MAS435 Algebraic Topology

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

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

In this course, we will study geometric objects from a topological point of view. This means we will be not be interested in the exact shape of a space or the distances between points, but rather the properties that are preserved under stretching. We will show how to capture these properties purely in terms of open sets, giving the notion of topological space. Our examples will include balls, spheres, surfaces, 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, but the focus will be on properties preserved by much more radical sorts of change (i.e., homotopy equivalence).

We will study a number of algebraic invariants (i.e. ways to associate algebraic objects to topological ones in which homotopy equivalent spaces give isomorphic algebraic objects). For each one, we need to show how to construct the invariant and how to calculate it. It then follows that if the algebraic objects associated to two spaces are not isomorphic, then the spaces are not homotopy equivalent.
The first invariant we consider is the fundamental group: this is a group constructed from a space by looking at loops in the space. It captures geometry in a rather accessible way, but is generally hard to calculate. On the other hand, the homology groups capture higher dimensional information, and are easier to calculate but harder to define. Once these invariants are defined we will calculate them for a range of spaces and give a variety of applications: Brouwer and Lefschetz fixed point theorems, hairy ball theorem, Ham Sandwich Theorem, Football Squashing Theorem, proof of the Fundamental Theorem of Algebra,

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

Prerequisites: MAS220 (Algebra); MAS331 (Metric Spaces)
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. Hairy balls, ham sandwiches and squashed footballs

Part B

    1. Simplicial complexes
    2. Chain complexes and homology.
    3. Homology of simplicial complexes
    4. Chain homotopy and cones
    5. The Mayer-Vietoris sequence, and examples
    6. The Lefschetz Fixed Point Theorem
    7. Simplicial approximation
    8. Homotopy invariance.



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 simplicial homology.
  • Calculate the homology of some simple spaces such as spheres, products of spheres and surfaces.
  • Understand and apply the Lefschetz Fixed Point Theorem.

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 five questions and the format will be "answer four of the five 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.

Reading list

Type Author(s) Title Library Blackwells Amazon
B Armstrong Basic Topology 513.83 (A) Blackwells Amazon
B Fulton Algebraic Topology: A First Course 514.2 (F) Blackwells Amazon
B Hatcher Algebraic Topology [free download from author's webpage] 513.83 (H) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.