MAGIC064: Algebraic Topology

Course details

A core MAGIC course

Semester

Spring 2014
Monday, January 20th to Friday, March 28th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Timetable

Tuesdays
09:05 - 09:55
Thursdays
09:05 - 09:55

Announcements

MAGIC064 Algebraic Topology: Assessment
The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. The paper has three questions. All questions carry equal weight and each question will be marked out of a total of 25 points. To pass the exam you will need  at least 40 points out of the total of 75 points.
The assessment period is from 13/04-27/04/2014. The exam paper will be online as pdf-file on the MAGIC website from Sunday, April 13th at 00:00. The deadline for online submission (as pdf- file, typed or scanned) of solutions on the MAGIC website is: Sunday, April 27th, 2014 at 23:59.

Description

Algebraic topology studies `geometric' shapes, spaces and maps between them by algebraic means. An example of a space is a circle, or a doughnut-shaped figure, or a Möbius band. A little more precisely, the objects we want to study belong to a certain geometric `category' of topological spaces (the appropriate definition will be given in due course). This category is hard to study directly in all but the simplest cases. The objects involved could be multidimensional, or even have infinitely many dimensions and our everyday life intuition is of little help. To make any progress we consider a certain `algebraic' category and a `functor' or a `transformation' from the geometric category to the algebraic one. We say `algebraic category' because its objects have algebraic nature, like natural numbers, vector spaces, groups etc. This algebraic category is more under our control. The idea is to obtain information about geometric objects by studying their image under this functor. Now the basic problem of algebraic topology is to find a system of algebraic invariants of topological spaces which would be powerful enough to distinguish different shapes. On the other hand these invariants should be computable. Over the decades people have come up with lots of invariants of this sort. In this course we will consider the most basic, but in some sense, also the most important ones, the so-called homotopy and homology groups.

Prerequisites

Algebra: Groups, rings, fields, homomorphisms, examples
Standard point-set topology: topological spaces, continuous maps, subspaces, product spaces, quotient spaces, examples

Syllabus

Content:
Homotopy: fundamental group and covering spaces, sketch of higher homotopy groups.
Singular homology: construction, homotopy invariance, relationship with fundamental group.
Basic properties of cohomology (not excision or Mayer-Vietoris yet), motivated by singular cohomology.
Relative (co)homology.
Connecting homomorphisms and exact sequences.
Excision.
The Mayer-Vietoris sequence.
Betti numbers and the Euler characteristic.
Options for additional content:
Thom spaces and the Thom isomorphism theorem, Cohomology of projective spaces and projective bundles, Chern classes.
Relationship with existing courses:
The cohomology part is constructed from the current MAGIC011.

Lecturer

  • FN

    Dr Frank Neumann

    University
    University of Leicester

Bibliography

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Assessment

Description

The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. The paper has three questions. All questions carry equal weight and each question will be marked out of a total of 25 points. To pass the exam you will need  at least 40 points out of the total of 75 points.
The assessment period is from 13/04-27/04/2014. The exam paper will be online as pdf-file on the MAGIC website from Sunday, April 13th at 00:00. The deadline for online submission (as pdf- file, typed or scanned) of solutions on the MAGIC website is: Sunday, April 27th, 2014 at 23:59.

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