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\centerline{\bf PART II}
\bigskip

\centerline{\bf ISOVARIANT HOMOTOPY, CLASSIFICATION PROBLEMS, AND}
\bigskip
\centerline{\bf GENERAL POSITION}
\bigskip
\bigskip
\bigskip
\centerline{\it Background material}
\bigskip
As in Part I we assume basic concepts in algebraic topology and 
transformation groups in Bredon, tom Dieck, Dovermann-Schultz,
Milnor-Stasheff, and Spanier.  Beyond this, we shall frequently mention
the concept of {\sl simple homotopy equivalence} as presented in 
Milnor's article [{\smc Mln2}] or M.~Cohen's book [Co], and we shall
also use data from the {\sl Sullivan-Wall surgery exact sequence}.  The 
standard reference for the latter is Section 10 of Wall's book, {\sl 
Surgery on Compact 
Manifolds} (:= [{\smc Wl}]). Most of the material of immediate interest 
in this article is summarized in [Brw2], and in particular the 
Sullivan-Wall sequence is presented in [Brw2, p. 29] with a minimum 
of technical diversions.  However, we 
shall use notation that differs slightly from that of [{\smc Wl}] and 
[Brw2], mainly because we shall also need to consider variants of 
the structure sets described in those references.  
\bigskip
Both [{\smc Wl}] and [Brw2] deal with
structure sets ${\Cal S}^{Diff}(X)$ of simple homotopy structures on a 
simple Poincar\'e complex $X$ and with certain algebraically defined 
surgery obstruction groups $L_k(\pi)$.  As noted in Section 17 of 
[{\smc Wl}] one can define analogous homotopy structure sets, surgery 
obstruction groups, and exact sequences for homotopy structures on a 
Poincar\'e complex; for this theory, equivalent structures are 
$h$-cobordant rather than diffeomorphic. To distinguish between the
two structure set theories in the smooth category we shall denote the 
simple homotopy objects by ${\Cal S}^{s,Diff}$ and $L^s_k(\pi,w)$, and 
we shall denote the ordinary homotopy objects by ${\Cal S}^{h,Diff}$and
$L^h_k(\pi,w)$; here $w$ refers to the homomorphism $\pi\to{\Bbb Z}_2$
defined by the first Stiefel-Whitney class.  We shall also use somewhat
different notation for the bordism classes of degree one normal maps 
that are called $N^{Diff}(X)$ in [Brw2].  Standard results in algebraic 
and geometric topology imply that $N^{Diff}(X)$ is isomorphic to the set
of homotopy classes $[X,F/O]$, where $F/O$ is the space classifying stable
fiber homotopy trivializations of stable vector bundles of $X$; this space
is considered in [Brw3, Sec. II.4] where it is called $G/O$ (we use $F/O$
rather than $G/O$ because the two names are essentially used interchangeably
in the literature and $G$ will frequently denote a finite group 
in the discussion that follows).  A full discussion of the isomorphism 
$N^{Diff}(X)\approx [X,F/O]$ appears in [Brw3, Thm. II.4.4, pp. 46--49].  
This description of $N^{Diff}(X)$ is useful because there is an exact 
sequence of
abelian groups
$$\cdots \to {\widetilde {KO}}(\Sigma X)\to \{\Sigma X,S^0\}\to [X,F/O]\to
{\widetilde {KO}}(X)\to {\widetilde {KSph}}(X)$$
where ${\widetilde{KO}}$ denotes reduced real $K$-theory, $\{-,S^0\}$
denotes stable cohomotopy, and ${\widetilde{KSph}}$ denotes the analog
of reduced $K$-theory for stable spherical fiber spaces.
