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
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.
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

NS
Prof Neil Strickland
 University
 University of Sheffield
Bibliography
No bibliography has been specified for this course.
Assessment
Attention needed
Assessment information will be available nearer the time.
Lectures
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