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

Code	Subject	
MMG1004	Morse Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	4	2+1+0	speciality	optional	

Teaching Staff in Charge	Assoc.Prof. PINTEA Cornel, Ph.D., cpintea@math.ubbcluj.ro	
Aims	he course presents the basic facts of Morse theory and some of its applications both for real valued and circular valued functions. Specifically, we pay some attention towards the spherical homotopic structure of differential manifolds and the Morse inequalities, as well as the Morse complex of a real valued Morse function and the Novikov complex of a circular Morse function.	
Content	I. ELEMENTS OF HOMOLOGY 
   1.1 Chain complexes
   1.2 The homology of chain complexes
   1.3 Exact sequences
   1.4 Betti numbers. Euler-Poincare characteristic

II. BASES OF MORSE THEORY
   1.1 The Morse lemma
   1.2 Passing a critical level
   1.3 The spheric homotopic structure of manifolds
   1.4 Morse inequalities

III. THE MORSE COMPLEX OF A MORSE FUNCTION
   3.1 The Morse complex for tranversal gradients
   3.2 The Morse complex for almost tranversal gradients
   3.3 The chain Morse equivalence 

IV CIRCLE-VALUED MORSE MAPS AND NOVIKOV COMPLEXES
   4.1 Completions of rings, modules and complexes
   4.2 Chain complexes over A[[t]]
   4.3 The Novikov comnplex of a circle-valued Morse map	
Bibliography	1. Burghelea D., Hangan, Th., Moscovici, H., Verona, A., Introducere in Topologia  
   Diferentiala, Editura Stiintifica, Bucuresti 1973.
2. Pajitnov, A.V., Circle-valued Morse Theory, Walter de Gruyter, 2006
3. Pintea, C., Geometrie. Geometrie Diferentiala. Geometrie Riemanniana. Grupuri si Algebre  
   Lie, Presa Universitara Clujeana, 2006
	
Assessment	Exam+Quiz+Reports