Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMG1003	Algebraic and Differential Topology	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	3	2+2+0	speciality	optional	

Teaching Staff in Charge	Lect. TOPAN Liana Manuela, Ph.D., ltopan@math.ubbcluj.ro	
Aims	The purpose of the course is to provide students with basic notions and classical results in algebraic and differential topology.	
Content	Locally Euclidean spaces, topological manifolds, differential manifolds. Examples of differentiable manifolds. Tangent spaces. Equivalent definitions for the tangent space of a differentiable manifold. The tangent map. The tangent bundle of a differentiable manifold. Vector bundles; examples, operations. The partition of unity. Submanifolds. The rank theorem. The inverse function theorem. The Sard@s theorem. The Whitney@s theorem. 
Manifolds with boundary. Submanifolds of the manifolds with boundary. The degree modulo two of a map.
Elements of Riemannian geometry. The existence of Riemann metrics on a manifold. Quotient topologies. Examples of quotient spaces. Attaching cells.
The groups &#960;_n(X, x_0). The fundamental group. The homotopic invariance of the homotopy groups. The computation of the fundamental group for the n-dimensional sphere and for the n-dimensional torus. Brower@s fixed point theorm. The fundamental theorem of algebra. Groups of relative homotopy. The exact homotopy sequence of a topological pointed pair. Covering spaces. Applications of the covering spaces for the computations of certain homotopy groups.
	
Bibliography	1. Conlon, L., Differentiable Manifolds, Birkhäuser, 2001
2. Craioveanu, M., Introducere în geometria diferenţială, Editura Mirton, 2004 
3. Gheorghiev, Gh., Oproiu, V., Varietăţi diferenţiabile finit şi infinit dimensionale,  
                                Editura Academiei, 1976
4. Hatcher, A., Algebraic Topology, Cambridge University Press, 2002
5. Nicolaescu, L.I., Lectures on the Geometry of Manifolds, World Scientific, 1996
6. Pop, I., Topologie algebrică, Editura Stiinţifică, Bucureşti, 1990
7. Postnikov, M., Leçons de géometrie. Varietés différentiables., MIR, 1990
8. Sandovici, P., Ţarină, M., Geometrie Diferenţială, Litografia UBB, Cluj-Napoca, 1974
9. Sharpe, R.W., Differential Geometry, Springer, 1996
	
Assessment	During the semester, th students will present individual homeworks, which provide 70% of the final grade. An oral exam contributes with the remaining 30%.