\bigskip
Finally, we shall also note the existence of {\sl relative structure 
sets} ${\Cal S}^{c,Diff}(X,\partial X)$ for Poincar\'e complexes with 
formal boundaries; here $c=s$ or $h$.  The basic idea is to take 
simple homotopy (resp. homotopy) equivalences $(M,\partial M)\to 
(X,\partial X)$ such that the map of boundaries is a diffeomorphism. There
is an extension of the Sullivan-Wall exact sequence to such objects 
modulo some adjustments; the Wall groups are simply those
for the fundamental group and first Stiefel-Whitney class of $X$, but the 
normal bordism set $N^{Diff}(X,\partial X)$ in this case is equivalent to
$[X\cup{\text{Cone}}(\partial X), F/O]$.
\bigskip
Recent work of M. Dawson [Daw] includes an independent proof of the main
results in Section 1 and applications to smooth variants of the
Cappell-Weinberger {\sl replacement theorems} ({\it e.g.}, see Theorem
5.1 below for a statement of one such result in the locally linear 
topological or PL categories).  Additional remarks on this work 
appear in Section 5.
\bigskip\bigskip\bigskip
\centerline{\bf 1. Isovariant homotopy structures}
\bigskip\bigskip
The main ideas of surgery theory began to emerge in the nineteen
fifties, and they became well established
with the work of M. Kervaire and J. Milnor [KM] on 
classifying smooth manifolds that are homotopy equivalent to spheres 
({\it i.e.}, homotopy spheres).  Subsequent work of W. Browder and 
S. P. Novikov yielded far reaching extensions of [KM] to existence and
classification questions for simply connected manifolds of a fixed 
homotopy type, and still further work of Wall extended the theory to 
manifolds with arbitrary fundamental groups [{\smc Wl}].
\bigskip
It soon became clear that surgery theory also yielded valuable information
on existence and classification questions for group actions on manifolds
({\it cf}. [Brw1]).  In particular, many striking applications to
free differentiable group actions on spheres were made during the nineteen
sixties ({\it e.g.}, see [HH], [Hs], [LdM]).  Systematic efforts to 
study nonfree
actions also began in the nineteen sixties with work of Browder and 
Petrie [BP] and Rothenberg and Sondow [RSo] on classifying smooth 
$G$-actions that are semifree and homotopically linear -- in other words, 
both $M$ and $M^G$ are closed manifolds that are homotopy equivalent to 
spheres (see also [Brw1] and [Sc3]).  Actions of this type can be 
viewed as smooth $G$-manifolds that
are equivariantly homotopy equivalent to a linear $G$ sphere $S(V)$ 
given by the unit sphere in some orthogonal, semifree representation of
$G$ on a finite dimensional real inner product space $V$.  In analogy 
with the Browder-Novikov-Wall extension of [KM] to arbitrary closed 
manifolds, it is natural to search for an extension of the Browder-Petrie
and Rothenberg-Sondow work to more general $G$-manifolds.
\bigskip
A major step in this direction was due to W. Browder, who presented his
ideas in a lecture at a conference in 1971 (see p. $vii$ in the book 
containing [MnY1]).  This work was later extended by F. Quinn and 
published jointly in [BQ].  The basic idea was to consider manifolds
that are {\sl isovariantly} homotopy equivalent to a given model such 
that the isovariant equivalence satisfies a {\sl transverse linearity}
condition. For the sake of simplicity we shall only describe this for
semifree actions. In such cases the isovariant homotopy equivalence 
$f:M\to N$ is supposed to be a map of triads from 
$(M;M\starbar M^G, D(\alpha_M))$ to $(N;N\starbar N^G, D(\alpha_N))$, 
where $\alpha_Y$ refers to the (equivariant) normal bundle 
of $Y^G$ in $Y$ as in Part I, and the induced map from $D(\alpha_M)$ 
to $D(\alpha_N)$ is assumed to be
(orthogonally) linear and fiber preserving.  If one specializes this 
theory to $G$-manifolds modeled by linear $G$-spheres, one obtains a 
theory that maps naturally into the Browder-Petrie and 
Rothenberg-Sondow theories and includes many infinite families of 
examples from (both of) the latter.
