
\magnification=\magstep1
\input amstex
%\documentstyle{imappt}
%\input usescript.tex
\documentstyle{amsppt}
\hsize=16true cm
\vsize=23.5true cm
\def\App{{\bf A}{\midbold PPENDIX}{\bf : }}
\def\bc{{\Bbb C}}
\def\bz{{\Bbb Z}}
\def\data{\operatorname{data\,}}
\def\Fix{\operatorname{Fix\,}}
\def\id{\operatorname{{\smc id}\,}}
\def\Int{\operatorname{Int\,}}
\def\Ker{\operatorname{Ker\,}}
\def\mh{M_{(H)}}
\def\mhv{M_{(H,V)}}
\def\Nonsing{\operatorname{Nonsing\,}}
\def\oB{\overline B}
\def\oC{\overline C}
\def\oh{\overline h}
\def\oV{\overline V}
\def\oX{\overline X}
\def\sgn{\operatorname{sgn\,}}
\def\Sing{\operatorname{Sing\,}}
\def\rel{\ {\text{\smc rel}}\ }
\def\RelSing{\operatorname{RelSing\,}}
\def\starbar{{\frac {*}{}}}
\def\tinycirc{{^{{\,}_{\text{\rm o}}}}}
\def\wY{\widetilde{Y}}
\def\fcek{\check f}
\def\brh{{}_{B\!R}H}
\def\gpi{\,{}_G\pi}
\def\iidd{{\text{\bf IsD}}}
\font\midbold=cmbx7
\def\ellll{\midbold L}
\def\gl{{\bf G}\ellll}
\def\wl{{{\bf W}\ellll}}
\def\abox{$\blacksquare$}
\NoBlackBoxes
\pageno=2
\document

{\tt This page should be discarded.}
\bigskip
{\tt It is needed so that a page number appears on page 3.}
\vfill\eject
\ \vskip 2 true cm
\centerline{\bf PART I}
\bigskip

\centerline{\bf INTRODUCTION TO ISOVARIANT HOMOTOPY THEORY}
\bigskip
\bigskip
\bigskip
\centerline{\it Background references and notation}
\bigskip
Although we shall include a few remarks on the basic facts and concepts
of differentiable transformation groups,
a detailed account of the subject's foundations is beyond the scope of
this article.  The following references contain most if not all of the 
relevant background material:
\bigskip
\itemitem{(1)} Chapters I--II and Sections VI.1--2 of Bredon, {\it Introduction 
to Compact Transformation Groups} ( := [Bre3]).
\bigskip
\itemitem{(2)} Chapter I and Sections II.1--2 of tom Dieck, {\it Transformation
Groups} ( := [tD2]).
\bigskip
\itemitem{(3)} Chapters I--II and the Summary of Dovermann--Schultz, {\it 
Equivariant Surgery Theories and Their Periodicity Properties} ( := [DoS2]).
\bigskip
Most of the algebraic topology that we use can be found in the standard
books by Spanier [Sp] and Milnor and Stasheff [MS].
\bigskip
We shall generally use standard notational conventions in transformation 
groups including $M^G$ for the fixed point set of a group $G$ acting on a
space $M$, $G_x$ for the isotropy subgroup of $G$ at $x$, and $M_{(H)}$ 
for the set of all points whose isotropy subgroups are conjugate to $H$,
and $|G|$ for the order of a finite group $G$. 
These (and many others) can be found in the references listed above.
\bigskip
As indicated by the title of this article, we shall deal mainly with
smooth group actions.  However, for purposes of comparison we shall
occasionally comment on other families of group actions that lie between
smooth and topological ({\it i.e.}, continuous) actions on manifolds.
In particular, we shall discuss results for group actions that are {\sl
locally linear} ($\equiv$ locally smooth in the sense of Bredon [Bre3]),
and if $G$ is finite we shall also discuss actions that are 
{\sl piecewise linear} (usually abbreviated to PL) or {\sl PL locally
linear} ($\equiv G$pl in [Ro, pp. 303--304]).  Further information on
these families can be found in the references cited, in [LaR], and in 
standard texts of PL topology such as Hudson [Hud] or Rourke and
Sanderson [RSa]; the logical relations among the various types of
actions are either obvious or 
implicit in work of S. Illman [{\smc Il3}]. 
\bigskip
Finally, to simplify the exposition we adopt the following default 
hypothesis:
\medskip
\itemitem{$(\star)$}{\it For all differentiable group actions, the group 
$G$ that acts is assumed to be finite unless there are statements or contexts
to indicate otherwise.}
\bigskip\bigskip\bigskip
\centerline{\bf 1. Equivariant differential topology}
\bigskip\bigskip
To a great extent the foundations of the theory of transformation groups
are developed by analogy. Specifically, given a fact or concept in some 
category of objects and morphisms 
$\Cal A$, one attempts to describe something parallel
in an equivariant category ${\Cal A}_G$ with the following sorts of data:
\bigskip
\itemitem{$(i)$} The objects of ${\Cal A}_G$ are pairs consisting of an 
object $X$ in $\Cal A$ and a homomorphism $\Phi: G\to {\text{Aut}}_{\Cal 
A}(X)$ that may be required to satisfy additional technical restrictions
({\it e.g.}, that a naturally associated map $\varphi:G\times X\to X$ lie
in $\Cal A$).

