Course details
A core MAGIC course
Semester
- Spring 2021
- Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th
Hours
- Live lecture hours
- 20
- Recorded lecture hours
- 0
- Total advised study hours
- 80
Timetable
- Tuesdays
- 09:05 - 09:55 (UK)
- Thursdays
- 09:05 - 09:55 (UK)
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.
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.
Prerequisites
Algebra: Groups, rings, fields, homomorphisms, examples.
Standard point-set topology: topological spaces, continuous maps, subspaces, product spaces, quotient spaces, examples.
Standard point-set 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 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
-
FN
Professor 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
The assessment for this course will be released on Monday 10th May 2021 at 00:00 and is due in before Monday 24th May 2021 at 11:00.
The assessment for MAGIC064 Algebraic Topology will be a take-away-home exam to be set on May 10 with deadline for submission being May 23, 23:59 (UK time). In order to pass the exam at least 50% need to be obtained.
Please note that you are not registered for assessment on this course.
Files
Only current consortium members and subscribers have access to these files.
Please log in to view course materials.
Lectures
Please log in to view lecture recordings.
