Abstracts for Topics in Topology

In this talk I will present some elementary computations in
equivariant (Bredon) homology and cohomology.  First I will give a
brief introduction to equivariant homotopy theory and Bredon
(co)homology.  For the computations we will focus on the case of a
cyclic group G and constant coefficients Z.  We will compute the
equivariant (co)homology of the one point compactification of any
finite dimensional real G-representation.  Non-equivariantly these
spaces are just spheres, but their Bredon (co)homology groups will

The classical de Rham isomorphism between the singular cohomology
of a smooth manifold $M$ and its de Rham cohomology is a well-known theorem.
In this talk we will consider Riemannian manifolds that admit a certain
triangulation $h: K \to M$, where $K$ is a simplicial complex in $\mathbb{R}^n$.
We will establish the notion of $L_p$-norms for differential forms on $M$
as well as for simplicial cochains on $K$. These define two cochain complexes
$L_p(K)$ and $L_p(M)$. Its cohomologies $H_p(K)$ and $H_p(M)$ are

The action of the mapping class group of a compact orientable surface of genus g on the first
homology group of the surface gives  the classical symplectic representation of dimension 2g.
Recently,  John Franks and Michael Handel proved that complex representations of dimension
$n \leq 2g-4$ of the mapping class groups are trivial. In this talk, I will show that all representations
of dimension $n \leq 2g-1$ are trivial, improving the result of Franks and Handel. I will also discuss t
he analogous problem for mapping class groups of nonorientable surfaces.

Any word in the free group on $d$ generators induces a word map from $G^d$ to $G$, for a given group $G$.

For finite simple groups, the following questions naturally arise:
1) Is this map surjective?
2) How the fibers of this map look like?
These questions have been investigated by Larsen, Liebeck, O'Brien,
Shalev, Tiep, the speaker and others.

For example, the Ore Conjecture, which had been open for more than 50 years, states that the commutator map is surjective on all finite simple groups.