Argomenti di Geometria e Topologia (2017/18-1S)

# Syllabus (preliminary)

The aim of the course is to take some classic topics in algebraic topology of simplicial complexes, introducing homology theory, cohomology theory and some aspects of homotopy theory. Contents: Simplicial complexes, homology and cohomology of polyhedra, triangulable manifolds, homotopy groups.

2017-11-17: La lezione purtroppo è rimandata a dicembre (vedi calendario sotto).

Prerequisites: Basic topics covered in bachelor courses of geometry and algebra

Main topics: 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.  Applications: computational homology, persistent homology, data analysis, dynamical systems.

# Book / Monography

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

Springer Page

E-book in unimib.it domain.

# Calendar

Calendario appelli (di verbalizzazione esame, studio docente):

INSEGNAMENTO: ARGOMENTI DI GEOMETRIA E TOPOLOGIA
CODICE: F4001Q083

2018-02-13 10:00
2018-03-13 10:00
2018-04-10 10:00
2018-06-19 10:00
2018-07-12 10:00
2018-09-11 10:00

+ su appuntamento



## Lectures

Lunedì    0830 - 1030, aula U5-2107 (cum tempore)
Mercoledì 1030 - 1230, aula U5-2109 (cum tempore)
Venerdì   0830 - 1030, aula U5-2109 (com tempore)


OCT
02 *01 Intro. Euclidean simplicial complexes.
04 *02 Abstract simplicial complex. Examples: Undirected graphs, K3,3, S^n,D^n. Join of complexes. Cone and suspension. Geometric realization of a complex.
06 *03 Simplicial maps. Geometric realization of a simplicial map. Example of a simplicial torus (homework).

09 *04 Csaszar torys (hint). Embedding of simplicial complexes in $R^{2n+1}$ and Vandermonde matrix/determinant. Simplicial chains and boundary.
11 *05 Simplicial boundary and simplicial chain complexes. Cycles, boundaries and homology groups. Examples (to compute the boundary of the $2$-simplex, i.e. $S^1$, as homework).
13 *06 Graphs and simplicial complexes: homology of a graph.

16 *07 Spanning tree, homology of a connected graph. Euler characteristic of a graph.
18 *08 Smith Normal Form and finitely generated abelian groups.
20 *09 Rank and tensor products of (f.g.) abelian groups. Structure theorem for f.g. abelian groups. Examples.

23 *10 Exact sequences. Introduction to category theory: categories and functors. Exact functor, tensoring by rationals. Euler Characteristic and exact sequences.
25 *11 Functors and categories: singular chain complex, chain complex. Homology functor (homology of a map). Diagram chasing: long exact sequences and zig-zag lemma.
27 *12 Snake lemma and proof of the L.E.S. Applications: Homology of a pair of simplicial complexes.

30 *13 Mayer-Vietoris sequence. Cones and Homology of simplicial spheres. Homology of a cone: chain homotopy.

NOV
01 *VAC* TUTTI I SANTI
03 *14 Categorical products and coproducts. Universal properties in Set, Ab, Gr, Top. Product of simplicial complexes and cylinder.

06 *15 Product of simplicial complexes, geometric realization. Induced morphism in simplicial chain complexes. Chain homotopy of i0 and i1.
08 *16 Mesh size of a euclidean simplicial complex. Subdivisions and barycentric subdivision. Mesh size of a barycentric subdivision.
10 *17 -STRIKE- POSTPONED

13 *18 Simplicial approximation theorem and Homology functor on the category of polyhedra and continuous functions.
15 *19 Homology of spheres and applications: Borsuk Ulam type theorems and degree (1)
17 *20 -PROBLEM- POSTPONED

20 *21 Homology of spheres and applications: Borsuk Ulam type theorems and degree (2)
22 *22 Tucker lemma, Borsuk-Ulam and triangulated antipodal spheres.
24 *23 Tensor product and hom-functors, half-exact functors. Homology with coeff & cohomology. Tor & Ext functors.

27 *24 Universal coefficients theorem. Kunneth theorem. Projective spaces, torus, examples of cellular homology.
29 *25 Homology and cohomology of real projective spaces. Cup product and cohomology ring (of P^n). Direct sum (MV) and cellular decomposition of orientable surfaces: homework.

DEC
01 *26 Direct sum and cellular homology of surfaces. Examples. Polygons.

04 *27 Classification theorem of closed surfaces: homework (induction on Euler characteristic).
06 *28 Homology, Poincaré Duality, Classification of surfaces and Poincaré homology sphere.
08 *VAC* IMMACOLATA

11 (rec) -POST-STRIKE- lecture *17 Topological Data Analysis: Ribs, Cech and filtration complexes.
13 (rec) -POST-PROBLEM- lecture *20 TDA: Persistent Homology, module structure, example and applications.
15 (rec) [OUT OF OFFICE]

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