Vector bundles over BG whose Euler classes are effective

In this thesis we show that certain finite groups have vector bundles over their classifying spaces that have effective Euler classes. In order to show this we give a characterization of [BG, BU(n)], for some finite groups G, in terms of characters. For these groups, we can find the vector bundle we are looking for by finding a character that has particular properties. Examples are given of finite groups for which this can be accomplished. Included in this work is a connection between Alperin's weak conjugation families and ample collections as defined by Dwyer. Also an obstruction theory is developed for monoids. This obstruction theory shows that in some cases performing the operation on elements of the monoid adds their corresponding obstructions. Applying this obstruction theory gives additional examples of finite groups G that have vector bundles over BG with effective Euler classes. The examples in this thesis include a new proof of the result of Adem and Smith that every rank 2 finite simple group, except PSL3&parl0;Fp&parr0; for odd primes p, has such a vector bundle. It is also proved that the groups PSL3&parl0;Fp&parr0; for odd primes p do not have such a vector bundle.