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.

**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.

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

E-book in **unimib.it**
domain.

Calendario appelli (di verbalizzazione esame, studio docente): INSEGNAMENTO: ARGOMENTI DI GEOMETRIA E TOPOLOGIA CODICE: F4001Q083 2017-02-14 10:00 2017-03-14 10:00 2017-04-11 10:00 2017-06-20 10:00 2017-07-18 10:00 2017-09-12 10:00 + su appuntamento

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

$8 \text{CFU} \times 7 \dfrac{\text{h}}{\text{CFU}} = 56 \text{h} = 28 \times 2 \text{h}$

- 2016-12-14 : Robert Ghrist, Barcodes: The persistent topology of data
- 2016-11-25 : Fundamental group of SO(3) and Dirac's scissors
- 2016-11-23 : ATTENZIONE: La lezione del venerdì 2 dicembre 2016 è rimandata (di tanto).
- 2016-11-21 : Homology of (simplicial spheres) and applications
- 2016-11-18 : Mesh, Barycentric Subdivision and Simplicial Approximation Theorem
- 2016-11-09 : Joins and Products of polyhedra
- 2016-11-09 : Ranks, Tensors and Abelian Groups
- 2016-11-04 : Graphs, trees and simplicial complexes
- 2016-10-27 : Computing the Smith Normal Form of a matrix (and homology?)
- 2016-10-14 : Extended note on Csaszar torus
- 2016-10-14 : Extended note on coproducts and products
- 2016-10-06 : Extended note on the counter-example for adjoint spaces.

OCT 03 *01 Introduction. Topology of function spaces: compact-open topology. Adjoint of a map. 05 *02 Counter-example for the adjunction map property. Continuity of the composition and avaluation map for locally compact Hausdorff spaces. 07 *03 Introduction to categories: objects and morphisms. Examples. Homotopy and relative homotopy: the fundamental groupoid. 10 *04 H-Top. Functors (covariant and contravariant). Examples. Free abelian group generated by a set, as functor. Hom (B,-) and Hom(-,B) as functors. 12 *05 Categorical co-products and products in Ab, Grp, Top, Top_*, ... Examples. 14 *06 Coproducts and comma category; posets; push-outs in Set and Top. 17 *07 Simplicial complexes: euclidean, abstract, and categorical. 19 *08 Examples of complexes: sphere, torus. Euler formula. Csaszar Torus. 21 *09 Geometric realization functor. Introduction to embeddings (of graphis). 24 *10 Immersions and embedding (linear). Pre-example of a chain complex (of $S^1$). 26 *11 Smith normal form and chain complexes: an example. 28 *12 Computational Smith Normal Form. Introduction to chain complexes. 31 *13 [YES] => [SEMI-SKIP] Question time. Just a few students: a long example of homotopy with and without mistakes. (it does not count as lect.) NOV 02 *14 Chain complexes, d^2=0, definition of simplicial homology groups. 04 *15 Computing homology. Graphs and trees. 07 *16 Homology of a graph. Euler-Poincaré characteristic (introd). Rank, dim, tensor product? 09 *17 An algebraic digression: rank, tensor product, free groups, exact sequence and Euler-Poinceré characteristic formula. Chain complex of a pair. 11 *18 Homology of a pair and LES of the pair. Mayer-Vietoris Long Exact Sequence. Join of simplicial complexes, of sphere, cone. Chain homotopy and homology of a cone. 14 *19 Homology of a cone. Functoriality and chain homotopy. Homology groups of spheres. 16 *20 Product of simplicial complexes and geometric realizations (proof). 18 *21 Chain homotopy. Barycentric subdivision. Mesh. 21 *22 Simplicial Approximation Theorem. Homology of maps and as functor on Top and HTop subcategory. Singular homology. 23 *23 Tor functor, tensor product and applications: Homology with Coefficients, Universal Coefficients Theorem and Künneth Theorem 25 *24 Computing homology of SO3, P^nR, cellular/block homology. Dirac's scissors. Fundamental group and homology. 28 *25 Homology of P^nR with coefficients in Z and Z2. Cohomology and left-exact hom-functor. Ext. UCT in cohomology. 30 *26 Z2-Cohomology of P2. Cup and cap products, ring structure. (cellular) homology of torus, Klein bottle, P2, torus of genus 2. DEC 02 *27 [SKIP] - "Didattica e Saperi Disciplinari conference" 05 *28 Cellular homology of closed surfaces. Classification theorem. Sketch of the proof. 07 *29 [NO: SANTAMBROGIO] 09 [NO] VAC. 12 *30 (27) Cutting surfaces. Fundamental group. Poincaré duality. Applications. 14 *31 (28) => PSEUDO-ULTIMA LEZIONE: LS Category and homology of projective spaces. Borsuk-Ulam and odd degrees. Hint of Data analysis and persistent homology. 16 *32 19 *33 21 *34 JAN 09 *35 11 *36 13 *37 16 *38 18 *39 [VERY LAST (?)] 20 *40 23 *41 25 *42 27 *43