Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Algebră şi Geometrie (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MG265	Elements of Algebraic and Differential Topology	

Semester	Hours: C+S+L	Category	Status	
1	2+1+1		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
ANDRICA Dorin	Ph.D.	Prof.	Geometry	*	*		

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The main purpose of this course consists in presentation of the basic problems of Differential Topology. Some important results concerning the smooth manifolds and smooth mapping as well some applications are given. Some results obtained by Chair of Geometry in the study of critical points of mappings beetwen differentiable manifolds are included in chapter 3.	

IV. Course contents
1. The global theory of smooth mappings between manifolds. Global constructions of 
smooth mappings (smooth partition of unity). Differentiable submanifolds. Manifolds
with boundary. Homotopy and isotopy of smooth mappings. The modulo 2 degree and some applications.
2. Fundamental results in Differential topolgy. Sets of zero measure on a manifold.
Sard'theorem and applications. Whitney'embeding theorem.
3. The study of critical points of mappings between manifolds. The critical set and the bifurcation set. Sufficient conditions for the infiniteness of the critical set. The 
G-equivariant and G-invariant cases.Some geometric applications.	

V. Bibliography
1. ANDRICA,D., Critical Point Theory and Some Applications, University of Ankara,1993
2. ANDRICA,D.,PINTEA,C., Spatii de acoperire cu aplicatii in teoria punctului critic, Universitatea "Babes-Bolyai" Cluj-Napoca, va apare
3. ANDRICA,D.,PINTEA,C., Functii cu numar minim de puncte critice, Presa Universitara Clujeana, in pregatire
4. CONLON,J., Differentiable Manifolds.A First Course, Birkhauser,1993
5. JOST,J., Riemannian Geometry and Geometric Analysis, Springer Verlag,1995	

VI. Thematic of didactic activities per weeks
Course 1.Smooth manifolds:locally Euclidean spaces, topological manifolds, smooth manifolds.
Seminar.Examples of manifolds.

Course 2.The tangent space.The tangent map.
Seminar.Equvalent definitions for the tangent space.

Course 3.The tangent bundle of a manifold.
Seminar.Vector bundles.Examples.

Course 4.Smooth partition of unity.
Seminar.Operations with vector bundles.

Course 5.Submanifolds.
Seminar.The rank theorem.The locally diffeomorphism theorem.

Course 6.Manifolds with boundary.
Seminar.The Sard’s theorem.

Course 7.Submanifolds of manifolds with boundary.
Seminar.The Whitney’s theorem .

Course 8.Modulo 2 degree of a mapping.
Seminar. Working paper.

Course 9.Elements of Riemannian geometry.
Seminar.The existence of Riemannian metrics on a manifold.

Course 10.Quotient topologies.
Seminar.Examples of quotient spaces.

Course 11.The fundamental group of a topological space.
Seminar.Coputations of the fundamental group.

Course 12.The homotopic invariance of the fundamental group.
Seminar.The fixed point theorem of Brower and the fundamental theorem of Algebra.

Course 13.Relative homotopy groups.
Seminar.Exact homotopy sequence of a topological pair.

Course 14.Fiber spaces and covering spaces.
Seminar.Applications.

VII. Didactic methods used
Presentations, dialogs, proofs, problem-solving, reports.	

VIII. Assessment
The final evaluation is given as follows:
-	the working paper during the semester 20%
-	the evaluation of the reports during the semester 10%
    -     the final exam 70%	

IX. Additional references
1.Burghelea,D.,s.a.,Introducere in topologia diferentiala,Ed.Stiintifica si enciclopedica,Bucuresti,1972.
2.Berger,M.,Gostieaux,B.,Geometrie Differentiale,Armand Colin,1972.
3.Gauld,D.,Differential Topology.An Introduction,Marcel Dekker,1982.
4.Guillemin,V.,Pollak,A.,Differential Topology,Prentice Hall,1974.
5Jost,J.,Riemannian Geometry and Geometric Analysis,Springer Verlag,1995.
6.Jurchescu,M.,Introducere in analiza pe varietati,Universitatea din Bucuresti,1980.
7.Pop,I.,Topologie algebrica,Ed.Stiintifica,Bucuresti,1990.

	Dean,							Chair head,

	Date,October 2/2008						Course responsible,
								Prof. ANDRICA Dorin, Ph.D.