Algebraic Topology (MAGIC064) |
GeneralThis course is part of the MAGIC core. Description
Algebraic topology studies `geometric' shapes, spaces and maps
between them by algebraic means.
An example of a space is a circle, or a doughnut-shaped 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 so-called homotopy and homology
groups.
SemesterSpring 2019 (Monday, January 21 to Friday, March 29) Hours
Timetable
PrerequisitesAlgebra: Groups, rings, fields, homomorphisms, examples
Standard point-set topology: topological spaces, continuous maps, subspaces, product spaces, quotient spaces, examples SyllabusContent:
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 Mayer-Vietoris yet), motivated by singular cohomology. Relative (co)homology. Connecting homomorphisms and exact sequences. Excision. The Mayer-Vietoris 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
Bibliography
Note: Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will 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.) AssessmentThe assessment for this course will be via a single take-home 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.
MAGIC064 Algebraic Topology Exam 2019
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