XVIII Encontro Brasileiro de Topologia (EBT 2012)
• Sankaran, P.
"An introduction to geometric group theory" (elementary) [course note]
(Institute of Mathematical Sciences, Chennai, India)
• Abstract: In these introductory lectures aimed at graduate students (at elementary level), the following topics are proposed to be covered in each lecture:

Lectures 1 & 2. Introduction to coarse geometry. Word metric on a group and quasi- isometry equivalence of a group and its Cayley graph. Examples.

Lecture 3. Group actions; Efromovich-Svarc-Milnor theorem; examples.

Lecture 4. Some quasi-invariants of groups such as ends, growth, etc. Emphasis will be more on conveying the ideas and doing examples rather than providing detailed proofs.

• Hillman, J. A.
"Poincaré duality in low dimensions" [course note]
(University of Sydney - Australia)
• Abstract: The goal of this course is to show how Poincaré duality imposes contraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic $\chi(M)$ and the fundamental group $\pi_1(M)$, for manifolds $M$ of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.)

Outline: In the first lecture we shall define the equivariant chain complex $C_∗ (\tilde X)$ of the universal cover of a finite cell complex $X$, and then define homology and cohomology of $X$ with coefficients in a $\mathbb{Z}[\pi_1 (X)]$-module. We then define Poin caré duality for a smooth manifold $M$ in terms of a chain homotopy equivalence e of chain complexes over $\mathbb{Z}[\pi_1 (M )]$. (We shall not need to know how the chain homotopy equivalence arises.)

In the second lecture we shall consider orientable $PD_3$-complexes. Turaév showed that such complexes have essentially unique factorizations into indecomposables, and Crisp later showed that indecomposable $PD_3$-complexes other than $S^1 \times S^2$ are either aspherical or have virtually free fundamental group. There are examples of the latter type which are not homotopy equivalent to 3-manifolds, but the corresponding question in the aspherical case is a major open problem.

In the third and fourth lectures we move to dimension 4. We recall that every finitely presentable group $\pi$ is the fundamental group of some orientable 4-manifold $M$, and simple estimates show that $\chi(M )$ is bounded below in terms of the Betti numbers of $\pi$. We shall focus on manifolds with group $\pi$ and minimal possible $\chi$. In particular, we shall give a criterion for a closed 4-manifold to be aspherical. If $\pi$ is free or is a surface group, such minimal 4-manifolds are familiar spaces. If time permits, we shall outline how these ideas lead to characterizations up to homotopy equivalence (or better) for most geometric 4-manifolds.

Background: The assumed background for participants in this course corresponds to a first course in algebraic topology: fundamental group and covering spaces, and some homology or homological algebra. In the final lecture we may use some more sophisticated algebraic topology. We shall concentrate on the essential points and key examples, and pass lightly over some details.

• Cohen, Fred
"An introduction to configuration spaces and their applications" [slides]
(University of Rochester - USA)
• Abstract: The main subject of these lectures are the structure and applications of configurations spaces of ordered q-tuples of distinct points in a space $M$ denoted $$Conf (M, q) = \{(m_1 , \ldots , m_q )~|~m_i \neq m_j \text{ if } i \neq j\}.$$ These spaces have been the subject of intense applications in many areas which trace back to Ptolemy as well as Gauss. These lectures will give definitions, together with elementary as well as more sophisticated properties for classical configuration spaces.

1. The initial setting of classical applications to collisions/non-collisions, Borsuk-Ulam Theorem, and coincidence points will be an introduction. Further versions of collisions, non-collisions and coincidences such as Borromean braids and links are given as well as how these structures provide a disguise for classical homotopy groups.

2.Geometric and algebraic invariants given by cohomology rings, homotopy groups as well as Pontrjagin rings of pointed loop spaces will be developed. These were first encountered by Ptolemy in construction of epicycles while trying to describe motions of planetary objects. These epicycles in turn give rise to 'tautologous cohomology classes' present in the cohomology of configuration spaces for 'most' manifolds.

3. Basic geometric properties such as fibration sequences, cofibration sequences, Thom spaces, and the cohomology of configuration spaces will be developed. Cohomology algebras will be given in many cases with some examples given by $M = N \times \mathbb{R}$. Some of the natural connections to classical representation theory and early work of P. Hall, and E. Witt will be described. Further connections to early work of Milnor as well as work of Toshitake Kohno on Vassiliev invariants will be described.

4. Stable decompositions of these and similar spaces together with some natural applications to other spaces such as spaces of packings will be developed. Some examples with configuration spaces having simple singularities will be given.