Course details
A core MAGIC course
Semester
 Spring 2011
 Monday, January 31st to Friday, April 1st; Tuesday, May 3rd to Tuesday, May 3rd
Hours
 Live lecture hours
 20
 Recorded lecture hours
 0
 Total advised study hours
 0
Timetable
 Tuesdays
 10:05  10:55
 Thursdays
 10:05  10:55
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. Relationship with existing courses:
The cohomology part is constructed from the current MAGIC011.
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. Relationship with existing courses:
The cohomology part is constructed from the current MAGIC011.
Lecturer

FN
Dr Frank Neumann
 University
 University of Leicester
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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