Course details
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
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Description
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
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.
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Lecturer

NS
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
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