\bigskip
One of the most important properties of the Browder-Quinn setting is
the existence of a surgery exact sequence that is formally parallel 
to the Sullivan-Wall sequence ({\it cf.} [BQ, Thm. 2.2, p. 29]):
$$\cdots\to L_{n+1}^{c,BQ}(X)\to {\Cal S}_G^{c,BQ}(X)\to [X/G,F/O]
\to L_{n}^{c,BQ}(X)$$
In this sequence $X$ is a closed smooth $G$-manifolds, the symbol 
$c$ denotes either $s$ (for equivariant simple homotopy; {\it cf.}
Illman [{\smc Il2}] or Rothenberg [Ro]) or $h$ 
(for ordinary equivariant homotopy), and 
the groups $L_{*}^{c,BQ}(X)$ are the {\sl Browder-Quinn 
surgery obstruction groups} as defined and studied in [BQ] and [DoS2,
Sec. 2].  Although these groups are written in terms of $X$, they 
are in fact determined by weaker data that is summarized in the 
{\sl geometric reference} $R_X$ of W.~L\"uck and I. Madsen [L\"uMa, 
Definitions (2.3) and (3.1), pp. 512 and 516].  As noted in [BQ] and
[DoS2], certain natural exact couples determine spectral sequences 
converging to the groups $L_{*}^{c,BQ}(X)$ such that the initial terms
are ordinary Wall groups $L_*^c$, and therefore one can view the 
terms in the Browder-Quinn surgery sequence as computable up to 
determination of the homotopy groups of $F/O$ and the appropriate 
Wall groups.  In analogy with the Sullivan-Wall sequence, there are
also relative versions of the Browder-Quinn sequence involving 
structure sets ${\Cal S}^{c,BQ}(X,\partial X)$ represented by 
transverse linear $G$-homotopy equivalences $(M,\partial M)\to
(X,\partial X)$, with $c=h$ or $s$ as usual, such that $\partial M$
maps to $\partial X$ by a diffeomorphism.
\bigskip
The basic aim of isovariant surgery theory is to provide a setting 
that is broad enough to include both the Browder-Quinn theory and 
the work of Browder-Petrie and Rothenberg-Sondow, but is also more
or less computable, at least in some cases beyond those of [BP], 
[RSo], and [BQ].
\medskip
We shall begin by relating [BQ] to [BP] and [RSo] in the case of 
homotopically linear semifree group actions on spheres.  The tangential
representation at a fixed point will be assumed to have the form 
$V\approx {\Bbb R}^k\oplus\alpha$, where the representation $\alpha$ 
has no trivial summands (hence $G$ acts freely on the unit sphere
$S(\alpha)$).  Deviating slightly from the 
the notation of [Sc3], let $CS_k(G,\alpha)$ be the set of equivariantly
oriented $h$-cobordism classes of 
homotopically linear semifree group actions as described above, where
the tangent space at a fixed point is $G$-isomorphic to the 
representation $V$; as noted in [RSo] these sets have natural 
abelian group structures if the dimension of the fixed point set is
at least 2.  There are also canonical abelian group structures on 
the relative Browder-Quinn structure sets 
$${\Cal S}^{h,BQ}_G(\,D(V),S(V)\,),$$ 
and there is a natural forgetful map ${\Cal S}^{h,BQ}_G(\,D(V),S(V)\,)
\to CS_k(G,\alpha)$, given by gluing a copy
of $D(V)$ to the boundary, that is additive.  As in Part I let 
$F_G(\alpha)$ be the space of equivariant self maps of 
the unit sphere $S(\alpha)$. The orthogonal centralizer of $\alpha$ is a 
compact subgroup of the topological monoid $F_G(\alpha)$ and will be 
denoted by $C_G(\alpha)$; it follows that the quotient space construction
defines a principal bundle 
$$C_G(\alpha)\subset F_G(\alpha)\to F_G(\alpha)/C_G(\alpha).$$
One can then define a {\sl knot invariant} homomorphism 
$$\omega: CS_k(G,\alpha)\to \pi_k(F_G(\alpha)/C_G(\alpha)\,)$$
as in [Sc3, top of p. 311] or [Sc4, Sec. 2].