\itemitem{$(ii)$} The morphisms $f:(X,\Phi)\to(X',\Phi')$ satisfy the 
equivariance identity $\Phi'(g)\tinycirc f = f \tinycirc \Phi(g)$ for all
$g\in G$.
\bigskip
In this article we are interested mainly in the category of smooth 
finite-dimensional manifolds.  Two basic concepts in this category 
are the notions of {\sl vector bundle} and {\sl smooth embedding}. 
Each of these has a natural analog in the $G$-equivariant category;
namely, a smooth $G$-vector bundle $E\downarrow M$ is taken to have a smooth
action of $G$ that sends the fiber over $x\in M$ linearly into the fiber
over $g\cdot x$ for all $(g,x)\in G\times M$ ({\it cf.} [Bre3, pp. 303--304]), 
and an equivariant smooth embedding is merely a smooth embedding that
is $G$-equivariant. In ordinary differential topology the Tubular
Neighborhood Theorems relate smooth embeddings to smooth vector bundles
in a fundamentally important manner, and one has a straightforward 
equivariant generalization with extremely far reaching consequences
for the theory of differentiable transformation groups:

\proclaim{Theorem 1.1} $(i)$ Let $M$ and $N$ be smooth (finite dimensional)
$G$-manifolds, and let $f:M\to N$ be a smooth equivariant embedding.  Then
there is an equivariant vector bundle $E\downarrow M$ and a smooth embedding
$F:E\to N$ whose restriction to the zero section is essentially $f$.

$(ii)$ Suppose that $f_j:E_j\to N$ (where $j=0,1$) satisfy $(i)$, and 
suppose we are given invariant riemannian metrics on $E_j$ with unit 
disk bundles $D(E_j)$.  Then there is a metric preserving $G$-vector 
bundle isomorphism $\varphi:E_0\to E_1$ covering the identity on $M$ and
a smooth equivariant ambient isotopy $H:N\times [0,1]\to N$ such that
$H_1\tinycirc f_o=f_1\tinycirc\varphi$.
\endproclaim

This theorem follows from the results of [Bre3, Sections VI.2--3].  
\medskip
If the source manifold $M$ in Theorem 1.1 is a homogeneous space $G/H$,
then the theorem reduces to the {\sl differentiable slice theorem}
[Bre3, Cor. VI.2.4, p. 308]:

\proclaim{Corollary 1.2} Let $M$ be a smooth finite dimensional $G$-manifold 
without boundary, let $x\in M$ be given, and let $H$ be the isotropy
subgroup $G_x$.  Then $x$ has a $G$-invariant neighborhood that is 
$G$-diffeomorphic to an associated fiber space $G\times_H V\downarrow
G/H$ for some finite dimensional $G$-representation $V$.$\blacksquare$
\endproclaim
\bigskip
\noindent{\smc Default convention. } Unless stated otherwise, all manifolds
are henceforth assumed to be finite dimensional.
\bigskip
For our purposes it is important to have an analog of Theorem 1.1 for smooth
$G$-manifolds with boundary. As in the nonequivariant category, the most 
important property of bounded smooth $G$-manifolds is the existence of {\sl
collar neighborhoods} for the boundary.

\proclaim{Proposition 1.3} Let $M$ be a smooth $G$-manifold with boundary
$\partial M$. Then $\partial M$ has a neighborhood that is equivariantly
diffeomorphic to $\partial M\times[0,1)$ -- with trivial action on the 
second coordinate -- such that $\partial M$ corresponds to $\partial M
\times \{0\}$. If $c_j$ are two smooth equivariant collar neighborhoods
(where $j=0,1$),
then $c_0|\partial M\times[0,{\frac 12}]$ and $c_1|\partial M\times
[0,{\frac 12}]$ are ambient isotopic.
\endproclaim

This result follows from the same sort of argument that appears in
M.~Hirsch's differential topology textbook
[Hi, Thm. 4.6.1, pp. 113--114].$\blacksquare$
\bigskip
As noted in [Hi, Ch. 4, Sec. 6], the analog of the Tubular Neighborhood
Theorem for bounded manifolds requires a suitably refined notion of 
embedding for manifolds with boundary.  Specifically, one needs to work
with {\sl neat embeddings} of $(M,\partial M)$ in $(N,\partial N)$ such
that boundary goes to boundary and interior to interior ({\it i.e.},
a proper embedding of manifolds with boundary) and some neighborhood of
$\partial M$ in $M$ meets $\partial N$ orthogonally along the boundary
(say with respect to some collar neighborhoods).  We then have the following 
equivariant analog of the results for bounded manifolds in [Hi, Thms. 4.6.3
and 4.6.5, pp. 114--116].

