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%  Author Package
%% Translation via Omnimark script a2l, September 19, 1996 (all in one day!)
%\controldates{4-OCT-1996,4-OCT-1996,4-OCT-1996,11-OCT-1996}
\documentclass{era-l}
%\issueinfo{2}{2}{OCT}{1996}
\pagespan{73}{81}
\PII{S 1079-6762(96)00010-8}
\usepackage{amscd}

%% Declarations:

\theoremstyle{plain}
\newtheorem*{theorem1}{Proposition 2.1}
\newtheorem*{theorem2}{Theorem 3.1}
\newtheorem*{theorem3}{Theorem 3.2} % (Teardrop Neighborhood Existence)}
\newtheorem*{theorem4}{Theorem 4.1} % (Stratified Sucking)}
\newtheorem*{theorem5}{Theorem 4.2} % (Stratified Straightening)}
\newtheorem*{theorem6}{Corollary 4.3} %(Controlled Stratified Isotopy Covering)}
\newtheorem*{theorem7}{Theorem 5.1} %(Multiparameter Isotopy  Extension)}
\newtheorem*{theorem8}{Theorem 5.2} %(Local Contractibility)}
\newtheorem*{theorem9}{Theorem 5.3} %(First Topological Isotopy)}
\newtheorem*{theorem10}{Theorem 5.4} %(Neighborhood Germ Classification)}
\newtheorem*{theorem11}{Theorem 5.5} %(MSAF Classification)}


%% User definitions:

\newcommand{\lra}{\rightarrow }
\newcommand{\ho}{\operatorname {holink}}
\newcommand{\hos}{\operatorname {{holink}_s}}
\newcommand{\inclusion}{\operatorname {inclusion}}
\newcommand{\proj}{\operatorname {proj}}
\newcommand{\cyl}{\operatorname {cyl}}
\newcommand{\ocyl}{\overset{\circ}{\cyl}}
\newcommand{\realnos}{\mathbb{R}}
\newcommand{\id}{\operatorname {id}}
\newcommand{\tp}{\operatorname {TOP}}
\newcommand{\btop}{\operatorname {BTOP}}
\newcommand{\level}{\operatorname {level}}


\begin{document}

\title[ Stratified spaces]{Geometric  topology of stratified spaces}
\author{Bruce Hughes}
\address{Department of Mathematics, Vanderbilt University, Nashville,
Tennessee 37240}
\email{hughescb@math.vanderbilt.edu}
\date{May 20, 1996}
\commby{Walter Neumann}
\keywords{Stratified space, approximate fibration, teardrop, locally
conelike, isotopy extension, strata, homotopy link, neighborhood germ}
\subjclass{Primary 57N80, 57N37; Secondary 55R65, 57N40}
\begin{abstract}A theory of tubular neighborhoods for strata in manifold
stratified spaces is developed. In these topologically stratified spaces,
manifold stratified approximate fibrations and teardrops play the role
that fibre bundles and mapping cylinders play in smoothly stratified
spaces. Applications include a multiparameter isotopy extension theorem,
neighborhood germ classification and a topological version of Thom's
First Isotopy Theorem. \end{abstract}
\thanks{Supported in part by NSF Grant DMS--9504759.}
\maketitle

\section*{1. Introduction
}Often spaces are studied which are not manifolds,
but which are composed of manifold pieces,
those pieces being called the {\em strata\/}
of the space. Examples include polyhedra,
algebraic varieties, orbit spaces of many group actions
on manifolds, and mapping cylinders of maps between manifolds.

The purpose of this note is to announce recent
progress in understanding the structure of neighborhoods
of strata in certain spaces, namely, the stratified spaces
proposed by Frank Quinn \cite{18}
and called by him `manifold homotopically stratified sets'.
Quinn's objective was `to give a setting for the study
of purely topological stratified phenomena'
as opposed to the smooth and piecewise linear
phenomena previously studied.

