%% filename: cite-xb.tex %% version: 1.00 %% date: 2004/06/30 %% %% American Mathematical Society %% Technical Support %% Publications Technical Group %% 201 Charles Street %% Providence, RI 02904 %% USA %% tel: (401) 455-4080 %% (800) 321-4267 (USA and Canada only) %% fax: (401) 331-3842 %% email: tech-support@ams.org %% %% Copyright 2004, 2010 American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c %% of this license or (at your option) any later version. %% The latest version of this license is in %% http://www.latex-project.org/lppl.txt %% and version 1.3c or later is part of all distributions of LaTeX %% version 2005/12/01 or later. %% %% This work has the LPPL maintenance status `maintained'. %% %% The Current Maintainer of this work is the American Mathematical %% Society. %% %% ==================================================================== \documentclass{amsart} \usepackage{amsrefs} \newtheorem{thm}{Theorem}[section] \begin{document} \title{Citation tests} \author{Michael Downes} The following examples are derived from \emph{Homology manifold bordism} by Heather Johnston and Andrew Ranicki (Trans.\ Amer.\ Math.\ Soc.\ \textbf{352} no 11 (2000), PII: S 0002-9947(00)02630-1). \bigskip \noindent \rule{\columnwidth}{0.5pt}\par \setcounter{section}{3} The results of Johnston \cite{Jo} on homology manifolds are extended here. It is not possible to investigate transversality by geometric methods---as in \cite{Jo} we employ bordism and surgery instead. The proof of transversality is indirect, relying heavily on surgery theory\mdash see Kirby and Siebenmann \cite{KS}*{III, \S 1}, Marin \cite{M} and Quinn \cite{Q3}. We shall use the formulation in terms of topological block bundles of Rourke and Sanderson \cite{RS}. $Q$ is a codimension $q$ subspace by Theorem 4.9 of Rourke and Sanderson \cite{RS}. (Hughes, Taylor and Williams \cite{HTW} obtained a topological regular neighborhood theorem for arbitrary submanifolds \dots.) Wall \cite{Wa}*{Chapter 11} obtained a codimension $q$ splitting obstruction \dots. \dots\ following the work of Cohen \cite{Co} on $PL$ manifold transversality. In this case each inverse image is automatically a $PL$ submanifold of codimension $\sigma$ (Cohen \cite{Co}), so there is no need to use $s$-cobordisms. Quinn \cite{Q2}*{1.1} proved that \dots \begin{thm}[The additive structure of homology manifold bordism, Johnston \cite{Jo}] \dots \end{thm} For $m\geq 5$ the Novikov-Wall surgery theory for topological manifolds gives an exact sequence (Wall \cite{Wa}*{Chapter 10}. The surgery theory of topological manifolds was extended to homology manifolds in Quinn \cites{Q1,Q2} and Bryant, Ferry, Mio and Weinberger \cite{BFMW}. The 4-periodic obstruction is equivalent to an $m$-dimensional homology manifold, by \cite{BFMW}. Thus, the surgery exact sequence of \cite{BFMW} does not follow Wall \cite{Wa} in relating homology manifold structures and normal invariants. \dots\ the canonical $TOP$ reduction (\cite{FP}) of the Spivak normal fibration of $M$ \dots \begin{thm}[Johnston \cite{Jo}] \dots \end{thm} Actually \cite{Jo}*{(5.2)} is for $m\geq 7$, but we can improve to $m\geq 6$ by a slight variation of the proof as described below. (This type of surgery on a Poincar\'e space is in the tradition of Lowell Jones \cite{Jn}.) \bibliographystyle{amsxport} \bibliography{jr} \end{document}