The language of the lecture will be Czech or English depending on the audience. (If there is an attendee who does not understand Czech, the lecture will be in English).
Research articles:
| Lecture | Contents | Literature |
|---|---|---|
| Lecture 1, 23.2. (MB) |
Introduction. Kneseger's graph KG(n,k), chromatic number of Kneser's graphs (just statement), the Ham Sandwich Theorem. fair cutting of convex bodies. Basic definitions from general topology: topological space, open set and closed set, Hausdorff space, subspace, continous mapping, homeomorphism, closure, interior and boundary, compact sets, connectivity, path-connectivity. (Strong) deformation retract.
Presentation [PDF],notes (in Czech) [PDF]. |
[M, Chap. 1]. |
| Lecture 2, 2. 3. (MB) | Homotopic mappings, homotopy of topological spaces, contractible space. Geometric simplicial complex, carrier (polyhedron). Triangulation of the sphere as a boundary of a simplex and as a boundary of a crosspolytope. Abstract simplicial complex, geometric representations of an abstract simplicial complex. Continuous mapping between polyhedra of simplicial complexes derived from a simplicial mapping, statement of the Simplicial approximation theorem barycentric subdivisions. Presentation [PDF], notes (in Czech) [PDF]. | [M, Chap. 1]. |
| Lecture 3, 9. 3. (MB) | The Borsuk-Ulam Theorem, proof of all the equivalences. Proof of Brouwer's Fixed Point Theorem using the Borsuk-Ulam Theorem. Presentation [PDF], notes [PDF]. | [M, Chap. 1]. |
| Lecture 4, 30. 3. (MT) | Applications of the Borsuk-Ulam theorem: Kneser graphs and the Lovász--Kneser theorem. Coloring hypergraphs and Doľnikov's theorem. Schrijver's graphs and Schrijver's theorem. Proof of Schrijver's theorem will appear at the next lecture. So far we stated Gale's lemma and sketched its proof. | [M, 3.3, 3.4, 3.5]. |
| Lecture 5, 13. 4. (MT) | Proof of Schirjver's theorem. Ham sandwich theorem: Motivation; a version for finite Borel measures; a version for finite point sets. | [M, 3.5, 3.1] |
| Lecture 6, 20. 4. (MT) | Ham sandwich theorem: An improved version for point sets in general position. Remarks on equipartition theorems. Applications of Ham sandwich: Akiyama-Alon theorem; necklace theorem. Hobby--Rice theorem (a proof expected at the tutorials). Tucker's lemma: Motivation, two equivalent formulations. Proof that Tucker's lemma is equivalent to the Borsuk--Ulam theorem (BU2b). | [M, 3.1, 3.2, 2.3] |
| Lecture 7, 27. 4. (MT) | Proof of Tucker's lemma (via happy simplices). The proof is for special triangulations but by earlier considerations, it implies (BU2b) and Tucker's lemma in general. Motivation for the topological Radon and Tverberg theorems. (The standard statements, affine versions and topological versions---so far only statements.) | [M, 2.3, 5.1, 6.5] |
| Lecture 8, 4. 5. (MT) | Joins of abstract simplicial complexes, examples (joins of 2-point discrete sets; the suspension of a triangulated sphere yields a triangulated sphere of dimension one more). Geometric joins. $|K_1 * K_2| \cong |K_1| * |K_2|$. (Proof skipped, expected at tutorials.) For geometric joins: if $X_1 \cong X_2$ and $Y_1 \cong Y_2$, then $X_1 * Y_1 \cong X_2 * Y_2$. (Only a sketch of a proof: It was explained what is the desired homeomorphism but its properties have not been verified.) Propostion (corollary of the earlier two): If $|K_1| \cong |K_2|$ and $|L_1| \cong |L_2|$ for simplicial complexes $K_1,K_2,L_1,L_2$, then $|K_1 * L_1| \cong |K_2 * L_2|$. Notation for points of the join; join of maps. $\mathbb Z_2$-spaces, $\mathbb Z_2$-action, free action and free space, $\mathbb Z_2$-maps. Examples: antipodality on $\mathbb R^n$ and $S^{n-1}$; (BU2a) is equivalent with non-existence of a certain $\mathbb Z_2$-map; two-fold product and join with transposition of coordinates; join of two $\mathbb Z_2$-spaces. Simplicial $\mathbb Z_2$-complex. Join of two simplicial $\mathbb Z_2$-complexes. Example: $n$-fold join of the discrete 2-point set. | [M, 4.2, 5.2] (Some details were formulated differently but the knowledge from the book should be, in general, sufficient.) |
| Lecture 9, 11. 5. (MT) | (Geometric) deleted join for a subset of $\mathbb R^d$. Simplicial deleted join. Lemma: There are $\mathbb Z_2$ maps between the geometric realization of the delted join and the deleted join of the geometric realization. (Proof of the nontrivial direction skipped but it will be proved during tutorials for a simplex.). Lemma: The deleted join of the join of two complexes is the join of their deleted joins. Lemma: The deleted join of an $n$-simplex is the $n$-sphere. (And the transposition action translates into antipodality.) The non-embeddability theorem. Proof of the topological Radon theorem. Remarks on other consequences of the non-embeddability theorem (stronger Kuratowski theorem for $K_{3,3}$; van Kampen-Flores theorem for generalized $K_{3,3}$---both without a proof). $G$-spaces, $G$-action and $G$-maps; $k$-connectedness. | [M, 5.5, 6.1, 4.3] (Some statements and proofs, e.g., the non-embeddability theorem, reformulated/simplified so that they do not require preliminaries from chapters 5.3 and 5.4.) |