A:C&T

Time and place: Thursdays, 8:45-12:15 , U Fribourg Pérolles, "Lonza" building.

Description: The study of topological objects associated to combinatorial structures is a relatively recent but fast developing two-way success story. Topological obstructions have played a major role in the solution of famous combinatorial problems (e.g., the Kneser conjecture) and, conversely, combinatorial techniques (e.g., discrete Morse theory) have led to new results in topology, both of computational and theoretical nature. This lecture takes mainly the second point of view and aims at offering an introduction to some techniques for the study of topological spaces with a strong combinatorial structure. Our main examples will include matroid complexes, order complexes of posets as well as subspace arrangements and their complements. A tentative list of the techniques that will be covered includes Quillen-type theorems, shellings, discrete Morse theory, acyclic categories. The syllabus can be adapted to the interests and the background of the participants.

Format: Lecture with some exercises. For those who need credit points, an oral examination. The exact format will be discussed during the first meeting.

Contact: SNSF-Prof. Emanuele Delucchi, emanuele.delucchi "at" unifr.ch

Lecture 1, February 21: introduction, simplicial complexes.

Bibliography:
J. Munkres, Elements of Algebraic Topology, Addison-Wesley, § 1 and § 2.
M. de Longueville, A course in topological combinatorics, Springer, Chapter B3.

Lecture 2, February 28: polytopal complexes, barycentric subdivisions.

Bibliography: see Lecture 1

Lecture 3, March 7: Partially ordered sets.

Bibliography: E.g., Stanley, Enumerative Combinatorics, vol. I, Chapter 3.

Lecture 4, March 14: Poset homotopy theorems.

Bibliography:
E. Spanier, Algebraic topology, Springer. [pp. 22-27] {Department's library code: D-1-75}
J. Walker, Homotopy type and Euler characteristic of partially ordered sets, European journal of combinatorics 9 (1988), 97-107. [Section 2]

Lecture 5, March 21: Poset homotopy theorems II

Lecture 6, March 28: Arrangements of hyperplanes.

Lecture 7, April 4: Shellable simplicial complexes

Lecture 8, April 11: Matroid complexes.

Lecture 9, April 18: Matroid complexes II

Lecture 10, May 2: CW complexes and posets.

Lecture 11, May 9: Shellability of regular CW complexes

Lecture 12, May 16: Discrete Morse theory

Lecture 13, May 23.