\proclaim{Theorem 1.4} $(i)$ Let $(M,\partial M)$ and $(N,\partial N)$ be 
smooth (finite dimensional) $G$-manifolds with boundary, and let $f:(M,
\partial M)\to (N,\partial N)$ be a smooth equivariant embedding satisfying 
the neatness condition described above (with respect to some invariant 
collar neighborhoods).  Then there is an equivariant vector bundle 
$E\downarrow M$ and a neat smooth embedding
$F:E\to N$ whose restriction to the zero section is essentially $f$.

$(ii)$ Suppose that $f_j:E_j\to N$ (where $j=0,1$) satisfy $(i)$, and 
suppose we are given invariant riemannian metrics on $E_j$ with unit 
disk bundles $D(E_j)$.  Then there is a metric preserving $G$-vector 
bundle isomorphism $\varphi:E_0\to E_1$ covering the identity on $M$ and
a smooth equivariant ambient isotopy $H:N\times [0,1]\to N$ such that
$H_1\tinycirc f_o=f_1\tinycirc\varphi$.$\blacksquare$
\endproclaim
\bigskip
The equivariant Tubular Neighborhood Theorems imply that certain invariant
subsets of a smooth $G$-manifold are also smoothly embedded submanifolds
or unions of submanifolds (where the dimension may vary from one component 
to the next).  For example, this holds for the fixed point set $M^G$ and 
the constant orbit type set $M_{(H)}$.  Since $M$ is the pairwise disjoint
union of these subsets $M_{(H)}$, where $(H)$ runs through the conjugacy
classes of subgroups of $G$, and the induced $G$-action on $M_{(H)}$ is
completely determined by fiber bundle considerations, we therefore have a
decomposition of $M$ into $G$-invariant pieces that can be studied effectively
by standard topological methods.  In order to understand things more 
thoroughly it is necessary to determine how these pieces fit together; a
fairly complete account of this can be found in work of M.~Davis [Dav].
As indicated in [DoS2, Sec. II.4], the process can be viewed as a special
case of the Thom-Mather theory of smoothly stratified sets.
\bigskip
{\sl Convention on abuse of language. }  If we are given an invariant
closed subset $N\subset M$ of a smooth $G$-manifold $M$ such that $N$ is 
a disjoint union of smoothly embedded submanifolds $N_j$, and the maps
$\varphi_j:E_j\to N_j$ define pairwise disjoint tubular ({\it i.e.}, 
vector bundle) neighborhoods as
in the preceding results, we shall often say that the disjoint union 
$$\tsize {\coprod \varphi_j:\coprod E_j\to\coprod N_j}\ (\approx N)$$
is an invariant tubular neighborhood of $N$ in $M$; if $\alpha$ is the
disjoint union of the associated vector bundles $\alpha_j$ (whose fibers
may have different dimensions!), then $D(\alpha)$ and $S(\alpha)$ will
denote the unions of the associated disk and sphere bundles 
$\coprod D(\alpha_j)$ and $\coprod S(\alpha_j)$ respectively.
\bigskip\bigskip\bigskip
\centerline{\bf 2. Equivariant homotopy theory}
\bigskip\bigskip
One of the main themes in algebraic topology is the use of cohomology
groups to analyze the homotopy classes of maps from one space to another.
The classical approach to this general question involves {\sl obstruction
theory} ({\it cf.} [Wh1]; for historical background see [{\smc Eil}] and
[Wh2]).  Although several new and powerful techniques for studying 
homotopy classes have emerged over the past four decades, in many cases
the obstruction-theoretic approach is still the most useful or illuminating.
The applicability of obstruction theory to equivariant homotopy was already
understood by the mid nineteen fifties ({\it cf.} [Dg]), especially
when the group action is {\smc free} ({\it i.e.}, all isotropy subgroups
are trivial).  Systematic investigations of equivariant homotopy theory 
began in the nineteen sixties.  A historical summary appears in the first 
part of [Sc10, Sec. 1]; we shall merely sketch the basic 
mathematical points here.

In principle, classical obstruction theory yields an algebraic setting
for describing homotopy classes of maps provided one has the following
data:
\medskip
\itemitem{$(i)$} Cellular decompositions for the source spaces.