Roughly, the stratified spaces of Quinn are spaces $X$
together with a finite filtration by closed subsets
\begin{equation*}X=X^{m} \supseteq X^{m-1} \supseteq \dots \supseteq X^{0} \supseteq X^{-1} = \emptyset ,\end{equation*}
such that the strata $X_{i} = X^{i}\setminus X^{i-1}$ are manifolds with
neighborhoods in $X_{i}\cup X_{k}$ (for $k>i$) which have the local
homotopy properties of mapping cylinders
of fibrations.
These spaces include the smoothly stratified spaces of
Whitney \cite{28}, Thom \cite{24}  and Mather \cite{16} (see e.g. \cite{9})
as well as the locally conelike stratified
spaces of Siebenmann \cite{21} and, hence, orbit
spaces of finite groups acting locally
linearly on manifolds.

Smoothly stratified spaces have the property 
that strata have neighborhoods which are
mapping cylinders of fibre bundles,
a fact which is often used in arguments
involving induction on the number
of strata.  Such neighborhoods fail to
exist in general for Siebenmann's locally
conelike stratified spaces.  For example,
it is known that a (topologically) locally
flat submanifold of a topological manifold
(which is an example of a locally conelike 
stratified space with two strata) may fail
to have a tubular neighborhood \cite{20}.
However, Edwards \cite{7}  proved that such
submanifolds do have neighborhoods
which are mapping cylinders of
manifold approximate fibrations
(see also \cite{14}). On the other hand,
examples of Quinn \cite{17} and
Steinberger-West \cite{23} show that
strata in orbit spaces of finite
groups acting locally linearly on
manifolds may fail to have mapping cylinder
neighborhoods. In Quinn's general setting,
mapping cylinder neighborhoods may fail
to exist even locally.

Our main result (Theorem 3.2) gives an effective
substitute for neighborhoods which are mapping
cylinders of bundles. Instead of fibre bundles,
we use `manifold stratified  approximate fibrations,'
and instead of mapping cylinders, we use `teardrops'.
This result should be thought of as a tubular neighborhood theorem
for strata in manifold stratified spaces.

Applications are discussed in Section 5. They
include a classification of neighborhood germs,
a multiparameter isotopy extension theorem, the
local contractibility of the homeomorphism  group 
of a compact stratified space, a topological version
of Thom's First Isotopy Theorem, and a generalization
of Anderson-Hsiang pseudoisotopy theory.

In related recent work, Weinberger \cite{27} has
developed a surgery theoretic classification
of manifold stratified spaces.
In fact, one of the proofs
envisioned by Weinberger for his theory relies
on our main result (Theorem 3.2).
The main result was first discovered
in the case of two strata in the course of joint work
with  Taylor, Weinberger, and  Williams \cite{15}
where the application to Weinberger's surgery theory is mentioned.
The book by Hughes and Ranicki \cite{12} contains a proof of the main result in
the special case of two strata with the lower stratum consisting of
a single point, and should be consulted for
further background, examples, historical remarks and applications.
The work of Steinberger and West \cite{22}, \cite{23} has been very influential
on the stratified point of view.
Other important recent work includes the thesis of Yan \cite{29},
the speculations of Quinn \cite{19} and the paper by Connolly and
Vajiac \cite{5}.

Complete proofs will appear elsewhere, most notably in \cite{11}.

\section*{2. Stratified spaces and stratified approximate fibrations
}We begin by recalling some definitions from Quinn \cite{18} 
(see also \cite{12}, \cite{15}).
A subset  $Y \subset X$ is {\em forward tame\/} in $X$
if there exist a neighborhood $U$ of $Y$ in $X$ and a 
homotopy $h:U \times I\lra X$ such that $h_{0} = \inclusion :U\lra X$,
$h_{t}|Y = \inclusion :Y\lra X$ for each $t\in I, h_{1}(U) = Y$, and 
$h((U\setminus Y)\times [0,1)) \subseteq X\setminus Y.$

Define the {\em homotopy link\/} of $Y$ in $X$ by
\begin{equation*}\ho (X,Y)=\{ \omega \in X^{I} ~|~ \omega (t) \in Y\hbox 
{ iff }t=0\}.\end{equation*}
Evaluation at $0$ defines a map
$q:\ho (X,Y)\lra Y$ called {\em holink evaluation\/}.

