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General


This 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.
Semester

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Tue 09:05 - 09:55
  • Thu 09:05 - 09:55

Lecturers


Frank Neumann (main contact)
Email fn8@mcs.le.ac.uk
Phone 01162522722
vcard
Photo of Frank Neumann
Profile: My major research areas are algebraic topology and algebraic geometry. I am very much interested in interactions between these subjects and in particular in applications of homotopy theory to algebraic geometry. My recent interests are especially in homotopy and cohomology of moduli stacks. This research has also direct links with arithmetic geometry.
John Hunton
Email jrh7@mcs.le.ac.uk
Phone (0116) 2525354
Interests Geometry and topology, quasicrystals and aperiodic order
vcard
Photo of John Hunton
Andrey Lazarev
Email al179@le.ac.uk
Phone (0116) 2523892
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Photo of Andrey Lazarev


Students


Photo of Bana Al Subaiei
Bana Al Subaiei
(Southampton)
Photo of Yumi Boote
Yumi Boote
(Manchester)
Photo of Tom Brookfield
Tom Brookfield
(Birmingham)
Photo of Giovanni Collini
Giovanni Collini
(Cardiff)
Photo of Chris Draper
Chris Draper
(York)
Photo of Aiman Elragig
Aiman Elragig
(Exeter)
Photo of Matthew Gadsden
Matthew Gadsden
(Sheffield)
Photo of Russell Goddard
Russell Goddard
(Nottingham)
Photo of Asma Ismail
Asma Ismail
(Exeter)
Photo of Andrew Jones
Andrew Jones
(Sheffield)
Photo of Jonas Lorenz
Jonas Lorenz
(Manchester)
Photo of Vladimir Lukiyanov
Vladimir Lukiyanov
(Loughborough)
Photo of Umberto  Lupo
Umberto Lupo
(York)
Photo of Liangang Ma
Liangang Ma
(Liverpool)
Photo of David OSullivan
David OSullivan
(Sheffield)
Photo of Linyu Peng
Linyu Peng
(Surrey)
Photo of Jeevan Rai
Jeevan Rai
(Surrey)
Photo of Robert Royals
Robert Royals
(East Anglia)
Photo of Dragomir Tsonev
Dragomir Tsonev
(Loughborough)
Photo of David Ward
David Ward
(Manchester)
Photo of David Wilding
David Wilding
(Manchester)
Photo of Marco Wong
Marco Wong
(Leeds)


Prerequisites


Algebra: Groups, rings, fields, homomorphisms, examples
Standard point-set 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 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.
Relationship with existing courses:
The cohomology part is constructed from the current MAGIC011.

Bibliography


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
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.)

Assessment


There will be a final exam at the end of the course to be marked with pass/fail. More details to be announced at later stage.

Assignments


MAGIC064Exam

Files:Exam paper
Deadline: Friday 27 April 2012 (722.1 days ago)
Instructions:MAGIC064 Exam:
This paper has four questions all of which should be attempted. All questions carry equal weight and each question is marked out of a total of 25 points. To pass the exam you will need > 50 points out of the total of 100 points.
The deadline for submission of solutions on the MAGIC website is: Friday, April 27th, 2012 at 23:59 p.m.