The Hauptvermutung Book

1. Front Matter

   Pages i-vi

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2. On the Hauptvermutung

   • A. A. Ranicki

   Pages 3-31

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3. Generalisations and applications of block bundles

   • A. J. Casson

   Pages 33-67

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4. Triangulating and Smoothing Homotopy Equivalences and Homeomorphisms. Geometric Topology Seminar Notes

   • D. P. Sullivan

   Pages 69-103

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5. The Princeton notes on the Hauptvermutung

   • M. A. Armstrong, C. P. Rourke, G. E. Cooke

   Pages 105-106

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6. The Hauptvermutung according to Lashof and Rothenberg

   • M. A. Armstrong

   Pages 107-127

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7. The Hauptvermutung according to Casson and Sullivan

   • C. P. Rourke

   Pages 129-164

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8. The Hauptvermutung according to Casson and Sullivan

   • G. E. Cooke

   Pages 165-187

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9. Coda: connection with the results of Kirby and Siebenmann

   • C. P. Rourke

   Pages 189-190

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10. Back Matter

    Pages 191-191

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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.

• Area
• Homeomorphism
• Homotopy
• Volume
• differential geometry
• manifold

` ... long awaited account of a major achievement of modern mathematics. ... highly recommended for all topologists and for those who are interested in the developments of major topological tools.'
Acta Scientificae Mathematicae, 63 (1997)

• Department of Mathematics and Statistics, The University of Edinburgh, Edinburgh, Scotland

  A. A. Ranicki

• Mathematics Department, University of California, Berkeley, USA

  A. J. Casson

• Mathematics Department, City University of New York, New York, USA

  D. P. Sullivan

• Mathematics Department, University of Durham, Durham, UK

  M. A. Armstrong

• Mathematics Institute, University of Warwick, Coventry, UK

  C. P. Rourke

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