Stephen Theriault

The odd primary

A compact, connected, simple Lie group G localized at an odd prime p is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of G is low. This holds for SU(n), for example, if n ≤ (p-1)(p-3). The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to ΩG. This is applied to prove useful information about the torsion in the homotopy groups of G, including an upper bound on its exponent.

H

We prove two homotopy decomposition theorems for the loops on simply-connected co-H-spaces, including a generalization of the Hilton-Milnor Theorem. Several examples are given.

-structure of low rank Lie groups and its application to exponents

In toric topology, to each simplicial complex K on m vertices one associates two key spaces, the Davis-Januskiewicz space DJK and the moment-angle complex ZK , which are related by a homotopy fibration ZK w̃ −→ DJK −→ ∏m i=1 CP∞. A great deal of work has been done to study properties of DJK and ZK , their generalisations to polyhedral products, and applications to algebra, combinatorics and geometry. In the first part of this paper we survey some of the main results in the study of the homotopy theory of these spaces. In the second part we break new ground by initiating a study of the map w̃. We show that, for a certain family of simplicial complexes K, the map w̃ is a sum of higher and iterated Whitehead products.

Decompositions of looped co-H-spaces

We give p-local homotopy decompositions of the suspensions of real toric spaces for odd primes p. Our decomposition is compatible with the one given by Bahri, Bendersky, Cohen, and Gitler for the suspension of the corresponding real moment-angle complex, or more generally, the polyhedral product. As an application, we obtain a stable rigidity property for real toric spaces.

Homotopy theory in toric topology

This is called the

Homotopy decomposition of a suspended real toric space

mod p

Homotopy commutativity in p-localized gauge groups

decomposition of

Self-Maps of Low Rank Lie Groups at Odd Primes

G

THE 3-PRIMARY CLASSIFYING SPACE OF THE FIBER OF THE DOUBLE SUSPENSION

. Observe that if

2-primary Anick fibrations

p\geq n