\medskip
\proclaim{Proposition 1.1} In the notation of the preceding paragraph,
there is a long exact sequence of the following form:
$$\cdots\pi_{k+1}(F_G(\alpha)/C_G(\alpha))\to
{\Cal S}^{h,BQ}_G(D(V),S(V))\to
CS_k(G,\alpha)\to 
\pi_k(F_G(\alpha)/C_G(\alpha))$$
\endproclaim
In other words, the forgetful map from the Browder-Quinn groups to 
the $CS_k(G,\alpha)$ groups is the ``homotopy fiber of the knot 
invariant.''$\blacksquare$
\medskip
\def\boldI{{\text{\bf I}}}
The next step in the comparison is to note that each element of 
$CS_k(G,\alpha)$ is canonically isovariantly homotopy equivalent to 
$S(V\oplus {\Bbb R})\approx D_+(V)\cup_\partial D_-(V)$.  This suggests
that the groups $CS(G,V)$ should be viewed as structure sets for 
relative $G$-isovariant homotopy structures on $(D(V),D(V))$.  In 
fact, it is possible to describe structure set theories for arbitrary
smooth semifree $G$-manifolds in the spirit of [BP] and [RSo].  As 
before there are two versions $\boldI{\Cal S}^s_G$ and 
$\boldI{\Cal S}^h_G$ for isovariant simple homotopy and ordinary 
isovariant homotopy equivalences respectively.  There are also relative
versions of these structure sets for isovariant homotopy structures 
that are diffeomorphisms on the boundary.
\bigskip
The exact sequence of [Sc3, (1.1), p. 311] plays an important role in many 
studies of the groups $CS_k(G,\alpha)$, and therefore one would like to 
have analogs of this for the structure sets $\boldI{\Cal S}^c_G(M)$.  
In order to do this it is necessary to generalize the
knot invariant, and this in turn requires a suitable analog of the 
homotopy group $\pi_k(F_G(\alpha)/C_G(\alpha)\,)$.  The approach below
is an adaptation of ideas from [Sc9, Secs. 2--3].
\medskip
\noindent{\bf Definition. }  Let $X$ be a $G$-space, let $A\subset X$ be
$G$-invariant, and let $\xi$ be a $G$-vector bundle over $X$.  A 
{\sl $G$-isovariant fiber homotopy linearization} of $\xi$ is a pair
$(\xi,h)$ consisting of a $G$-vector bundle $\omega\downarrow X$ and a
$G$-isovariant fiber homotopy equivalence $h:S(\omega)\to S(\xi)$ that
is an orthogonal isomorphism over $A$.  The set $F/O_{G,iso}(\xi
\ {\text{rel}}\ A)$ is the set of Such objects modulo the equivalence
relation generated by fiber preserving orthogonal vector bundle
isomorphisms $S(\omega')\to S(\omega)$ and isovariant homotopies 
$H_t:S(\omega)\to S(\xi)$ that are orthogonal over $A$.

{\sl Note. } If $G$ acts semifreely on $M$ with $X=M^G$ and
$\xi=\alpha_M$, then $G$ acts freely on $S(\xi)$ and the isovariance
condition merely requires that $G$ act freely on $S(\omega)$.