Let $X=X^{m} \supseteq X^{m-1} \supseteq \dots \supseteq X^{0} \supseteq X^{-1}=\emptyset $ be a space with a finite filtration by closed
subsets.
Then $X^{i}$ is the $i$-{\em skeleton\/}, the difference $X_{i} =
X^{i}\setminus X^{i-1}$ is called the $i$-{\em stratum\/}, and
$X$ is said to be a {\em space with a stratification\/}.
A subset $A$ of a space $X$ with a stratification is called a {\em pure\/} 
subset if $A$ is  closed and a union of components of strata of $X$.
For example, the skeleta are pure subsets.

The {\em stratified homotopy link\/} of $Y$ in $X$, denoted
by $\hos (X,Y)$, consists of all $\omega $ in $\ho (X,Y)$
such that $\omega ((0,1])$ lies in a single stratum of $X$.
The stratified homotopy link has a natural filtration
with $i$--skeleton $\hos (X,Y)^{i} =\{\omega | \omega (1)\in X^{i}\}$.
The holink evaluation (at $0$) restricts to a map
$q:\hos (X,Y)\lra Y$.

If $X$ is a filtered space, then a map
$f:Z\times A \lra X$ is {\em stratum preserving along\/}
$A$ if for each $z\in Z$, $f(\{ z\} \times A)$
lies in a single stratum of $X$. In particular, a map
$f:Z\times I\lra X$ is a {\em stratum preserving homotopy\/}
if $f$ is stratum preserving along $I$.

A filtered space  $X$ is a {\em manifold stratified space\/}
if the following four conditions are satisfied:
\begin{itemize}
\item[(i)] {\bf Manifold strata.} $X$ is a locally compact, separable metric
space and each stratum $X_{i}$ is a topological manifold
(without boundary).

\item[(ii)] {\bf Forward tameness.} For each $k>i$, the stratum $X_{i}$ is
forward tame in $X_{i} \cup X_{k}$.

\item[(iii)] {\bf Normal fibrations.} For each $k>i$, the holink evaluation
$q:\ho (X_{i}\cup X_{k},X_{i})\lra X_{i}$ is a fibration.

\item[(iv)] {\bf Finite domination.} For each $i$ 
there exists a closed subset $K$
of the stratified homotopy link $\hos (X,X^{i})$ such that
the holink evaluation map $K\lra X^{i}$ is proper, together with 
a stratum preserving homotopy 
\begin{equation*}h: \hos (X,X^{i})\times I\lra \hos (X,X^{i}),\end{equation*} 
which is also fibre preserving over $X^{i}$ (i.e., $qh_{t}=q$ for each
$t\in I$), such that $h_{0} =
\id $ and $h_{1}(\hos (X,X^{i}))\subseteq K$.
\end{itemize}

For $x\in X^{i}$, the subset $q^{-1}(x)\subseteq \hos (X,X^{i})$ is called the
{\em stratified local holink at\/} $x$. Note that condition (iv) implies that
the stratified local holinks are finitely dominated.

Quinn \cite{18} calls a filtered space satisfying (ii) and (iii)
a `homotopically stratified set', and he calls such a 
space a `manifold homotopically stratified set' if it 
additionally satisfies (i) 
(he also allows the manifold strata to have boundaries). 
These four conditions are not independent;
in fact, (iv) follows from the other conditions assuming
some fundamental group properties. Quinn implicitly assumes
these properties so that our manifold stratified spaces
are essentially the same as Quinn's manifold homotopically
stratified sets (cf. \cite[9.15--18, 10.13--14]{12}). 

Next we generalize the definition of an approximate fibration
(as given in \cite{13}) to the stratified setting. Let
$X=X^{m} \supseteq \dots \supseteq X^{0} $
and $Y=Y^{n}\supseteq \dots \supseteq Y^{0} $
be filtered spaces and let $p:X\lra Y$ be a map ($p$
is not assumed to be stratum preserving).
Then $p$ is said to be a {\em stratified approximate fibration\/}
provided given any space $Z$ and any commuting diagram
\begin{equation*}\begin{CD}
Z @>f>> X \\
@V{\times 0}VV   @VVpV \\
Z\times I @>F>> Y 
\end{CD}\end{equation*}
where $F$ is a stratum preserving homotopy, there exists a
{\em stratified controlled solution\/}, i.e., a map 
$\tilde F : Z\times I\times [0,1) \lra X$
which is stratum preserving along  $I \times [0,1)$
such that $\tilde F (z,0,t) = f(z)$ for each
$(z,t) \in Z \times [0,1)$ and the function
$\bar F : Z \times I \times [0,1] \lra Y$
defined by $\bar F|Z \times I \times [0,1) = p\tilde F$
and $\bar F|Z \times I \times \{ 1 \} = F \times \id _{\{1\}}$
is continuous.

