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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."
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."
This book presents a definitive account of the applications of the
algebraic L-theory to the surgery classification of topological
manifolds. The central result is the identification of a manifold
structure in the homotopy type of a Poincare duality space with a
local quadratic structure in the chain homotopy type of the
universal cover. The difference between the homotopy types of
manifolds and Poincare duality spaces is identified with the fibre
of the algebraic L-theory assembly map, which passes from local to
global quadratic duality structures on chain complexes. The
algebraic L-theory assembly map is used to give a purely algebraic
formulation of the Novikov conjectures on the homotopy invariance
of the higher signatures; any other formulation necessarily factors
through this one. The book is designed as an introduction to the
subject, accessible to graduate students in topology; no previous
acquaintance with surgery theory is assumed, and every algebraic
concept is justified by its occurrence in topology.
This book presents a definitive account of the applications of the
algebraic L-theory to the surgery classification of topological
manifolds. The central result is the identification of a manifold
structure in the homotopy type of a Poincare duality space with a
local quadratic structure in the chain homotopy type of the
universal cover. The difference between the homotopy types of
manifolds and Poincare duality spaces is identified with the fibre
of the algebraic L-theory assembly map, which passes from local to
global quadratic duality structures on chain complexes. The
algebraic L-theory assembly map is used to give a purely algebraic
formulation of the Novikov conjectures on the homotopy invariance
of the higher signatures; any other formulation necessarily factors
through this one. The book is designed as an introduction to the
subject, accessible to graduate students in topology; no previous
acquaintance with surgery theory is assumed, and every algebraic
concept is justified by its occurrence in topology.
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