Istituzione di Geometria Superiore 1

 

 

topologia algebrica: omologia e coomologia singolare:

 

Simplessi e complessi simpliciali, omologia simpliciale, omologia singolare, complessi di catene, sequenze esatte, sequenze esatte brevi di complessi di catene, lunga sequenza esatta di omologia, simplessi piccoli, sequenza esatta di Mayer-Vietoris, excision, assioma di omotopia, assiomi dell’ omologia singolare;

 

omologia delle sfere, teorema del punto fisso di Brower, grado di mappe fra sfere, Jordan curve theorem e invariance of domain, isomorfismo tra omologia singolare e omologia simpliciale, caratteristica di Euler, classificazione e omologia delle superfici;

 

omologia con coefficienti e coomologia, funtori Tor e Ext, coefficienti universali per omologia e coomologia, dualità di Poincaré

 

 

algebraic topology: singular homology and cohomology:

 

simplices and simplicial complexes, simplicial homology, singular homology, chain complexes, exact sequences, short exact sequences of chain complexes, long exact homology sequence, small simplices, exact sequence of Mayer-Vietoris, excision, axiom of homotopy, axioms of singular homology;

 

homology of spheres, fixed point theorem of Brower, degree of maps between spheres, Jordan curve theorem and invariance of domain, isomorphism between singular and simplicial homology, Euler characteristic, classification and homology of surfaces;

 

homology with coefficients and cohomology, functors Tor and Ext, universal coefficients for homology and cohomology, Poincaré duality

 

 

Riferimenti (textbooks):

 

main textbooks:

 

J. R. Munkres, Elements of Algebraic Topology. Addison-Wesley Publishing Company 1984

 

A. Hatcher, Algebraic Topology. Cambridge University Press 2002

(http://www.math.cornell.edu/~hatcher/AT/ATpage.html)

 

 

additional references:

 

E. H. Spanier, Algebraic Topology. McGraw-Hill 1966

 

R. Stoecker, H. Zieschang, Algebraische Topologie. B. G. Teubner Stuttgart 1994