Mini-Workshop: The Hauptvermutung for High-Dimensional Manifolds

Abstract

The Mini-Workshop \emph{The Hauptvermutung for High-Dimensional Manifolds}, organised by Erik Pedersen (Binghamton) and Andrew Ranicki (Edinburgh) was held August 13th--18th, 2006. The meeting was attended by 17 participants, ranging from graduate students to seasoned veterans. The manifold Hauptvermutung is the conjecture that topological manifolds have a unique combinatorial structure. This conjecture was disproved in 1969 by Kirby and Siebenmann, who used a mixture of geometric and algebraic methods to classify the combinatorial structures on manifolds of dimension $>4$. However, there is some dissatisfaction in the community with the state of the literature on this topic. This has been voiced most forcefully by Novikov, who has written ``In particular, the final Kirby-Siebenmann classification of topological multidimensional manifolds therefore is not proved yet in the literature." (http://front.math.ucdavis.edu/math-ph/0004012) At this conference we discussed a number of questions concerning the Hauptvermutung and the structure theory of high-dimensional topological manifolds. These are our conclusions: We found nothing fundamentally wrong with the original work of Kirby and Siebenmann \cite{ks1}, which is solidly grounded in the literature. Their determination of $TOP/PL$ depends on Kirby's paper on the Annulus Conjecture and his `torus trick'. It was noted that Kirby's paper is based on the well-documented work on $PL$ classification of homotopy tori (Hsiang and Shaneson, Wall) and Sullivan's identification of the $PL$ normal invariants with $[-,G/PL]$, but does not depend on any other work of Sullivan, documented or undocumented. This classification can be reduced to the Farrell Fibering Theorem \cite{far}, the calculation of $\pi_i(G/PL)$ (Kervaire and Milnor \cite{km}), and Wall's non-simply connected surgery theory \cite{wall}. There are modern proofs determining the homotopy type of $TOP/PL$ using either the bounded surgery of Ferry and Pedersen \cite{fp} or a modification of the definition of the structure set. Sullivan's determination of the homotopy type of $G/PL,$ which is well-doc\-u\-ment\-ed (for instance, in Madsen and Milgram \cite{mm}) is used to determine the homotopy type of $G/TOP$ and is fundamental to understanding the classification of general topological manifolds. The 4-fold periodicity of the topological surgery sequence established by Siebenmann \cite[p.283]{ks1} contains a minor error having to do with base points. This is an easily corrected error, and the 4-fold periodicity is true whenever the manifold has a boundary. The equivalence of the algebraic and topological surgery exact sequence as established by Ranicki \cite{ran} was confirmed. Sullivan's characteristic variety theorem, however it is understood, is not essential for the Kirby-Siebenmann triangulation of manifolds. The following papers have been commissioned: \begin{itemize} \item W. Browder, ``$PL$ classification of homotopy tori'' \item J. Davis, ``On the product structure theorem'' \item I. Hambleton, ``$PL$ classification of homotopy tori'' \item M. Kreck, ``A proof of Rohlin's theorem'' \item E.K. Pedersen, ``Determining the homotopy type of $TOP/PL$ using bound\-ed surgery'' \item A. Ranicki, ``Siebenmann's periodicity theorem'' \item M. Weiss, ``Identifying the algebraic and geometric surgery sequences'' \end{itemize} The Hauptvermutung website {\it http://www.maths.ed.ac.uk/$\sim$aar/haupt} will re\-cord further developments. \begin{thebibliography}{99} \bibitem{far} F.~T.~Farrell, \emph{The obstruction to fibering a manifold over a circle}, Yale Ph.D. thesis (1967), Indiana Univ. Math. J. {\bf 21}, 315-346 (1971) \bibitem{fp} S.~Ferry and E.~K.~Pedersen, {\it Epsilon surgery}, in {\it Novikov conjectures, Index Theorems and Rigidity}, Vol. 2, LMS Lecture Notes {\bf 227}, Cambridge, 167--226 (1995) \bibitem{km} M.~Kervaire and J.~Milnor, \emph{Groups of homotopy spheres}, Ann. of Maths. {\bf 77}, 504--537 (1963) \bibitem{ks1} R.~Kirby and L.~Siebenmann, {\it Foundational essays on topological manifolds, smoothings, and triangulations}, Ann. of Maths. Studies {\bf 88}, Princeton University Press (1977) \bibitem{mm} I.~Madsen and J.~Milgram, {\it The classifying spaces for surgery and cobordism of manifolds.} Ann. of Maths. Studies {\bf 92}, Princeton University Press (1979) \bibitem{ran} A.~Ranicki, {\it Algebraic $L$-theory and topological manifolds}, Tracts in Mathematics {\bf 102}, Cambridge University Press (1992) \bibitem{wall} C.~T.~C. Wall, \emph{Surgery on {C}ompact {M}anifolds}, Academic Press (1970) \end{thebibliography}