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Argomenti di Geometria e Topologia (2014-1S)

Testo / Monografia

Ferrario, Piccinini: Simplicial structures in topology. CMS Books in Mathematics, Springer, New York, 2011. xvi+243 pp. ISBN: 978-1-4419-7235-4

Pagina Springer

E-book scaricabile dal dominio unimib.it.

Calendario

Syllabus (preliminary)

Fundamental concepts: topological spaces, connectedness, compactness, function spaces, general ideas on Categories, push-out diagrams. Euclidean and abstract simplicial complexes. Introduction to homological algebra. Homology with coefficients. Category of polyhedra. Cohomology of polyhedra. Cohomology ring, cap product. Triangulable manifolds. Surfaces and classification. Poincaré Duality. Fundamental group of polyhedra. Fundamental group and homology. Homotopy groups. Obstruction theory.

Calendario

Calendario appelli (di verbalizzazione esame, studio docente):

2014-02-18 1100 
2014-06-17 1100 
2014-07-01 1100
2014-07-15 1100
2014-09-16 1100
2015-02-17 1100
+ su appuntamento 

Lezioni

OCT:
 02 *1 Intro; Function spaces; Compat-open topology; Loc. cpt. spaces and adjoints. 
 07 *2 Duality and adjoint in function spaces; quotient maps; Categories.
 09 *3 Htop; Functors; Suspension and cylinder functors; Colimits. 
 14 *4 Universal property; Push-out; Co-equalizer and coproducts and existence of pushout. 
 16 *5 [Grad-Day]
 21 *6 Euclidean and abstract simplicial complexes. Join, cone and suspension. 
 23 *7 Simplicial maps, ordered complexes and product of simplicial complexes. 
 28 *8 Examples of ordered complexes (Torus with 7 vertices, sphere); Simplicial sets. 
 30 *9 Simplicial set associated to an abstract simplicial complex. Faces and degeneracies. 

NOV:
 04 *10 Simplicial sets, geometric realization and direct product. 
 06 *11 Abstract simplicial complex and chain complex. 
 11 *12 Homology functor. 
 13 *13 Homology groups: some examples (homology of *, S^0, \Delta^2, \boundary \Delta^2, a graph). Augmentation \epsilon. 
 18 *14 Trees, graphs and generators of H_1 of a connected graph. 
 20 *15 Reduced homology, homology of disjoint union, $H_0$ and $\tilde H_0$; homology of the torus. 
 25 *16 Long Exact Sequence of a short exact sequence of chain morphisms, connecting homomorphism, diagram chase, homology of a pair. 
 27 *17 Mayer-Vietoris long sequence, homology of spheres. 

DEC:
 02 *18 Chain complex homotopy, homology of a cone, homotopy and homology.
 04 *19 Chain complex homotopy (direct definition, acyclic carriers) of $K\times I$ cylinder.
 09 *20 Homology of polyhedra: barycentric subdivision, simplicial approximation theorem. 
 11 *21 Simplicial approximation theorem and consequences.
 16 *22 Homology of retractions, deformation retractions, contractible spaces. S^n. EP-characteristic is homotopy-type invariant.  
 18 *23 Tor functor, tensor product, Universal Coeff. Thm, Kunneth thm, Hom functor. 

JAN:
 08 *24 Hom functor, Ext, Cohomology and UCT in cohomology. Example: k-forms. 
 13 *25 Cup product. Example: 2-Torus.
 15 *26 Skew-commutativity in 2-torus and concrete cup products. 
 20 *27 Cap product. Manifolds. Connected sums. Closed surfaces and their classification. 
 22 *28 Cellular/block homology and classification of surfaces. 
 27 *29 Closed surfaces, orientability, Poincaré duality.
 29 *30 Fundamental group, homology and Poincaré homology sphere.