\itemitem{$(ii)$} Suitably defined cohomology groups for the source 
spaces, in general with twisted coefficients, that can be computed 
from small cochain complexes reflecting the cell structure.  The 
coefficients are determined by the homotopy type of the target 
space and some information involving fundamental groups.
\bigskip
In [Bre2] G. Bredon described general and usable equivariant analogs of
$(i)$ and $(ii)$. The objects in $(i)$ were forerunners of {\sl 
$G-CW$ complexes} ({\it e.g.}, see [tD2, Sec. II.1]), and the cohomology
functors in $(ii)$ evolved into the so-called {\sl Bredon 
(equivariant) cohomology groups}; the latter were first defined by Bredon 
[Bre2], and a close relation of these
to ordinary singular cohomology follows from an alternative definition due
to Th.~Br\"ocker [Br\"o] (in this connection also see [{\smc Il1}]).  
Equivariant coefficient systems 
in Bredon cohomology are more complicated than ordinary coefficient 
groups and require families of abelian groups indexed by the components of 
each fixed point set $Y^H$ (where $H$ is a subgroup of $G$), but one still 
has small cochain complexes
for computing the Bredon cohomology groups $\brh^*(X, {\Cal A})$.
\bigskip
The following results reflect the strong analogies between ordinary and
equivariant homotopy theory.

\proclaim{Property 2.0}  {\rm ($G$-homotopy extension property)} 
Let $G$ be a compact Lie group, let $X$ be 
a $G-CW$ complex, let $A\subset X$ be a subcomplex, let $f:X\to Y$
be a continuous equivariant map into some $G$-space $Y$, and let 
$h_t:A \to Y$ be an equivariant homotopy such that $h_0=f|A$.  Then
$h_t$ extends to an equivariant homotopy $H_t:X\to Y$ such that 
$H_0=f$.$\blacksquare$
\endproclaim
\bigskip
\proclaim{Property 2.1} {\rm (Extension theorem)}  Let $G$, $A$, $X$, 
$Y$ be as above, and let $h:A\to Y$ be continuous and equivariant.
Then $f$ extends to a continuous equivariant map from $A$ to $Y$ if 
a sequence of obstructions valued in the $i$-dimensional Bredon cohomology 
groups of $(X,A)$, with coefficients in the $(i-1)$-dimensional homotopy 
groups of fixed set components of $Y$, is trivial.$\blacksquare$
\endproclaim
\bigskip
\proclaim{Property 2.2} {\rm (Classification theorem)}  Let $G$, $X$, 
$Y$ be as above, and let $h_j:X\to Y$ be continuous and equivariant for 
$j=0, 1$.  Then $h_0$ and $h_1$ are equivariantly homotopic if 
a sequence of obstructions valued in the $i$-dimensional Bredon cohomology 
groups of $X$, with coefficients in the $i$-dimensional homotopy 
groups of fixed set components of $Y$, is trivial.$\blacksquare$
\endproclaim
Note the difference in coefficient groups between 2.1 and 2.2.
\bigskip
\proclaim{Property 2.3} {\rm (Barratt-Federer spectral sequence)}  Let $G$, 
$X$, $Y$ be as above, and also assume $X$ is 
finite-dimensional.  Modulo some mildly exceptional behavior in 
dimensions $1$ and $0$ there is a spectral sequence such that 
$$E^2_{i,j} = \brh^{-i}_G\left(X;\gpi_j(Y)\right)$$
where $\gpi_j(Y)$ is a coefficient system derived from the homotopy
groups of components of fixed point sets $Y^H$ (where $H$ runs 
through subgroups of $G$)
and $E^\infty_{i,j}$ gives a series for $\pi_{i+j}(F_G(X,Y))$,
where $F_G(X,Y)$ is the space of $G$-equivariant continuous maps from 
$X$ to $Y$ with the compact open topology.$\blacksquare$
\endproclaim
\bigskip
Precise statements of 2.1--2.3 appear in [DuS, Sec. 1].  Unfortunately,
the preceding results are computationally less useful than their
nonequivariant counterparts because Bredon cohomology groups are far more
difficult to compute than ordinary singular cohomology.  One the other
hand, results from [Sc2] yield alternatives to 2.1--2.3 that involve 
ordinary cohomology groups; not surprisingly, there is a price to pay for
this -- roughly speaking, each Bredon cohomology group is replaced by a
finite list of ordinary cohomology groups. In particular, the analog of
Property 2.3 given by [Sc2] is the Barratt-Federer/F\'ary spectral 
sequence in [DuS, Thm. 1.5].

