MAS6370 Algebraic Topology I

Both semesters, 2017/18 20 Credits
Lecturer: Prof John Greenlees Home page
Aims Outcomes Assessment Full Syllabus

In this course, we will study spaces from a topological point of view. This means we will be interested in some notion of the "shape" of a space rather than distances between points, so the emphasis will no longer be on metrics. We will show how to generalise the notion of metric space to achieve this, giving the notion of topological space. Our examples will include balls, spheres, the n-holed torus, the Möbius strip, the Klein bottle, other surfaces, knots, projective spaces. We will define what it means for two spaces to be homeomorphic, and introduce the more subtle and expressive notion of homotopy equivalence, with some interesting examples. We will study two methods for using algebra to analyse the properties of a space: the fundamental group, and homology. For any space we will define a group, called the fundamental group, which is a beautiful and powerful way of using algebra to detect topological features of spaces; for example we can sometimes use the fundamental group to check whether two spaces are homotopy equivalent. We will calculate the fundamental groups of a number of spaces and give some applications, including a proof of the Fundamental Theorem of Algebra, and the classification of surfaces. In the second part of the course we will study homology groups, which give a more tractable method than homotopy groups for studying higher-dimensional properties of spaces.


Note that one of the suggested books is available online free from http://www.math.cornell.edu/ ∼ hatcher/

There are no prerequisites for this module.
No other modules have this module as a prerequisite.


Outline syllabus

Part A

    1. Motivation.
    2. Reminder on metric spaces.
    3. Topological spaces.
    4. Homotopy.
    5. Fundamental group.
    6. Covering spaces.
    7. Van Kampen Theorem.
    8. Applications.
    9. Higher homotopy.

Part B

    1. Introduction to chain complexes and homology.
    2. Low dimensions.
    3. Singular homology and homotopy invariance.
    4. Abelianisation of the fundamental group and some group theory.
    5. Exact sequences and some homological algebra.
    6. Quotients vs relative homology.
    7. Axiomatisation.



Aims

  • To teach the basic ideas of topological spaces, the fundamental group and homology.
  • To illustrate these ideas by reference to a range of examples, including surfaces.
  • To show how to calculate fundamental groups, chain complexes and homology of various topological spaces.

Learning outcomes

  • Understand the idea of a topological space.
  • Understand the idea of a continuous map and a homeomorphism between topological spaces.
  • Understand the idea of a homotopy between two maps, and a homotopy equivalence between two spaces.
  • Understand the definition of the fundamental group and prove its basic properties.
  • Calculate the fundamental group of some simple spaces such as Rn, balls, spheres, projective space, n-holed torus, Klein bottle and other surfaces.
  • Understand the proofs of the Fundamental Theorem of Algebra and the Brouwer Fixed Point Theorem using the fundamental group.
  • Understand the definition of singular homology.
  • Calculate the homology of some simple spaces such as spheres, products of spheres and surfaces.

40 lectures, no tutorials

Assessment

Short weekly tests in lectures (worth 20 percent), and one formal 2.5 hour written examination (worth 80 percent). The exam will have four questions and the format will be "answer all questions".

Full syllabus

Part A


1. Motivation.
Intuitive topology by pictures. Motivation for considering high-dimensional examples that cannot be visualised. Some obvious statements that are hard to prove. Translating problems into algebra.
2. Reminder on metric spaces.
Definitions and examples. Continuity and homeomorphism. Open and closed sets, images and preimages, characterisations of continuity.
3. Topological spaces.
Definition and relationship with metric spaces. Examples without proof, to include Rn, balls, spheres, projective space, n-holed torus, products and quotients. Continuous maps and examples. Paths, loops, path components and path connectedness.
4. Homotopy.
Idea in pictures. Definition of homotopy, path homotopy, homotopy equivalence, contractibility. Idea of homotopy type, homotopy invariance.
5. Fundamental group.
Idea in pictures. Definition and some preliminary examples without proof. Reminders of groups and introduction of categories and functors. The fundamental group functor. Dependence on basepoint and homotopy invariance. Fundamental group of circle. Notion of simply connected space.
6. Covering spaces.
Idea in pictures. Definition and lifting correspondence. Universal covers and some pictorially vivid examples. Application to fundamental group.
7. Van Kampen Theorem.
Idea of building up spaces by glueing. Statement of Van Kampen Theorem and examples of its use. Reminder on normal subgoups and quotient groups, and definition of free product. Sketch proof of Van Kampen Theorem and application to cell complexes and classification of surfaces.
8. Applications.
Fundamental Theorem of Algebra and Brouwer Fixed Point Theorem.
9. Higher homotopy.
Introduction to the idea of higher homotopy groups and classification of homotopy types.

Part B

1. Introduction to chain complexes and homology.
Chain complexes, cycles and boundaries, compare and contrast with homotopy.
2. Low dimensions.
Some direct calculations from familiar low-dimensional cell-complexes: torus, Klein bottle, projective plane, circle, sphere. Introduction to simplices.
3. Singular homology and homotopy invariance.
Singular chains, reduced homology, chain homotopy, homotopy invariance.
4. Abelianisation of the fundamental group and some group theory.
Some Abelian group theory: commutator subgroup, abelianisation, finitely generated groups, fundamental theorem of finitely-generated Abelian groups. The relationship between the first homology group and the Abelianisation of the fundamental group.
5. Exact sequences and some homological algebra.
Quotients and relative homology, the long exact sequence of homology associated to a short exact sequence of chain complexes.
6. Quotients vs relative homology.
The relationship betwen the homology of a quotient and relative homology for CW-complexes. Examples. The Mayer-Vietoris sequence.
7. Axiomatisation.
Homology via functors and natural transformations. Wedge sums. Moore spaces.