Our book on “Two-dimensional Homotopy and Combinatorial Group Theory (LMS 197)” appeared 24 years ago. Since the turn of the millennium we were encouraged to publish in a similar style the (remarkable) progress that has been obtained in the meantime. On the other hand, none of the fundamental open questions like the Andrews-Curtis Conjecture and the Whitehead-Asphericity problem have been (fully) solved, and almost no prognosis is possible about when it will finally happen. We hope that mathematicians, who are in danger of leaving the subject because of this perspective, will be kept active by this sequel publication; and new ones will be gained for our important field.
Again, we mix surveys and original material in the chapters, which can be read by students and experts alike. Ongoing research is mentioned as work in progress. There are ample references to the LMS 197 volume which will be cited as [I] including the bibliography and the index, where new publications and new notions are listed.
Since the publication of LMS 197 there have been workshops of the authors at least every fifth year. We met in Germany, Italy, Russia and the USA, and the fruits of these meetings are the basis of this book. During a meeting close to Naumburg (Saale) we made the decision to start what you can read now.
A mathematically energetic member of the team of authors was NancyWaller. But when the actual writing had to start, she unexpectedly died. Her preprint [BeWa13] together with F. Rudolf Beyl could not be finished, and in particular it was impossible without her to write a planned chapter on the geometric realization problem for algebraic 2-complexes at the same time. Hence “Nancy's Toy”, which she hoped that would result in a counterexample, has been left to the future. What can be said at present, is collected in an appendix to chapter I by F. Rudolf Beyl and Jens Harlander. This book can be considered as serving for Nancy Waller's memory.