smooth manifold invariants. Using this setup, we develop Milnor's original
smooth invariant and a generalisation of it to a wider class of manifolds. We give a brief
introduction to Morse theory, which we use to characterise topological spheres. Having set
up the necessary background, we construct a number of examples of exotic spheres. We first
present Milnor's original example, and then develop a more general tool, plumbing disk bundles,
to give a much larger class of examples. Finally, we turn to the classiffication of smooth
structures on spheres of dimension greater than four, developing the necessary background
to state Milnor and Kervaire's classiffication results on homotopy groups of spheres, before
indicating a number of future directions of study to the reader, as this thesis is ultimately
intended to be an introduction to a vast field.
| Item Type: |
Thesis
(Masters)
|
| Keywords: |
Smooth Structures; Spheres; |
| Academic Unit: |
Faculty of Science and Engineering > Mathematics and Statistics |
| Item ID: |
17275 |
| Depositing User: |
IR eTheses
|
| Date Deposited: |
06 Jun 2023 11:04 |
| URI: |
|
| Use Licence: |
This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available
here |
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