A stratified approximate fibration between manifold stratified 
spaces is a {\em manifold stratified approximate fibration\/}
if, in addition, it is a proper map (i.e., inverse images
of compact sets are compact).
The following result suggests that there is a relationship
between manifold stratified spaces and manifold stratified  
approximate fibrations.

\begin{theorem1}
Let $p:X \lra Y$ be a map 
between manifold stratified spaces. Then the open mapping cylinder
$\ocyl (p)$ is a manifold stratified space if and only if $p$ is a 
manifold stratified approximate fibration. 
\end{theorem1}


If $Y = Y^{n}\supseteq \dots \supseteq Y^{0}$ and $X = X^{m}\supseteq \dots \supseteq X^{0}$, then in the proposition above $\ocyl (p)$ is
filtered so that the strata are given by
\begin{equation*}(\ocyl (p))_{i} = \begin{cases}Y_{i} 
& \text{if $0\leq i\leq n$},\\
                 X_{i-n-1}\times (0,1) & \text{if $n+1\leq i \leq m+n+1$}.
\end{cases}
\end{equation*}
Note that $Y$ is a pure subset of the open mapping cylinder
$\ocyl (p)$.

\section*{3. Teardrop structure on neighborhoods}
Given spaces $X$, $Y$ and a map $p:X\lra Y\times \realnos $,
the {\em teardrop\/} of $p$ (see \cite{15}) is the space denoted by 
$X\cup _{p} Y$ whose underlying set is the disjoint union
$X\amalg Y$ with the minimal topology such that
\begin{itemize}
\item[(i)] $X\subseteq X\cup _{p} Y$ is an open embedding, and

\item[(ii)] the function $c: X\cup _{p} Y\lra Y\times (-\infty ,
+\infty ]$ defined by
\begin{equation*}c(x) = \begin{cases}p(x) & \text{if $x\in X$},\\
             (x,+\infty ) & \text{if $x\in Y$},
\end{cases}
\end{equation*}
is continuous.
\end{itemize}The map $c$ is called {\em the tubular map of the teardrop\/}
or {\em the teardrop collapse\/}. The tubular map terminology comes from
the smoothly stratified case (see \cite{16}, \cite{25}, \cite{6}). 
This is a generalization of the construction of the open
mapping cylinder of a map $g: X\lra Y$. Namely, $\ocyl (g)$
is the teardrop $(X\times \realnos ) \cup _{g\times \id } Y$.
The following result is an analogue  of
Proposition 2.1 for teardrops.

\begin{theorem2}
If $X$ and $Y$ are manifold stratified spaces
and $p: X \lra Y \times \realnos $ is a manifold stratified  approximate
fibration, then $X\cup _{p} Y$ is a manifold stratified space with $Y$ a
pure subset.
\end{theorem2}


In this statement, $Y \times \realnos $ and $X\cup _{p} Y$ are given
the natural stratifications.

The main result is a kind of converse to this proposition.
First, some more definitions. A subset $Y$ of a space $X$
has a {\em teardrop neighborhood\/} if there exist a neighborhood
$U$ of $Y$ in $X$ and a map $p: U\setminus Y\lra Y \times \realnos $
such that the natural function $(U\setminus Y)\cup _{p} Y \lra U$ is 
a homeomorphism. In this case, $U$ is the {\em teardrop
neighborhood\/} and $p$ is the restriction of the tubular map.

\begin{theorem3}[Teardrop neighborhood existence] 
Let $X$ be a manifold
stratified space such that all components of strata have
dimension greater than $4$, and let $Y$ be a pure subset.
Then $Y$ has a teardrop neighborhood whose tubular map 
$c:U\lra Y\times (-\infty ,+\infty ]$ is a
manifold stratified  approximate fibration.
\end{theorem3}


Note that it follows that the restriction 
$c|:U\setminus Y\lra Y\times \realnos $ is also a manifold stratified
approximate fibration. This is what was established for the two strata
case in \cite{15}, so that 3.2 is a slight improvement of \cite{15} even for
two strata.
Recall from the introduction that $Y$ need not have a mapping cylinder
neighborhood in $X$.

