MATH 3024: Homotopy and Surfaces


Semester: 2, Credits: 15
Prerequisites: MATH1022 (Introductory Group Theory) AND
either MATH1035 (Analysis) or MATH1050 (Calculus and Mathematical Analysis)
Please see me if are an Erasmus student, or you have not got these prerequisites for some other reason.

Module details: 2011--12

Lecturer: Prof. J.C. Wood, 8.20e, Mathematics; e-mail: j.c.wood@leeds.ac.uk

Module home page: http://www1.maths.leeds.ac.uk/Pure/staff/wood/MATH3024/MATH3024.htm
which should also be accessible from Blackboard - VLE: https://vlebb.leeds.ac.uk/webapps/portal/frameset.jsp

Lectures:
Mondays: 0900--1000, Roger Stevens LT 3
Wednesdays: 1200--1300, Roger Stevens LT 16
Thursdays: 1600--1700, Roger Stevens LT 16

Five of these sessions will be workshops. If you want help, please come and see me: fix a time when you see me at lectures or by e-mail.

Attendance monitoring From the School of Mathmematics webpages: http://www.maths.leeds.ac.uk/school/students/Student_Attendance.html where you can find more details:
`Students are expected to attend all teaching activities, including examples classes.
Attendance at lectures will be monitored on several randomly chosen ocassions chosen by the administrative staff.

Module description: http://webprod1.leeds.ac.uk/banner/dynmodules.asp?Y=201112&M=MATH-3024

Booklist
1. L.C.Kinsey, Topology of surfaces, Springer, 1993/1997
2. C. Kosniowski, A first course in algebraic topology, Cambridge Univ Press, 1980
3. M. A. Armstrong, Basic Topology, McGraw-Hill, 1979
4. H. B. Griffiths, Surfaces, Cambridge Univ Press, 1976
5. E.D.Bloch, A first course in geometric topology and differential geometry, Birkhäuser, 1997.

Assessment: 100% examination (2.5 hours long) at end of semester.

Private Study: There are 33 hours of lectures; about 5 of these will be Examples Classes. There are 117 hours of private study, so you should expect to spend about 10 hours a week on private study for this module.




Syllabus
1. Subsets of products and quotients of such subsets, continuous maps and homeomorphisms between these.
2. Polyhedral surfaces. Representation by sentences. Equivalent sentences. Classification of sentences by canonical words.
3. Geometrical description of surfaces represented by canonical words, Euler characteristic.
4. Fundamental group. Path space. Homotopy of paths. Composition of paths.
5. Homotopies of maps. Deformation retracts.
6. Calculation of (S1). Path lifting theorem. Homotopy lifting theorem.
7. Applications from among: Brouwer's fixed point theorem. Borsuk-Ulam theorem. Fundamental theorem of algebra. Jordan curve theorem, Pancake and Ham Sandwich theorems.
8. Computations of fundamental groups. Finitely presented groups. Van Kampen's theorem. Classification of surfaces.