\bigskip
If $f:M\to N$ is an isovariant homotopy equivalence of semifree smooth
$G$-manifolds, then one can define a generalized knot invariant of $f$
in $F/O_{G,iso}(\alpha_N)$ as follows:  By the results of Section I.3 we
can deform $f$ isovariantly so that $f$ maps $S(\alpha_M)$ to
$S(\alpha_N)$, and the construction yields a unique isovariant homotopy
class of maps $S(\alpha_M) \to S(\alpha_N)$; this map can be further
deformed, again uniquely up to isovariant homotopy, to a map $f'$ such
that the following diagram commutes:
$$
\CD
S(\alpha_M) @>{f'}>> S(\alpha_N)\\
@VVV @VVV\\
M^G @>{f^G}>> N^G\\
\endCD
$$
Since $f^G$ is a homotopy equivalence there is a unique $G$-vector
bundle $\beta$ (up to isomorphism) such that $\alpha_M\cong
\{f^G\}^*\beta$, and it follows that $f'$ factors through an isovariant
fiber homotopy equivalence $\eta:S(\beta)\to S(\alpha_M)$; by
construction the class of $(\beta,\eta)$ in the set $F/O_{G,iso}
(\alpha_N)$ is well defined, and this is the generalized knot 
invariant of $f$.
\medskip
A similar construction is valid for relative homotopy structures on a
compact smooth semifree $G$-manifold with boundary, and in this case 
the knot invariant lies in the relative set
$F/O_{G,iso}(\alpha_N\ {\text{rel}}\ \partial N)$.
The following result is a natural extension of Proposition 1.1 to
arbitrary smooth semifree $G$-manifolds:
\proclaim{Theorem 1.2}  If $X$ is a closed smooth semifree $G$-manifold
such that each component of $X^G$ is at least $5$-dimensional,
then there is an exact sequence of structure sets
$$\cdots F/O_{G,iso}(\alpha_X\times I\ {\text{rel}}\ X\times\{0,1\})
\to{\Cal S}^{c,BQ}(X)\to
\boldI{\Cal S}^c_G(X)\to F/O_{G,iso}(\alpha_X)$$
that extends infinitely to the left.  All objects to the left of 
${\Cal S}^{c,BQ}(X)$ are groups, and all maps are compatible with
group structures as in the Sullivan-Wall exact sequence.$\blacksquare$
\endproclaim
{\it Reminder. } In the Sullivan-Wall exact sequence the source and
target for the surgery obstruction map $[X,F/O]\to L^c_n(\pi,w)$ 
are abelian groups but the map itself is not additive in general.
\bigskip
The preceding sequence relates ${\Cal S}^{c,BQ}(X)$ to
$\boldI{\Cal S}^c_G(X)$.  There is also an exact sequence for 
$\boldI{\Cal S}^c_G(X)$ that generalizes the exact sequence for
homotopy linear actions in [Sc3, (1.1), p. 311].  Before stating this 
result we need a notational convention:
\medskip
\itemitem{}If $Y$ is a compact bounded manifold then $\partial^*: {\Cal
S}^c(Y)\to{\Cal S}^c(\partial Y)$ is given by restriction to the
boundary.
\bigskip
\proclaim{Theorem 1.3}  Let $X$ satisfy the conditions of Theorem $1.2$.
Then there is an exact sequence of sets
$$
\CD
\cdots{\Cal S}^{c,Diff}(X-{\text{Int}}(D(\alpha_X))/G,S(\alpha_X)/G) @>>>
\boldI{\Cal S}^{c}_G(X) \\
@. @VVV \\
@. {\Cal S}^{c,Diff}(X^G)\times F/O_{G,iso}(\alpha_X) \\
@. @VVV\\
@. {\Cal S}^{c,Diff}(S(\alpha_X)/G)/{\text{Image}}\ \partial^* 
\endCD
$$
that extends infinitely to the left.  All objects to the left of 
the raised dots
are groups and the corresponding maps are compatible with group
structures as in Theorem $1.2$.$\blacksquare$
\endproclaim
\medskip
As usual, there is a variant of this exact sequence for relative
structure sets.
