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author | Karl Berry <karl@freefriends.org> | 2007-10-14 22:19:57 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2007-10-14 22:19:57 +0000 |
commit | 6e5c0acada6a9c582fe33af8763348a18eb2a0f6 (patch) | |
tree | b9c5791b76f3c25d0cd5207cc35c3bf01f609395 /Master/texmf-dist/source/latex/amsrefs/cite-xb.tex | |
parent | 53b3eebc589bf2a49e91bd9c9d8112f2b38bce11 (diff) |
amsrefs update
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diff --git a/Master/texmf-dist/source/latex/amsrefs/cite-xb.tex b/Master/texmf-dist/source/latex/amsrefs/cite-xb.tex new file mode 100644 index 00000000000..5b7aa30f8d2 --- /dev/null +++ b/Master/texmf-dist/source/latex/amsrefs/cite-xb.tex @@ -0,0 +1,98 @@ +\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} |