The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds (K-Monographs in Mathematics, 1)

the Hauptvermutung Is The Conjecture That Any Two Triangulations Of A Polyhedron Are Combinatorially Equivalent. This 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 Further 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. Then, The Development Of Surgery Theory Led To The Disproof Of The High-dimensional Manifold Hauptvermutung In The Late 1960s. Up To Now, The Published Record Of The Hauptvermutung Has Been Incomplete. This Volume Brings Together The Original Papers Of Casson And Sullivan (1967), And The `princeton Notes On The Hauptvermutung' Of Armstrong, Rourke And Cooke (1968/1972). They Include Several Results Which Have Become Part Of Mathematical Folklore, But Of Which Proofs Had Never Been Published. The Material Is Complemented By An Introduction On The Hauptvermutung And An Account Of Recent Developments In The Area. Also, References Have Been Updated Wherever Possible. Audience: This Book Will Be Valuable To All Mathematicians Interested In The Topology Of Manifolds, Geometry, And Differential Geometry. booknews the Hauptvermutung Is The Conjecture That Any Two Triangulations Of A Polyhedron Are Combinatorially Equivalent. Formulated At The Turn Of The Century, This Conjecture Was A Central Problem Of Topology Until Its Resolution. This Collection Of Papers On The Topology Of Manifolds Brings Together The Original Papers Of Casson And Sullivan (1967) And The Princeton Notes On The Hauptvermutung Of Armstrong, Rourke, And Cooke (1968/1972). They Include Several Results Which Have Become Part Of The Mathematical Folklore, But Of Which Proofs Had Never Been Published. The Material Is Complemented By An Introduction On The Haupvermutung And An Account Of Recent Developments In This Area. Includes Updated References Where Possible. Lacks An Index. Annotation C. By Book News, Inc., Portland, Or. 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.