Middle East Technical University   Department of Mathematics     Spring 2002
Elementary Geometric Topology
Math 422

Homework I (deadline: March 1 )    dvi-file     ps   ps-zipped
Homework II ( deadline: March 1 5)    dvi-file    ps     ps-zipped

Homework III (deadline: April 9 )    dvi-file    ps     ps-zipped

Homework IV (deadline: April 22)    dvi-file    ps     ps-zipped
Midterm I       dvi-file   ps   ps-zipped           Results

Midterm II    



Instructor Prof. Sergey Finashin

Reference Books

E.D.Bloch    A First Course in Geometric Topology and Differential Geometry, Birkhauser 1997
(Chapters 1-3 only)

I.M.Singer, J.A.Thorpe    Lecture Notes on Elementary Topology and Geometry, Springer (Chapters  1-4)

V.V.Prasolov    Intuitive Topology,    AMS 1995

D.W.Farmer, T.B.Stanford    Knots and Surfaces (A Guide to Discovering Mathematics),   AMS 1996

N.D.Gilbert, T.Porter      Knots and Surfaces,   Oxford Univ. Press 1995



Syllabus     

1. Plane Curves
- homeomorphisms and ambient homeomorphisms of sets in Euclidean space
- topological circles and arcs, Jordan theorem, the rooted tree and the code of a multi-component plane curve
- the Besout theorem for algebraic curves
- the winding number (index with respect to a point) and the rotation number (the Whitney index) for immersed curves
- the chord diagrams for generic immersed curves
- Reidemeister's moves and Arnold's invariants
Homework I

2. Knots and Links
- knot and links, their diagrams, Reidemeister moves, characterization of unknots and unlinks
-the notion of a knot invariant, the Gordion (unknotting) number, the minimal crossing number
- connected sums, prime and composite knots
- examples of knots (torus knots, cable knots,Whitehead double of a knot)
 - linking number
Homework II

3. Spans for knots and links
- construction of spans (Seifert surfaces), orientation of a surface, the genus of a knot
- generating cycles on a span, Seifert matrix
- the Alexander polynomial and signature of a knot
- the Arf invariant

4. Braids
- presentation of knots as closed braids (Alexander's theorem), colored and non colored braids
- the Braid group, its generators and relations
Homework III

5. Surfaces
- surfaces with and without boundary, orientability,
- connected sums, the genus (as a number of handles), classification
- cutting and pasting surfaces
- sketching graphs on surfaces and their Euler characteristic
- example: recognizing spanning surfaces of knots
- presentation of Seifert surfaces by graphs, calculation of the Seifert form for a graph

Midterm I   March 30
6. Vector Fields on Surfaces
-  vector fields, the index of  a singularity, relation to the Euler characteristic (proof for the gradient vector fields)
- the spine graph for a gradient vector field on a surface

Homework IV

7. Point Set Topology in an Euclidean Space
- definition of a topological space, and of the induced subspace topology on a subset, metric spaces
- position of a point with respect to a set (interior, exterior, boundary)
- the operations Int, Cl, Fr (interior, closure, boundary)
- open and closed sets, relatively open and closed sets
- continuous maps, open maps
- homeomorphisms,
- topological properties, compactness and connectedness

8. Constructions  
- Products, wedges, disjoint unions, cylinders, cones, suspensions, joins,
- contraction of a subset, quotient map, partitions, quotient space (its topological invariance)
- gluing of two sets along subsets, gluing via a map,
- presentation of  two-dimensional cell complexes by a set of words (gluing from polygons)
Midterm II      (Take-home) May 10 - 17
9. Simplicial Complexes and Cellular Complexes

10. Homology Groups and the Fundamental Group of  Simplicial and Cellular Complexes