\section*{4. The main tools
}There are two tools which are important in the proof of the 
Teardrop Neighborhood Existence Theorem, `stratified sucking' and `stratified 
straightening.' These generalize unstratified results of Chapman
\cite{4}, Hughes \cite{10}, and Hughes-Taylor-Williams \cite{13}.
Stratified sucking gives a condition for a map to be close to a 
manifold stratified  approximate fibration, whereas stratified straightening
can be thought of as a uniqueness result which gives a
condition for two manifold stratified  approximate fibrations
to be controlled homeomorphic.

Let $X = X^{m}\supseteq \dots \supseteq X^{0}$ and
$Y = Y^{n} \supseteq \dots \supseteq Y^{0}$ be filtered spaces.
Let $\beta $ be an open cover of $Y$. Then a map $p: X \lra Y$
is a {\em stratified $\beta $--fibration\/} provided given any space
$Z$ and any commuting diagram
\begin{equation*}\begin{CD}
Z @>f>> X \\
@V{\times 0}VV   @VVpV \\
Z\times I @>F>> Y 
\end{CD}\end{equation*}
where $F$ is a stratum preserving homotopy, there exists a
stratum preserving homotopy $\tilde F : Z \times I \lra X $
such that $\tilde F (z,0) = f(z)$ for each $z \in Z$ and
$p\tilde F $ is $\beta $ -close to $F$.

For the remainder of the section, suppose $X$ and $Y$ are manifold stratified
spaces such that all components of strata have dimension greater
than 4. For undefined terms related to the {\em controlled category\/}
see \cite{13}.

\begin{theorem4}[Stratified sucking] 
For every open cover $\alpha $ of $Y$  there exists an open cover $\beta $
of $Y$ such that if $p:X\lra Y$ is a proper stratified $\beta $-fibration,
then $p$ is properly $\alpha $-homotopic to a manifold stratified
approximate fibration.
\end{theorem4}


We remark that there are also relative and $\Delta ^{k}$-parameter versions
of stratified sucking.

\begin{theorem5}[Stratified straightening]
Suppose $p:X\times \Delta ^{k} \lra Y\times \Delta ^{k}$ is a map
which is fibre preserving over $\Delta ^{k}$ and stratum preserving
along $\Delta ^{k}$. Suppose further that for each $t \in \Delta ^{k}$,
$p_{t}:X\times \{ t\}\lra Y\times \{ t\}$ is a manifold stratified  approximate
fibration. Then there exists a homeomorphism $h: X\times \Delta ^{k}
\times [0,1)\lra X\times \Delta ^{k} \times [0,1)$ such that
\begin{itemize}
\item[(i)] $h$ is fibre preserving over $\Delta ^{k} \times [0,1)$,
\item[(ii)] $h$ is stratum preserving along $\Delta ^{k} \times [0,1)$,
\item[(iii)] $h(x,0,s) = (x,0,s)$ for each $(x,s) \in X\times [0,1)$,
\item[(iv)] $h$ is a controlled map from $p_{0}\times \Delta ^{k}$ to $p$
where $0\in \Delta ^{k}$ is a vertex; i.e., the function $\bar {h}:
X\times \Delta ^{k}\times [0,1]\lra Y\times \Delta ^{k}$ defined by $\bar {h}|X\times \Delta ^{k} \times [0,1) = (p_{0} \times \id _{\Delta ^{k}}) \circ \proj \circ h$
and $\bar {h}|X\times \Delta ^{k} \times \{1\} = p$ is continuous.
\end{itemize}\end{theorem5}


A useful consequence of the straightening principle is the fact that 
manifold stratified approximate fibrations have an isotopy covering
property which we now state.