\bigskip
\centerline{\it Extensions to more general actions}
\medskip
Theorems 1.2 and 1.3 provide a means for analyzing isovariant structure
sets in terms of ordinary structure sets and equivariant/isovariant
homotopy theory, provided the group action is semifree.  Each of these
extends to actions with more complicated orbit structure.  In
particular, a generalization along the lines of 1.2 was considered in
earlier work by the author [Sc13]; the necessary 
modifications include 

\itemitem{$(i)$} an extension of $F/O_{G,iso}(-)$ from vector bundles
to the vector bundle systems (known as $\Pi$-{\sl bundles} in the papers
of Dovermann-Petrie-Rothen\-berg [DP1--2, DR]) over Sing$(X)$,

\itemitem{$(ii)$} the definition of a generalized knot invariant
for an isovariant homotopy equivalence, taking its value in the set
described above.

\noindent
With this machinery in place, it is a formal exercise to prove that 
the forgetful map ${\Cal S}^{BQ,c}(X)\to\boldI{\Cal S}^c_G(X)$ is 
essentially the homotopy fiber of the knot invariant constructed by
$(ii)$.
\medskip
Theorem 1.3 is essentially a means for analyzing isovariant homotopy
structures on $X$ by splitting them into two pieces; namely, pieces over
a tubular neighborhood of $X^G$ and pieces over the free $G$-manifold
$X\starbar X^G$.  There are several ways of extending this
to more general actions; we shall only discuss two extreme cases here.
The first approach is to split an arbitrary smooth $G$-manifold into a
smooth equivariant regular neighborhood ${\Cal R}_X$ of the singular set
Sing$(X)$ and the free $G$-manifold $X\starbar{\text{Sing}}(X)$.  This
approach was discussed in [Sc6]; we shall not attempt to provide a
precise description because it requires a notion of isovariant structure
set for the singular set Sing$(X)$, which is generally not a smooth 
$G$-manifold (with a possibly noneffective group action).  To describe
a complementary approach, we shall assume for the sake of simplicity
that all isotropy subgroups are normal ({\it e.g.}, this happens if $G$
is abelian).  Suppose that $H$ is a maximal isotropy subgroup, and let
$\alpha_H$ be the equivariant normal bundle of $X^H$ in $X$.  Then one
has the following analog of Theorem 1.3:
\medskip
\proclaim{Theorem 1.4} Let $X$ satisfy the conditions of the preceding
paragraph.  Then there is an exact sequence of sets
$$
\CD
\cdots\boldI{\Cal S}^{c}_G(X-{\text{Int}}(D(\alpha_X)),
S(\alpha_X)) @>>>
\boldI{\Cal S}^c_G(X) \\
@. @VVV \\
@. \boldI{\Cal S}^{c}_{G/H}(X^H)\times F/O_{G,iso}(\alpha_X) \\
@. @VVV\\
@. \boldI{\Cal S}^{c}_G(S(\alpha_X))/{\text{Image}}\ \partial^* 
\endCD
$$
that extends infinitely to the left.  All objects to the left of 
the raised dots
are groups and the corresponding maps are compatible with group
structures as in Theorems $1.2-1.3$.$\blacksquare$
\endproclaim
This result provides a means for analyzing isovariant structure sets
inductively with respect to the number of orbit types, for the two
structure sets in the sequence aside from $\boldI{\Cal S}^{BQ,c}_G(X)$
have fewer orbit types than the original action, and the same is true
for the data in $F/O_{G,iso}(-)$.  In subsequent work we shall study
special cases of this sequence in connection with questions from Section
3 below.
