About the subject
A manifold is an abstraction which generalizes the concept of embedded surface in R^3 and is the basic object studied in differential geometry. A manifold looks locally like R^n, i.e. there are maps which identify parts of the manifold with the flat space, R^n, and if two maps describe overlapping regions, there is a unique smooth/continuous way to identify the overlapping points. In “geometry and topology” we study the general shape, normally referred to as the “number of holes”, of manifolds and more general spaces. Here we regard manifolds as malleable objects which can be twisted and bent (but not be torn apart) and try to describe properties which are preserved though these deformations.
The broad “aim” of topology of manifolds is to classify manifolds up to some reasonable equivalence (normally, homeomorphisms or homotopies). That failing, one tries to, at least, obtain enough information to distinguish different manifolds in several cases of interest. The general approach is to cook up some algebraic object from geometric data and hope that this algebraic object
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1)captures enough information about the manifold (geometric object) to distinguish different manifolds;
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2)is computable.
Examples of such objects will be the fundamental group and the higher homotopy groups as well as homology and cohomology groups.
This course will cover the following concepts:
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• topological spaces and cell complexes,
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• definition and examples of manifolds,
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• quotients,
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• fiber bundles,
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• homotopies,
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• fundamental group,
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• covering spaces,
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• homotopy groups,
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• homology,
The course will also cover the following important results relating the concepts above:
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• van Kampen theorem,
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• homotopy groups of a fibration (hopefully),
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• Mayer–Vietoris sequence,
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• excision theorem
Pre-requisites
I will assume that you are familiar with the contents of the courses Group theory and Introduction to topology.
Material covered so far from Hatcher’s book
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•Chapter 0
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•Chapter 1, sections 1.1, 1.2 and 1.3 up to and inclusive “The classification of covering spaces”.
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•Classification of surfaces (The ZIP proof)
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•Chapter 2, sections 2.1 and 2.2.
Material for the final exam:
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•Chapter 0
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•Chapter 1, sections 1.1, 1.2 and 1.3 up to and inclusive “The classification of covering spaces”.
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•Classification of surfaces (The ZIP proof)
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•Chapter 2, sections 2.1 and 2.2.
Friday 25 April 2014 (Week 17)
Outlook of the course. Introduction to homotopies.
Monday 28 April 2014 (Week 18)
CW complexes and constructions of spaces.
Friday 2 May 2014 (Week 18)
Chapter 0: Wedge product, smash product, attaching spaces via maps, homotopy extension property. Chapter 1: definition of the fundamental group.
Friday 9 May 2014 (Week 19)
Fundamental group : Fundamental group of the circle continued and applications.
Monday 12 May 2014 (Week 20)
No Lecture today
Friday 16 May 2014 (Week 20)
Fundamental group: Van Kampen theorem and computations of fundamental groups.
Monday 19 May 2014 (Week 21)
Fundamental group: proof of van Kampen’s theorem and fundamental group of cell complexes
Friday 23 May 2014 (Week 21)
Classification of surfaces.
Monday 26 May 2014 (Week 22)
Covering spaces: definition, lifting of maps and the induced map of fundamental groups.
Friday 30 May 2014 (Week 22)
Covering spaces: proved that there is a bijection between isomorphism classes of pointed covering spaces and subgroups of the fundamental group.
Monday 2 June 2014 (Week 23)
Higher homotopy groups: definition.
Friday 6 June 2014 (Week 23)
Exam 1. From 9:00 to 12:00 in room 611 (Maths building)
Friday 13 June 2014 (Week 24)
Homology: Simplicial and singular homology (definition) — Crainic.
Monday 16 June 2014 (Week 25)
Homology: Exact sequences, excision and Mayer–Vietoris sequence — Crainic
Friday 20 June 2014 (Week 25)
Homology: Proofs of Exact sequences, excision and Mayer–Vietoris sequence — Leer-Duran
Monday 23 June 2014 (Week 26)
Homology: Degree and cellular homology.
Friday 27 June 2014 (Week 26)
Homology: Cellular homology and coefficients.
Outlook.
Wednesday 2 July 2014 (Week 27)
Exam: Ruppert Paars from 13:30 to 16:30.
