Alejandro Adem

Twisted Orbifold K-Theory

Abstract: We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K*orb(X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient.

Periodic complexes and group actions

In this paper we show that the cohomology of a connected CW-complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on In x Sm; we construct nonstandard free actions of rank-two simple groups on finite complexes Y Sn x Sm; and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p)3.

Commuting elements, simplicial spaces and filtrations of classifying spaces

Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.

Stable splittings, spaces of representations and almost commuting elements in Lie groups

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits

The geometry and cohomology of the Mathieu group M12

\Rep(\Z^n, SU)

Essential cohomology of finite groups

,

The cohomology of the Mathieu group M22

\Rep(\Z^n,U)

Characters and K-theory of discrete groups

and

A classifying space for commutativity in Lie groups

\Rep(\Z^n, Sp)

Commuting elements in central products of special unitary groups

are explicitly described as finite products of Eilenberg-MacLane spaces.