\begin{theorem6}[Controlled stratified isotopy covering] 
Let $p: X\lra Y$ be a manifold stratified  approximate fibration,
and let $H: Y\times \Delta ^{k} \lra Y\times \Delta ^{k}$ be a stratum 
preserving isotopy; i.e., $H$ is a homeomorphism such that $H$ is
fibre preserving over $\Delta ^{k}$, stratum preserving along $\Delta ^{k}$,
and $H_{0} = \id :Y\times \{ 0\}\lra Y\times \{ 0\}$.
Then there exists a homeomorphism
\begin{equation*}\tilde H :X\times \Delta ^{k} \times [0,1)\lra X\times \Delta ^{k}
\times [0,1)\end{equation*} such that
\begin{itemize}
\item[(i)] $\tilde H$ is fibre preserving over $\Delta ^{k} \times [0,1)$,

\item[(ii)] $\tilde H$ is stratum preserving along $\Delta ^{k} \times [0,1)$,

\item[(iii)] $\tilde {H}(x,0,s) = (x,0,s)$ for all $(x,s)\in X\times [0,1)$,

\item[(iv)] $\tilde H$ is a controlled map from $p\times \id _{\Delta ^{k}}$ to
$H\circ (p\times \id _{\Delta ^{k}})$.
\end{itemize}\end{theorem6}


\begin{proof} 
Apply Theorem 4.2 to the map $H\circ (p\times \id _{\Delta ^{k}})$.
\end{proof}


In the two stratum case the proof in \cite{15} of the 
Teardrop Neighborhood Existence Theorem relied on the  sucking
principle whereas the corresponding uniqueness result relied on the
straightening principle, both principles in the  manifold (unstratified)
case. The proof of 3.2 in the multiply stratified case involves a
complicated induction on the number of strata in the pure subset $Y\subseteq X$ and the number of strata in the complement $X\setminus Y$, and
the stratified straightening principle must be proved as part of the
induction.

\section*{5. Applications
}One of Quinn's main results in \cite{18} is an isotopy extension theorem
for manifold stratified spaces, a result which is quite useful for the theory
of group actions (see \cite{26} and \cite{2}). Quinn's methods only
apply to a single isotopy at a time. On the other hand, Siebenmann \cite{21}
had earlier established a {\em multiparameter\/} isotopy extension
theorem for locally conelike stratified spaces. Our first application
is a generalization to manifold stratified spaces.

\begin{theorem7}[Multiparameter isotopy extension]
Let $X$ be a manifold stratified space such that all components of strata have
dimension greater than $4$, let $Y$ be a pure subset of $X$, 
let $U$ be a neighborhood of $Y$ in $X$, and let
$h: Y\times \Delta ^{k} \lra Y\times \Delta ^{k}$ be a $k$-parameter
stratum preserving isotopy. Then there exists a $k$-parameter stratum
preserving isotopy $\tilde h :X\times \Delta ^{k} \lra X\times \Delta ^{k}$
extending $h$ and supported on $U\times \Delta ^{k}$.
\end{theorem7}


Siebenmann's main goal in studying locally conelike stratified spaces was 
to  provide a setting for generalizing the Edwards-Kirby \cite{8}
and Cernavskii \cite{3} result on the local contractibility 
of the homeomorphism group of a compact manifold.
Siebenmann's proof is adequate for manifold stratified spaces in 
general, so we have the following result.

\begin{theorem8}[Local contractibility]
Let $X$ be a compact manifold stratified space such that all components
of strata have dimension greater than $4$. Then the group
of all stratum preserving self-homeomorphisms of $X$ is locally
contractible in the compact-open topology. 
\end{theorem8}


The next result is a topological analogue of Thom's First
Isotopy Theorem \cite{24}. This can be viewed as a first step
towards a topological theory of topological stability.

\begin{theorem9}[First topological isotopy]
Let $X$ be a manifold stratified space and let $p: X\lra \realnos ^{n}$
be a map such that
\begin{itemize}
\item[(i)] $p$ is proper,

\item[(ii)] for each stratum $X_{i}$ of $X$, $p|:X_{i} \lra \realnos ^{n}$
is a topological submersion,