\bigskip\bigskip\bigskip
\centerline{\bf 2. Isovariance and the Gap Hypothesis}
\bigskip\bigskip
During the nineteen seventies and early eighties, Petrie and several
other topologists (beginning with S. Straus [Str]) found many striking
applications of surgery to smooth $G$-manifolds satisfying the following
basic condition:
\medskip
{\smc Gap Hypothesis. } A smooth $G$--manifold $M$ is said 
to satisfy the (standard version of the)
{\sl Gap Hypothesis} if for each pair of 
isotropy subgroups $H \supsetneqq K$ and each pair of 
components $B\subset M^H$, $C\subset M^K$ such that
$B\subsetneqq C$ we have
$$
\dim B < {\tsize\frac12}(\dim C ).\tag{\ddag}
$$
\bigskip
This is basically a general position condition.  Its usefulness arises
because surgery theory involves the existence of smoothly embedded
spheres whose dimensions are no more than half the dimensions of the
ambient manifolds.  If the Gap Hypothesis holds, then one can choose the
appropriate embedded spheres in each fixed set component $C\subset M^K$ 
to miss all the components $D\subset M^H$ such that $D\subsetneqq C$.
This means that all the constructions involving embedded spheres can be
done equivariantly on the set $M^K\cap M_{(K)}$, which has only one
isotropy type (namely, $K$).  In effect, this reduces an equivariant
surgery problem to a sequence of nonequivariant problems over the orbit 
spaces $M_{(K)}/G$.  A similar reduction arises in the Browder-Quinn
theory even if the Gap Hypothesis does not hold; this follows directly
from the isovariance and transverse linearity conditions of [BQ].
\bigskip
Most of Petrie's work dealt with the existence of smooth $G$-actions on
disks and spheres with properties quite unlike those of orthogonal
actions (compare [Pet1--2] and [DPS]).  In a somewhat different 
direction, K. H.
Dovermann and M. Rothenberg modified Petrie's approach to construct
classification theories for $G$-manifolds in a given equivariant
homotopy type {\sl provided the Gap Hypothesis holds} (see [DR] and 
[L\"uMa]).

One of the central problems in equivariant surgery is to understand the
role of the Gap Hypothesis more clearly ({\it cf.} [Sc12, Sec. 4]), and thus
it is natural to seek relationships between the isovariant homotopy
structure theory of Section 1, which does not require the Gap
Hypothesis, and equivariant surgery theories that somehow rely on the
Gap Hypothesis as in [DR] or [L\"uMa] (related examples are also
discussed in [DoS2, Sec. II.3]).  As noted in [Daw], the isovariant
structure sets of Section 1 lie somewhere between such 
equivariant surgery theories and the Browder-Quinn theories.
The following result of S. Straus [Str] and W. Browder [Brw4] establishes 
a stronger and more precise relationship; in
particular, the theories of [DR] and [L\"uMa] are equivalent to the
theories of Section 1 when the Gap Hypothesis holds.
\medskip
\proclaim{Theorem 2.1}  Let $f:M\to N$ be an equivariant homotopy
equivalence of closed smooth $G$-manifolds that satisfy the Gap
Hypothesis.  Then $f$ is equivariantly homotopic to an isovariant
homotopy equivalence.  Furthermore, if $M\times[0,1]$ satisfies the Gap
Hypothesis then this isovariant homotopy equivalence is unique up to
isovariant homotopy.
\endproclaim
\medskip
This result and the machinery of Sections I.4 and II.1 suggest a two
step approach to analyzing smooth $G$-manifolds within a given
equivariant homotopy type if the Gap Hypothesis does not necessarily
hold; namely, the first step is to study the obstructions to isovariance
for an equivariant homotopy equivalence and the second step is to study
the isovariant structure sets of the preceding section.
\bigskip
\noindent{\bf Sketch of the proof of Theorem 2.1. }  We shall only deal
with semifree $G$-manifolds in order to illustrate the ideas without 
addressing the bookkeeping problems that arise for more general actions;
furthermore, for the sake of simplicity we shall use a slightly stronger
version of the Gap Hypothesis with $\dim B +\varepsilon < 
{\tsize\frac12}(\dim C )$ for some small positive integer
$\varepsilon$.  Finally, we shall only consider the existence question;
the uniqueness result follows by applying similar methods to $M\times
[0,1]$.

