max planck institut
informatik # Topological Methods in Geometry (SS 2011)

## Basic Information

Lecture time: Wednesday 14-16 023, Campus E 1.4 Wednesday 16-17 Saurabh Ray, Nabil H. Mustafa The course counts as computer science lecture (6 CP). The lecture will be given in English. The first lecture will take place on Wenesday, April 13. There will be weekly exercises. TBA TBA Using the Borsuk-Ulam Theorem. Most of the material is also available here. Some parts of the book is available on this page.

### Description

Topological methods are becoming increasingly useful in proving results in seemingly unrelated areas. There are many important results
in Discrete Geometry and Combinatorics for which only a topological proof is known. In this course we aim to give an introduction to the
use of topological methods in Discrete Geometry. We will also present some of the recent results in this area. Prior knowledge of
Topology or Discrete Geometry is not required.

## Lectures

Date Topic Readings Homework Lecturer
Apr 13 Introduction - Intermediate value theorem, Sperner's lemma, Brouwer's Fixed Point theorem.
-
ex1.pdf
Saurabh
Apr 20
Basic concepts of Metric Spaces, Point Set Topology and Geometric Simplicial Complexes.
1.1,1.3
ex2.pdf
Saurabh
Apr 27
Abstract Simplicial Complexes, Simplicial Maps and Affine Extentions,
Borsuk Ulam Theorem and its various equivalent forms.
1.4,1.5,2.1
ex3.pdf
Saurabh
May 04
Tucker's Lemma and its equivalence with Borsuk Ulam theorem,
Combinatorial Proof of Tucker's Lemma.
2.3
ex4.pdf
Saurabh
May 11
Another proof of the Borsuk Ulam theorem.
2.4
ex5.pdf
Saurabh
May 18
Applications of Borsuk Ulam theorem - Ham Sandwich theorem, Lovasz-Kneser theorem,
Dolnikov's theorem.
3.1,3.3,3.4
ex6.pdf
Saurabh
May 25
Deformation Retraction, Homotopy Equivalence, Joins of Topological spaces,
Simplicial Complexes and Maps, Z_2 Spaces, Z_2 Maps, Z_2 Index.
1.2,4.1,4.2
ex7.pdf
Saurabh
June 1
Radon's theorem, geometric proof, re-statement as affine function from simplex,
topological Radon's theorem, proof for d=1, inductive proof via Borsuk-Ulam ,
Z_2 spaces, Z_2 maps, Z_2 indices, transitivity, index of S^n, Z_2 action on product
spaces, deleted products, proof of topological Radon's for d=1 using deleted products.
5.2, 5.3, 5.4
Notes
ex8.pdf
Nabil
June 8
Deleted Joins, Z_2 index of the deleted join of B^d, d-simplex,
topological Radon's theorem via deleted joins.
5.5
Notes
ex9.pdf
Nabil
June 15
Z_2-simplicial complexes, Barycentric subdivision, order complexes of posets,
Sarkaria's inequality, Proof of van Kampen-Flores theorem.
5.7
Notes
ex10.pdf
Nabil
June 22
Plan outline of Lovasz's original proof of Kneser conjecture, hypergraph of minimal nonfaces,
upper-bounding \Delta(L0\L1) with the chromatic number of this hypergraph, Proof of Van-Kampen,
non-planarity of K{3,3}, and another proof of Kneser conjecture.
5.8, 5.9
Notes
ex11.pdf
Nabil
June 29
Lecture cancelled, as Nabil is travelling.

Nabil
July 8
Bier Spheres, Proof of von Kampen-Flores theorem. 5.6

Weijia
July 13
Graph homomorphisms, Box complexes, lower-bound on chromatic number 5.9

Shay
July 20
k-connectedness, k-connectedness implies high Z_2 index. 4.3, 5.3

Vijay
July 29
Group actions on topological spaces, Z_p index, k-fold deleted joins,
Topological Tverberg's theorem, Sarkaria's inequality of Z_p index,
Kneser hypergraphs for k-fold joins, Colored Tverberg's theorem.
Chapter 6

Nabil

## References

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