  Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson,
represent the culmination of a long evolution. The basic philosophy behind the
process was formulated almost 25 years ago by Rector in his vision of a
homotopy theoretic incarnation of Lie group theory. What was then technically
impossible has now become feasible thanks to modern advances such as Miller's
proof of the Sullivan conjecture and Lannes's division functors. Today, with
Dwyer and Wilkerson's implementation of Rector's vision, the tantalizing
classification theorem seems to be within grasp. Supported by motivating
examples and clarifying exercises, this guide quickly leads, without ignoring
the context or the proof strategy, from classical finite loop spaces to the
important definitions and striking results of this new theory.