One can in fact extend nearly all the basic concepts and results in 
algebraic topology to the category of $G-CW$ complexes, including
an equivariant version of the Whitehead theorem for recognizing 
equivariant homotopy equivalences
(see [Bre2]), Postnikov decompositions, and localization at a subring of 
the rationals [MMT].  This can be interpreted as a special case of a more 
general observation (see 
[DF, p. 131]; in this connection also see [May] and [DuS, statement
(1.8)]).  There is also a corresponding analog of equivariant stable
homotopy theory; references for the latter include tom Dieck's book on 
the Burnside ring [tD1], an exhaustive account by G. Lewis, J. P. May, and 
M. Steinberger [LMS], and a more recent survey article by G. Carlsson
[Car].
\bigskip
\bigskip
\centerline{\it Application to smooth $G$-manifolds}
\medskip
Of course, if we wish to apply the preceding machinery to compact smooth 
$G$-manifolds it is necessary to find appropriate interpretations of the 
latter as $G-CW$ complexes. There are two ways of doing this.  Results of
A. Wassermann [Wa] yield a version of Morse Theory for $G$-invariant
smooth functions on smooth $G$-manifolds, and from this it is 
elementary to show that smooth $G$-manifolds have the $G$-homotopy type
of finite-dimensional $G-CW$ complexes; in fact, for compact smooth 
$G$-manifolds one can choose the $G-CW$ complexes to be finite
(also see [Ko] for a self-contained account of these results).  On 
the other hand, for many purposes it is more useful to invoke a stronger
result due to S.~Illman  [{\smc Il3}]:  {\sl Every smooth $G$-manifold 
($G$ finite) has a $G$-equivariant triangulation that is smooth in the
sense of J. H. C. Whitehead} (see [Mun] for the corresponding result in
ordinary differential topology).
\bigskip\bigskip\bigskip
\centerline{\bf 3. Isovariant homotopy theory}
\bigskip\bigskip
Recall that a finite group action on a reasonable (say paracompact
Hausdorff) space $X$ is {\sl free} if the isotropy subgroups at all
points are trivial.  Basic results in topology state that a free action
of a finite group $G$ on such a space is determined by its orbit space
$X/G$ and a homotopy class of maps from $X/G$ to a universal base space
$BG\simeq K(G,1)$.  In [Pa] R. S. Palais extended this to a classification
theory for $G$-spaces that are not necessarily free; this involves a 
special class of maps that Palais called {\smc isovariant}.  Specifically,
a $G$-equivariant map $f:X\to Y$ is said to be isovariant if for all 
$x\in X$ the isotropy subgroups satisfy $G_x=G_{f(x)}$; an equivariant
map $f$ automatically satisfies $G_x\subset G_{f(x)}$, but it is easy
to see that equality often does not hold.  During the nineteen sixties
and seventies isovariant maps were also discussed in connection with 
various topological problems, and questions about isovariant homotopy
arose naturally; much of this is discussed at various points of 
[DuS].  The usefulness of isovariant homotopy for classifying 
$G$-manifolds became explicit in the work of Browder and Quinn [BQ]
on stratified surgery theory.  Various {\sl ad hoc} techniques for dealing
with isovariant homotopy theory gradually emerged, and by the mid 
nineteen eighties it was clear that one could analyze isovariant homotopy
effectively by the basic techniques of algebraic topology.  The key 
idea is expressed in [DuS] as follows:
\medskip
\itemitem{}{\sl Isovariant homotopy for smooth $G$-manifolds is 
essentially equivalent to equivariant homotopy theory for suitable 
diagrams of smooth $G$-manifolds.}
\medskip
We shall explain this statement in several steps, beginning with
a discussion of diagram categories.
If ${\Bbb D}$ is a small category and ${\Cal S}$ is some category
of topological spaces, then a {\sl ${\Bbb D}$-diagram with values in 
${\Cal S}$} is merely a covariant functor ${\Bbb D}\to {\Cal S}$ 
and a {\sl morphism of ${\Bbb D}$-diagrams} is a natural transformation
of functors.  Results of W. Dwyer and D. M. Kan [DK]
show that one can extend much of classical homotopy theory and 
obstruction theory to categories of ${\Bbb D}$-diagrams with 
values in the category of CW complexes, and subsequent generalizations
of E. Dror Farjoun [DF] yield corresponding results for suitable 
categories of ${\Bbb D}$-diagrams of $G$-CW complexes.  More precisely,
we need to choose a small category associated to some finite partially
ordered set ${\Bbb P}$ that is closed under taking greatest lower
bounds, and we also must restrict attention to ${\Bbb P}$-diagrams
satisfying a few simple admissibility conditions (see [DuS, conditions
$(i)-(iii)$ preceding (1.6)].  The basic results of equivariant 
algebraic topology extend directly to such equivariant diagrams and 
are summarized in [DuS, (1.6)--(1.10)].

