Course details
A core MAGIC course
Semester
 Spring 2019
 Monday, January 21st to Friday, March 29th
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
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.
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
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 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.
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
Description
The assessment for this course will be via a single takehome 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.
Assessment not available
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Lectures
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