Course details
A core MAGIC course
Semester
 Spring 2021
 Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th
Hours
 Live lecture hours
 20
 Recorded lecture hours
 0
 Total advised study hours
 80
Timetable
 Tuesdays
 09:05  09:55
 Thursdays
 09:05  09:55
Course forum
Visit the MAGIC064 forum
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 MÃ¶bius 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.
An example of a space is a circle, or a doughnutshaped figure, or a MÃ¶bius 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.
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
 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 MayerVietoris yet), motivated by singular cohomology.
 Relative (co)homology.
 Connecting homomorphisms and exact sequences.
 Excision.
 The MayerVietoris 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.
Lecturer

FN
Dr Frank Neumann
 University
 University of Leicester
Bibliography
Follow the link for a book to take you to the relevant Google Book Search page
You may be able to preview the book there and 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.
 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, )
Assessment
Attention needed
Assessment information will be available nearer the time.
Files
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Lectures
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