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## General

This course is part of the MAGIC core.

#### 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.

#### Semester

Spring 2019 (Monday, January 21 to Friday, March 29)

#### Hours

• Live lecture hours: 20
• Recorded lecture hours: 0
• Total advised study hours: 80

#### Timetable

• Tue 09:05 - 09:55
• Thu 09:05 - 09:55

#### 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.
Thom spaces and the Thom isomorphism theorem, Cohomology of projective spaces and projective bundles, Chern classes.

## Lecturer

 Email fn8@le.ac.uk Phone 01162522722
Profile: My major research areas are algebraic topology and algebraic geometry. I am very much interested in interactions between these subjects and in particular in applications of homotopy theory to algebraic geometry. My recent interests are especially in homotopy and cohomology of moduli stacks. This research has also direct links with arithmetic geometry and mathematical physics.

## Bibliography

 Algebraic topology from a homotopical viewpoint Aguilar, Gitler and Prieto Algebraic topology tom Dieck Algebraic topology: a first course Fulton Algebraic Topology Book Hatcher A concise course in algebraic topology May A basic course in algebraic topology Massey Topology And Groupoids Brown Basic Topology Armstrong Elements of Topology T. B. Singh Homotopical Topology Fomenko, Fuchs

Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

## Assessment

The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. There will be 3 questions of 25 points each with 75 points in total and you will need the equivalent of 40 points to pass.

MAGIC064 Algebraic Topology Exam 2019

 Files: Exam paper Released: Sunday 28 April 2019 (148.3 days ago) Deadline: Sunday 12 May 2019 (133.3 days ago) Instructions: The exam paper has three questions. All questions carry equal weight and each question is marked out of a total of 25 points. To pass the exam you will need at least 40 points out of a total of 75 points. The deadline for online submission (as pdf-file, typed or scan of readable manuscript) of solutions on the MAGIC website is: Sunday, May 12th, 2019 at 23:59:59

## Files

Files marked L are intended to be displayed on the main screen during lectures.

Week(s) File 1-20 lectures1-4.pdf L 1-20 lectures1-4Handouts.pdf 1-20 lectures13-16.pdf L 1-20 lectures13-16Handouts.pdf 1-20 lectures17-20.pdf L 1-20 lectures17-20Handouts.pdf 1-20 lectures5-8.pdf L 1-20 lectures5-8Handouts.pdf 1-20 lectures9-12.pdf L 1-20 lectures9-12Handouts.pdf 1-20 magicalgtop.pdf