Before discussing the types of diagrams we need, a technical remark is
in order.  Strictly speaking, the results of [DuS] only deal with a 
restricted class of group actions, but this class includes all actions of 
the cyclic groups ${\Bbb Z}_{p^r}$, where $p$ is a prime, and all fixed 
point free actions of ${\Bbb Z}_{pq}$, where $p$ and $q$ are distinct primes.
Extensions to more general actions will appear in a sequel to [DuS].  We 
shall first discuss the types of diagrams needed in an important special
case and then outline an inductive extension to the general case.
\bigskip
\centerline{\it The semifree case}
\bigskip
The conclusions of [DuS] apply directly to actions that are {\sl semifree}
(= free off the fixed point set), so we shall describe the appropriate
diagrams of $G$-manifolds in this important special case.  If $M$ is a
compact smooth $G$-manifold, let $M^G$ denote its fixed point set as 
usual, let $\alpha_M$ be the equivariant normal bundle of $M^G$ in $M$,
let $D(\alpha_M)$ and $S(\alpha_M)$ be the associated unit disk and sphere
bundles respectively, and let $M\starbar M^G$ denote the closure of 
$M-D(\alpha_M)$.  Then the relevant diagram of spaces, which is denoted
by $B({\text{\bf QF}}_M)$ in [DuS], is the following partially ordered set:
$$
\CD
S(\alpha_M) @>>> D(\alpha_M) @<<< M^G\\
@VVV @VVV \\
\tsize M\starbar M^G @>>> M
\endCD
$$
\bigskip
In the preceding special case the main result of [DuS, Sec. 4] can be
stated as follows:
\bigskip
\proclaim{Theorem 3.1}  {\rm (compare [DuS, Thm. 4.5])} Let $G$ be a 
finite group, let $X$ and $Y$ be 
compact smooth semifree $G$-manifolds, and let
$B({\text{\bf QF}}_X)$ and $B({\text{\bf QF}}_Y)$ be the diagrams
described above. Then there is a canonical 
isomorphism:
$$
\bmatrix
G-\text{equivariant}\\
\text{homotopy classes of}\\
\text{continuous equivariant}\\
\text{diagram morphisms}\\
B({\text{\bf QF}}_X)\to B({\text{\bf QF}}_Y)\endbmatrix
\qquad\cong\qquad\bmatrix
G-\text{isovariant}\\
\text{homotopy classes of}\\
\text{continuous isovariant}\\
\text{maps of spaces}\\
X\longrightarrow Y\endbmatrix
$$
\abox
\endproclaim
The proof of this result is a fairly straightforward application of 
standard compactness considerations and the covering homotopy 
property for fiber bundles; details appear in [DuS, Sects. 2--4].
\bigskip
Results of Dwyer and Kan [DK] and E. Dror Farjoun [DF] show that 
one can ``do algebraic topology'' in the diagram category corresponding
to the left hand side of the correspondence displayed in the theorem.
In particular, the associated obstruction theories in the diagram 
categories ({\it cf.} [DuS, Sec. 1]) are essentially obstruction theories
for isovariant homotopy theory.  Furthermore, as noted in [DuS, 
(1.8)--(1.10)] one also has Postnikov decompositions, localization at 
subrings of the rationals, and spectral sequences for the homotopy 
groups of isovariant function spaces analogous to those of J. M{\o}ller
and the author [M{\o}, Sc2].  At the end of [DuS, Sec. 4] these
ideas are applied to prove isovariant analogs of the Whitehead Theorems
for recognizing (ordinary or equivariant) homotopy equivalences.
Here is one special case:
\bigskip
\proclaim{Theorem 3.2}  Let $G$ be a finite group, let $X$ and $Y$ be
compact, {\smc unbounded}, semifree smooth $G$-manifolds, 
let $f:X\to Y$ be an isovariant map, and also assume that all fixed
point sets $X^H$ and $Y^H$ are orientable.  Then $f$ is an isovariant 
homotopy equivalence if and only if 
for each isotropy subgroup $H$ the map $f$ induces homotopy 
equivalences from $X^H$ to $Y^H$ and from $X^H-{\text{\rm Sing}}(X^H)$ 
to $Y^H-{\text{\rm Sing}}(Y^H)$.
\endproclaim
{\bf Notation and remark. }  The {\sl singular set} of $M^H$, denoted by 
${\text{\rm Sing}} (M^H)$, is the set of all points in $M^H$ whose 
isotropy subgroups {\smc properly} contain $H$.  By definition an 
isovariant map $X\to Y$ automatically takes $X^H-{\text{\rm Sing}}(X^H)$ 
to $Y^H-{\text{\rm Sing}}(Y^H)$.\abox
\bigskip
Related isovariant analogs of the Whitehead Theorem also appear in [DuS,
Thm. 4.10 and Cor. 4.11]).
\bigskip
\centerline{\it Actions with more general isotropy structures}
\bigskip
The preceding results extend to arbitrary compact differentiable 
$G$-manifolds (where $G$ is finite as usual).  In principle the argument
combines the techniques used in the semifree case with an inductive 
framework that we shall describe below; details will appear in a sequel
to [DuS].
\bigskip
{\bf Definition. } If $G$ is a compact Lie group acting with finitely
many orbit types ({\it e.g.}, if $G$ is finite), then the {\sl isotropy 
depth} of the action is the largest nonnegative integer $d$ such that 
one has a sequence of isotropy subgroups
$$H_0\subsetneqq H_1\subsetneqq \cdots \subsetneqq H_d.$$
In particular, an action with only one isotropy type ({\it e.g.}, a free
action) has isotropy depth zero, and a semifree action has isotropy 
depth one; the results of [DuS] include extensions of 3.1 and 3.2 to 
actions with isotropy depth one.  Aside from semifree actions, the 
class with isotropy depth one also includes fixed point free, effective
actions of ${\Bbb Z}_{pq}$, where $p$ and $q$ are distinct primes 
(since the group in question contains exactly two nontrivial proper 
subgroups, and neither contains the other).
\bigskip
One of the main steps in extending Theorem 3.1 is the description of 
an analog to $B({\text{\bf QF}}_M)$ for actions of isotropy depth greater 
than one.  Suppose we are given a compact smooth $G$-manifold $M$; we 
begin by choosing a $G$-invariant metric on $M$ and using it to construct 
a system of invariant tubular neighborhoods of the fixed point sets 
$M^H$ where $H$ runs through the isotropy subgroups of the action.  If
we are given an action of isotropy depth zero, then the finiteness of
$G$ implies that $M$ splits into a
disjoint union of codimension zero submanifolds $M^K$ where $K$ runs
through the conjugates of $H$, and the {\sl isovariance diagram} 
$\iidd (M)$ is given by the partially ordered set whose members are the 
sets $M^K$.  One then assumes that $\iidd(Y)$ has been defined for
smooth actions of isotropy depth less than $d$ such that $\iidd(Y)$
satisfies some appropriate conditions ({\it e.g.}, it contains all 
fixed point sets $Y^H$ where $H$ runs through the isotropy subgroups 
of the action).