\item[(iii)] for each $t\in \realnos ^{n}$, the filtration of $X$
restricts to a filtration of $p^{-1}(t)$ giving $p^{-1}(t)$
the structure of a manifold stratified space such that all components of strata
have dimension greater than $4$.
\end{itemize}Then $p$ is a bundle and can be trivialized by a stratum preserving 
homeomorphism; that is, there exists a stratum preserving 
homeomorphism $h: p^{-1}(0)\times \realnos ^{n} \lra X$ 
such that $ph$ is projection.
\end{theorem9}


The next application concerns the classification of 
neighborhoods of pure subsets of a manifold stratified space.
Given a manifold stratified space $Y$, a {\em stratified neighborhood\/}
of $Y$ consists of a manifold stratified space containing $Y$
as a pure subset. Two stratified neighborhoods 
$X, X'$ of $Y$ are {\em equivalent\/} if there exist 
neighborhoods $U, U'$ of $Y$ in $X, X'$, respectively,
and a stratum preserving 
homeomorphism $h: U\lra U'$ such that $h|Y = \id $.
A {\em neighborhood germ\/} of $Y$ is an equivalence class
of stratified neighborhoods of $Y$.

\begin{theorem10}[Neighborhood germ classification]
Let $Y$ be a manifold stratified space such that all components
of strata have dimension greater than $4$. Then the
teardrop construction induces a one-to-one correspondence
from controlled, stratum preserving homeomorphism classes
of manifold stratified approximate fibrations over
$Y\times \realnos $ to neighborhood germs of $Y$.
\end{theorem10}


When $Y$ has just one stratum (i.e., $Y$ is a manifold),
one can use the following result (which generalizes
\cite{13}, \cite{14}) to give a classifying space description
of the manifold stratified approximate fibrations which
occur in the theorem above.
For notation, let $B$ be a connected $i$-manifold and
let $q: V\lra \realnos ^{i}$ be a manifold stratified
approximate fibration where all components of strata
of $V$ have dimension greater than $4$. A 
manifold stratified approximate fibration $p: X\lra B$ has
{\em fibre germ\/} $q$ if there exists an embedding
$\realnos ^{i} \subseteq B$ such that $p|: p^{-1}(\realnos ^{i})
\lra \realnos ^{i}$ is controlled, stratum preserving
homeomorphic to $q$. Fibre germs are unique
up to controlled homeomorphism if the embedding
$\realnos ^{i}\subseteq B$ is orientation preserving in the
case $B$ is orientable. Let $\tp ^{\level }(q)$ denote the simplicial
group of self-homeomorphisms of the mapping cylinder
$\cyl (p)$ which preserve the mapping cylinder levels 
and are stratum preserving with respect to 
the induced stratification
of $\cyl (q)$. Note that there is a restriction homomorphism
$\tp ^{\level }(q)\lra \tp _{i}$.

\begin{theorem11}[MSAF classification]
Controlled, stratum preserving homeomorphism classes of 
manifold stratified approximate fibrations over $B$ with fibre germ
$q$ are in one-to-one correspondence with homotopy classes
of lifts of the  map
$\tau : B\lra \btop _{i}$
which classifies the tangent bundle of $B$,
to $\btop ^{\level }(q)$.
\end{theorem11}


Actually, Theorems 5.4 and 5.5 are just corollaries of deeper
theorems which give homotopy equivalences between simplicial
sets. Then 5.4 and 5.5 are just the statements of the
results on the $\pi _{0}$ level.

Finally, we mention that the teardrop technology announced
in this paper allows the stratified pseudoisotopy
theory of Anderson-Hsiang \cite{1} (which is valid in the
locally conelike case) to be generalized to manifold stratified spaces.
Like those of Anderson-Hsiang, our results are valid
for the full space of pseudoisotopies, whereas Quinn's work \cite{18}
only gives information about $\pi _{0}$ of that space.

\section*{Acknowledgements }The author thanks his collaborators 
Andrew Ranicki, Larry Taylor, Shmuel Weinberger and Bruce Williams
for many useful conversations which have helped to crystalize
the results announced here. During various periods of this research 
the author was supported by  grants from
the Vanderbilt University Research Council,
the National Science Foundation, and the Science and
Engineering Research Council of the United Kingdom,
and was a Fulbright
Scholar at the University of Edinburgh.

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