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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• 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.
Material covered so far: Chapter 0 and Chapter 1.1, 1.2 and 1.3 up to proofs of Galois correspondence, Chapter 2.1 (except Delta complexes and the equivalence of simplicial and singular homology) and Chapter 2.2 (except homology with coefficients).
Material for the midterm exam: Chapter 0 and Chapter 1.1, 1.2 and 1.3 up to statement of Galois correspondence.
Material for the final exam: Chapter 0 and Chapter 1.1, 1.2 and 1.3 up to proofs of Galois correspondence, Chapter 2.1 (except Delta complexes and the equivalence of simplicial and singular homology) and Chapter 2.2 (except homology with coefficients).
Tuesday 9 February 2016 (Week 6)
Outlook of the course. Introduction to homotopies.
Thursday 11 February 2016 (Week 6)
Chapter 0: CW complexes, examples and properties.
Tuesday 16 February 2016 (Week 7)
Chapter 1: Definition of the fundamental group, basic properties and start of the computation of the fundamental group of the circle.
Thursday 18 February 2016 (Week 7)
Fundamental group : Fundamental group of the circle and applications.
Tuesday 23 February 2016 (Week 8)
Fundamental group: applications of the fundamental group of the circle, induced homomorphisms and homotopy invariance.
Thursday 25 February 2016 (Week 8)
Fundamental group: products, amalgamated product, statement of van Kampen’s Theorem and examples.
Tuesday 1 March 2016 (Week 9)
Fundamental group: and fundamental group of cell complexes & proof of van Kampen’s Theorem.
Thursday 3 March 2016 (Week 9)
Classification of surfaces.
Tuesday 8 March 2016 (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 10 March 2016 (Week 10)
Covering spaces: started the proof of the correspondence between subgroups of the fundamental group and isomorphism classes of covering spaces.
Tuesday 15 March 2016 (Week 11)
Covering spaces: started the proof of the correspondence between subgroups of the fundamental group and isomorphism classes of covering spaces.
Thursday 17 March 2016 (Week 11)
Higher homotopy groups: definition.
Exam 1. From 15:00 to 18:00 at UNNIK 220.
Tuesday 29 March 2016 (Week 13)
Homology: Introduction, definition & basic properties up to homotopy invariance.
Thursday 31 March 2016 (Week 13)
Homology: Relative homology, reduced homology, long exact sequence of a pair, excision. Computed homology of spheres.
Tuesday 5 April 2016 (Week 14)
Homology: Excision, good pairs and homology of spheres.
Thursday 7 April 2016 (Week 14)
Homology: Mayer--Vietoris, degree and cellular homology.
Tuesday 12 April 2016 (Week 15)
Cellular homology continued. Homology with coefficients, Euler characteristic. Overview.
Thursday 14 April 2016 (Week 15)
Cohomology and outlook.
Exam week (Week 16)
