The Hauptvermutung is the conjecture that any two triangulations of a
poly- hedron are combinatorially equivalent. The conjecture was
formulated at the turn of the century, and until its resolution was a
central problem of topology. Initially, it was verified for
low-dimensional polyhedra, and it might have been expected that furt her
development of high-dimensional topology would lead to a verification in
all dimensions. However, in 1961 Milnor constructed high-dimensional
polyhedra with combinatorially inequivalent triangulations, disproving
the Hauptvermutung in general. These polyhedra were not manifolds,
leaving open the Hauptvermu- tung for manifolds. The development of
surgery theory led to the disproof of the high-dimensional manifold
Hauptvermutung in the late 1960's. Unfortunately, the published record
of the manifold Hauptvermutung has been incomplete, as was forcefully
pointed out by Novikov in his lecture at the Browder 60th birthday
conference held at Princeton in March 1994. This volume brings together
the original 1967 papers of Casson and Sulli- van, and the 1968/1972
'Princeton notes on the Hauptvermutung' of Armstrong, Rourke and Cooke,
making this work physically accessible. These papers include several
other results which have become part of the folklore but of which proofs
have never been published. My own contribution is intended to serve as
an intro- duction to the Hauptvermutung, and also to give an account of
some more recent developments in the area. In preparing the original
papers for publication, only minimal changes of punctuation etc.
