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) via using 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 though 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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• fiber bundles,
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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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• 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
Hatcher: Chapter 0, Chapter 1.1, 1.2 &1.3, Chapter 2.1 (minus delta complexes).
Classification of surfaces.
Thursday 5 February 2015 (Week 6)
Outlook of the course. Introduction to homotopies.
Tuesday 10 February 2015 (Week 7)
Chapter 0: CW complexes, example and properties.
Thursday 12 February 2015 (Week 7)
Chapter 1: Definition of the fundamental group and basic properties.
Tuesday 17 February 2015 (Week 8)
Fundamental group : Fundamental group of the circle.
Thursday 19 February 2015 (Week 8)
Fundamental group: applications of the fundamental group of the circle, induced homomorphisms and homotopy invariance.
Tuesday 24 February 2015 (Week 9)
Fundamental group: products, amalgamated product, statement of van Kampen’s Theorem and examples.
Thursday 26 February 2015 (Week 9)
Fundamental group: and fundamental group of cell complexes & proof of van Kampen’s Theorem.
Tuesday 3 March 2015 (Week 10)
Classification of surfaces.
Thursday 5 March 2015 (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.
Tuesday 10 March 2015 (Week 11)
No lecture today
Thursday 12 March 2015 (Week 11)
Higher homotopy groups: definition.
Tuesday 17 March 2015 (Week 12)
Exam 1. From 9:00 to 12:00 at Educatorium Beta zaal.
Thursday 19 March 2015 (Week 12)
Covering spaces: finish the proof of the correspondence between subgroups of the fundamental group and isomorphism classes of covering spaces.
Tuesday 24 March 2015 (Week 13)
Homology: Introduction, definition & basic properties up to homotopy invariance.
Thursday 26 March 2015 (Week 13)
Homology: Relative homology, reduced homology, long exact sequence of a pair, excision. Computed homology of spheres.
Tuesday 31 March 2015 (Week 14)
Homology: Mayer-Vietoris, degree and overview of homology so far.
Thursday 2 April 2015 (Week 14)
Homology: Degree and cellular homology and coefficients.
Tuesday 7 April 2015 (Week 15)
Cohomology and relation between homology and cohomology.
Thursday 9 April 2015 (Week 15)
Outlook.
Thursday 16 April 2015 (Week 16)
Exam: Educatorium Alpha from 13:30 to 16:30.