Consider next a smooth action on $M$ 
with isotropy depth equal to $d$, and let $F_d\subset M$ be the
set of points whose isotropy subgroups are maximal.  It follows that 
$F_d$ is an invariant union of smooth submanifolds (neatly embedded if
$\partial M\neq\varnothing$).  Let $S\langle d\rangle$ and $D\langle d
\rangle$ be the unit sphere and disk bundles for an invariant tubular
neighborhood of $F_d$ (with the abuse of language convention at the 
end of Section 1), and let $C\langle d\rangle := M\starbar F_d$ be 
the closure of the complement of $D\langle d\rangle$.  The diagram 
$\iidd (M)$ of $G$-invariant subspaces of $M$ is then constructed in
pieces as follows:  Since $S\langle d\rangle$ and $C\langle d\rangle$
have isotropy depth $\leq d-1$ we can take $\iidd (M;M\starbar F_d)$
to be the union of $\iidd(S\langle d\rangle)$ and $\iidd(C\langle d
\rangle)$.  The set $F_d$ is a pairwise disjoint union of the subsets 
$M^{H}$ where $H$ runs through the maximal isotropy subgroups of the 
action, so define $\iidd (M; F_d)$ to be the family of all such sets 
$M_{(K)}$.  Next, define $\iidd(M;D\langle d\rangle)$ to be the family
of all subsets $D\langle d\rangle^H$; the partially ordered set
$\iidd(M)$ is then defined to be a family of subsets generated from 
$\iidd (M;M\starbar F_d)$ and $\iidd(M; D\langle d\rangle)$ by adjoining 
suitable unions $P\cup Q$ where $P\in \iidd (M;M\starbar F_d)$ and 
$Q\in \iidd(C\langle d \rangle)$.  

