Course details
A core MAGIC course
Semester
 Spring 2022
 Monday, January 31st to Friday, March 25th; Monday, April 25th to Friday, May 6th
Hours
 Live lecture hours
 20
 Recorded lecture hours
 0
 Total advised study hours
 80
Timetable
 Tuesdays
 14:05  14:55
 Thursdays
 14:05  14:55
Description
Algebraic topology studies `geometric' shapes, spaces and maps between them by algebraic means. An example of a space is a circle, or a doughnutshaped 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 socalled homotopy and homology groups.
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 socalled homotopy and homology groups.
Prerequisites
Algebra: Groups, rings, fields, homomorphisms, examples.
Standard pointset topology: topological spaces, continuous maps, subspaces, product spaces, quotient spaces, examples.
Standard pointset topology: topological spaces, continuous maps, subspaces, product spaces, quotient spaces, examples.
Syllabus
 Introduction to the properties of cohomology rings
 Survey of some interesting spaces, and description of their cohomology (without proof)
 The construction of singular cohomology
 Basics of homotopy theory; homotopy invariance of cohomology
 Exact sequences and the Snake Lemma. Subdivision and the MayerVietoris sequence
 Cohomology of spheres. Applications: Invariance of Domain, the Brouwer Fixed Point Theorem, the Fundamental Theorem of Algebra.
 Proofs for the cohomology of various other spaces, developing various techniques along the way. Depending on the available time we may cover the Thom Isomorphism Theorem, the Projective Bundle Theorem, Chern classes and Euler classes for vector bundles, the theory of mapping degrees and Poincare duality.
Lecturer

Professor Neil Strickland
 University
 University of Sheffield
Bibliography
No bibliography has been specified for this course.
Assessment
The assessment for this course will be released on Monday 9th May 2022 and is due in by Sunday 22nd May 2022 at 23:59.
Assessment for all MAGIC courses is via takehome exam which will be made available at the release date (the start of the exam period).
You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).
If you have kept uptodate with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.
Please note that you are not registered for assessment on this course.
Lectures
Please log in to view lecture recordings.