Algebra and Topology Seminar 

We say that a finite CW-complex $X$ embeds up to homotopy in a sphere $S^n$ if there exists a subpolyhedron $K\subset S^n$ having the same homotopy type as $X$. In this talk, I will explain a sufficient condition for the existence of such an embedding in a given codimension that we obtained in the particular case where $X$ is a two-cone (that is, $X$ is the homotopy cofibre of a map between two suspensions). As an application of this result, we will see that there is no rational obstruction to embeddings up to homotopy of a two-cone in codimension 3.

The open unit ball in a complex Banach space $A$ is a bounded complex domain, the holomorphic structure of which leads to the existence of a closed subspace $A_s$ of $A$ and a triple product $\{...\}:A\times A_s\times A\to A$ which is symmetric and bilinear in the outer variables, conjugate linear in the middle variable and, for $a$, $b$ and $d$ in $A_s$ and $c$ in $A$, satisfies the Jordan triple identity,

Let $A_k^n$ denote the affine space of dimension $n$ over a field $k$ and $X\subset A_k^n$ be an arrangement of linear subvarieties. Set $R=k[x_1,\dots ,x_n]$ and let $I\subset R$ denote an ideal which defines $X$. If $k$ is a field of characteristic zero, the local cohomology modules modules $H_I^r(R)$ are known to have a module structure over the Weyl algebra $A_n(k)$, and one can therefore consider their characteristic cycles, denoted $\mathrm{CC}(H_I^r(R))$ in this talk. In either the real or the complex case, we shall determine the Betti numbers of the complement of the arrangement $X$ in terms of the multiplicities of the local cohomology modules $H_I^r(R)$.

A central problem in algebraic topology has been to single out the homotopy theoretical properties which characterize compact Lie groups. The right definition of a $p$-local version of a compact Lie group came with Dwyer and Wilkerson who introduced the so called $p$-compact groups and proved that they possess many nice properties. For example a $p$-compact group has a maximal torus, a maximal torus normalizer and a Weyl group.

For odd primes $p$, we give a classification of p-compact groups by proving that they are determined by their maximal torus normalizers. In particular the Weyl group data gives a bijection between connected $p$-compact groups and finite $p$-adic reflection groups.

A number of corollaries follow easily from this classification, for example we give an affirmative answer to the maximal torus conjecture for finite loop spaces up to $\mathbb{Z}[1/2]$-localization.

We present certain twisted wreath constructions in equivariant cohomology, modifying some constructions by L. Evans. We use these constructions to answer classical questions in representation theory and present new multiplicative transfers in cohomology that compute Chern classes of induced representations and induced vector bundles.