About the subject
In “geometry and topology” we study the general shape, normally referred to as the “number of holes”, of topological spaces (particularly CW-complexes) by means of algebraic gadgets. Here we regard topological spaces as malleable objects which can be twisted and bent (but not be torn apart) and try to describe properties which are preserved through these deformations.
The broad “aim” of algebraic topology is to classify these spaces up to some reasonable equivalence (normally, homeomorphisms or homotopies). That failing, one tries to, at least, obtain enough information to distinguish different spaces 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 space (geometric object) to distinguish different spaces;
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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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• homotopies,
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• fundamental group,
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• quotients,
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• covering spaces,
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• homotopy groups,
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• singular homology,
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• singular cohomology.
The course will also cover the following important results relating the concepts above:
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• van Kampen theorem,
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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.
Tuesday 7 February 2017 (Week 6)
Outlook of the course. Introduction to homotopies. this lecture will finish a little before 11:30.
Thursday 9 February 2017 (Week 6)
Chapter 0: CW complexes, examples and properties.
Tuesday 14 February 2017 (Week 7)
Chapter 1: Definition of the fundamental group, basic properties and start of the computation of the fundamental group of the circle.
Thursday 16 February 2017 (Week 7)
Fundamental group : Fundamental group of the circle and applications.
Tuesday 21 February 2017 (Week 8)
Fundamental group: applications of the fundamental group of the circle, induced homomorphisms and homotopy invariance.
Thursday 23 February 2017 (Week 8)
Fundamental group: products, amalgamated product, statement of van Kampen’s Theorem and examples.
Tuesday 28 February 2017 (Week 9)
Fundamental group: and fundamental group of cell complexes & proof of van Kampen’s Theorem.
Thursday 2 March 2017 (Week 9)
Classification of surfaces.
Tuesday 7 March 2017 (Week 10)
Covering spaces: definition, lifting of maps and the induced map of fundamental groups. First steps towards the proof that there is a bijection between isomorphism classes of pointed covering spaces and subgroups of the fundamental group.
Thursday 9 March 2017 (Week 10)
Covering spaces: started the proof of the correspondence between subgroups of the fundamental group and isomorphism classes of covering spaces.
Tuesday 14 March 2017 (Week 11)
Covering spaces: started the proof of the correspondence between subgroups of the fundamental group and isomorphism classes of covering spaces.
Thursday 16 March 2017 (Week 11)
Higher homotopy groups: definition.
Exam 1. From 15:00 to 18:00 at BBG 169.
Tuesday 21 March 2017 (Week 12)
Homology: Introduction, definition & basic properties up to homotopy invariance.
Thursday 23 March 2017 (Week 12)
Homology: Relative homology, reduced homology, long exact sequence of a pair, excision. Computed homology of spheres.
Tuesday 28 March 2017 (Week 13)
Homology: Excision, good pairs and homology of spheres.
Thursday 30 March 2017 (Week 13)
Homology: Mayer--Vietoris, degree and cellular homology.
Tuesday 4 April 2017 (Week 14)
Cellular homology continued. Homology with coefficients, Euler characteristic. Overview.
Thursday 6 April 2017 (Week 14)
Cohomology and outlook.
Exam week (Week 15)