With these definitions of isovariance
diagrams Theorem 3.1 generalizes to a similar canonical isomorphism:
$$
\bmatrix
G-\text{equivariant}\\
\text{homotopy classes of}\\
\text{continuous equivariant}\\
\text{diagram morphisms}\\
\iidd(M)\to \iidd(N)\endbmatrix
\qquad\cong\qquad\bmatrix
G-\text{isovariant}\\
\text{homotopy classes of}\\
\text{continuous isovariant}\\
\text{maps of spaces}\\
M\longrightarrow N\endbmatrix
$$
where $M$ and $N$ are compact smooth $G$-manifolds.  Roughly speaking,
this can be done inductively by first adjusting the map on $D\langle 
d\rangle$ and then using an inductive hypothesis to adjust the map on 
$M\starbar F_d$ leaving $S\langle d\rangle$ fixed.
%\bigskip\bigskip\bigskip
\vfill\eject
\centerline{\bf 4. Isovariance versus equivariance}
\bigskip\bigskip
One basic question in equivariant homotopy theory is to determine 
whether a continuous map between two $G$-spaces is homotopic to an 
equivariant map ({\it cf.}  [LW], [MP], and [La]).
In this section we are interested in a corresponding question for
isovariant homotopy theory; namely, whether a continuous equivariant
map is equivariantly homotopic to an isovariant map.  Several special 
cases have been studied in by geometric methods.  One example
will be discussed in Section II.2; other results include Illman's 
work on isovariance and equivariant general position [{\smc Il}4]
and results with applications to embedding manifolds in the metastable
range (see Haefliger [Hae, Prop. 2, p. 245/06] and Harris [Har, Prop.
13, p. 24]).  Each approach seems to yield insights not apparent from
the others.
\bigskip
{\sl Default hypothesis: }  To keep the notation relatively
uncomplicated we shall only consider semifree actions on connected
manifolds whose fixed point sets are connected.
\medskip
Similar results also hold for more general actions, but the 
terminology quickly becomes far more complicated.
\bigskip
We begin with an elementary observation.
\proclaim{Proposition 4.1}  Let $M$ and $N$ be as above, let $\alpha_M$
and $\alpha_N$ be the equivariant normal bundles of the fixed point
sets, let $D(-)$ denote an associated unit disk bundle, and let $f:M\to
N$ be an equivariant map.  Then $f$ is equivariantly homotopic to a map
$h$ such that $h(D(\alpha_M))\subset D(\alpha_N)$.  Furthermore, if
$h_0$ and $h_1$ are two equivariant maps with this property and $\Phi_t$
is an equivariant homotopy from $h_0$ to $h_1$, then there is an
equivariant homotopy $\Psi_t$ from $h_0$ to $h_1$ such that
$\Psi_t(D(\alpha_M))\subset D(\alpha_N)$ for all $t\in
[0,1]$.
\endproclaim
\medskip
This is essentially a special case of [DuS, Prop. 5.1].$\blacksquare$
\bigskip
Theorem 3.1 and Proposition 4.1 combine to yield the following lifting
condition for finding an isovariant map in an equivariant homotopy
class:
\proclaim{Theorem 4.2}  Let $M$ and $N$ be as above, and let $f:M\to N$
be a continuous equivariant map satisfying the conditions of Proposition
$4.1$.  Then $f$ is equivariantly homotopic to an isovariant map if the 
associated maps $S(\alpha_M)\to D(\alpha_N)$ and $M\starbar M^G\to N$
lift -- equivariantly and compatibly -- to $S(\alpha_N)$ and $N\starbar
N^G$ respectively.
\endproclaim
This is a special case of [DuS, Thm. 5.3].$\blacksquare$
\bigskip
One can obtain cohomological isovariance obstructions by combining
Theorem 4.2 with obstruction theory in several different ways.  Typical
results along this line are given in [DuS, Thms. 5.4--5.5].  The remarks
at the end of [DuS, Sec. 5] discuss variants and special cases of these
theorems.
\bigskip
\centerline{\it Comparative computations}
\medskip
Another way of studying the difference between equivariant and isovariant
homotopy is to compare the homotopy groups of an equivariant function 
space $F_G(X,Y)$ with those of the corresponding isovariant function 
space $IF_G(X,Y)$, say if $X$ and $Y$ satisfy the basic assumptions of
[DuS].  For example, one could use the appropriate Barratt-Federer spectral 
sequences from [Sc2] and [DuS] in each case, and as a first step
it might be worthwhile to take tensor products with the rationals and
obtain information about rational homotopy groups.

The final section of [DuS] provides a comparison along these 
lines when $G={\Bbb Z}_p$ and $X=Y=S(V)$, where $p$ and $q$ are distinct
odd primes and $V$ is a linear $G$-representation such that $V$ contains
at least two free irreducible summands.  If $F_G(V)$ is the space of 
equivariant self maps of $S(V)$ then the Barratt-Federer spectral 
sequence of [Sc2] shows that $\pi_k(F_G(V))\otimes{\Bbb Q} = 0$
for all but finitely many $k$.  On the other hand,  if $IF_G(V)$ is 
the space of isovariant self maps of $S(V)$ then the results of 
[DuS, Sec. 6] show that each rational homotopy group 
$\pi_k(IF_G(V))\otimes{\Bbb Q}$ is finite dimensional, but the 
sequence of dimensions $d_k$ satisfies limsup${}_{k\to\infty}d_k/k^n 
=\infty$ for every positive integer $n$.  There is an explanation
for this difference in terms of the Barratt-Federer spectral sequences:
In the equivariant setting one has cohomology groups with coefficients
in the homotopy groups of spheres, and rationally these vanish in all
but at most two dimensions.  On the other hand, 
in the isovariant setting one has cohomology groups with coefficients
in the homotopy groups of wedges of spheres, and the ranks
of these groups tend to grow exponentially as the dimension increases.
\bye