The original proofs of Straus and Browder rely heavily on methods and 
results from nonsimply connected surgery.  The argument presented here
does not completely eliminate geometric topology, but it only requires
simple considerations involving embeddings in the general position range
and transversality.  Of course it would be interesting to know if the
proof can be done entirely with homotopy theoretic machinery.
\bigskip
The first step in the proof is to deform $f$ equivariantly so that it
maps $D(\alpha_M)$ isovariantly to $D(\alpha_N)$ such that 
$S(\alpha_M)$ is sent to $S(\alpha_N)$.  This will follow quickly if 
$S(\alpha_M)$ and $S(\{f^G\}^*\alpha_N)$ are equivariantly fiber
homotopy equivalent.  To prove the latter, one first uses a result of 
K.~Kawakubo [Ka] to show that the equivariant tangent bundle $\tau_M$
is stably equivariantly fiber homotopy equivalent to $f^*\tau_N$.
Restricting to fixed point sets, we conclude next that the restrictions
of these bundles to $N^G$ are also equivariantly stably fiber homotopy
equivalent; in other words, $\{f^G\}^*\tau_{N^G}\oplus \{f^G\}^*\alpha_N$ 
is equivariantly stably fiber homotopy equivalent to 
$\tau_{M^G}\oplus \alpha_M$.  The classifying space versions of the
standard splittings for equivariant stable homotopy theory ({\it e.g.},
the discussion at the end of [Se]) then imply that $\{f^G\}^*\alpha_N$ 
is equivariantly stably fiber homotopy equivalent to $\alpha_M$.  Since
the dimensions of the latter bundles are at least somewhat larger than the 
dimensions of $M^G$ and $N^G$, the stable range theorems of [Sc1]
and [Sc5]
imply that the unit sphere bundles of $\{f^G\}^*\alpha_N$ and $\alpha_M$
are already equivariantly fiber homotopy equivalent before stabilization.
\medskip
The second step is to analyze the set of points where the modified map
fails to be isovariant.  We can apply transversality on the complement
of $D(\alpha_M)$, without changing the map on $S(\alpha_M)$, so that
a further equivariant deformation of $f$ is transverse to $N^G$ on the
complement of $M^G$.  It follows immediately that the set $Y$ of
nonisovariant points is a smooth invariant submanifold such that 
$\dim Y=\dim M^G$.  Using general position and the fact that $f$ is an
equivariant homotopy equivalence, one can then show that $Y$ lies in 
some tubular neighborhood of the fixed point set (the inclusion of 
$Y$ can be deformed into $D(\alpha)$ because $f$ is an equivariant
homotopy equivalence, and by general position one can modify this into
an isotopy of $Y$ into $D(\alpha)$).
\medskip
The third step is to show that the map obtained in the previous step is
equivariantly homotopic to an isovariant map if and only if the class of
the nonisovariant set in an appropriate bordism theory vanishes.  By the
results of Section I.4, the obstruction to deforming the map $f_2$
obtained thus far is the obstruction to finding an equivariant lifting
of $f_2|M\starbar M^G$ from $N$ to $N\starbar N^G$.  Because the Gap
Hypothesis holds, one can use the Blakers-Massey Theorem to view this
lifting obstruction as the obstruction to finding an equivariant
nullhomotopy for the composite of $f_2|M\starbar M^G$ with the collapse
map $N \to N/N\starbar N^G\approx D(\alpha_N)/S(\alpha_N)$.
\medskip
The next to last step is to notice that the obstruction from the preceding 
step need not vanish, but it has a canonical indeterminacy given by the
possible choices of the equivariant fiber homotopy equivalence from
$\alpha_M$ to $\{f^G\}^*\alpha_N$.   In fact, since we are in the stable
range the homotopy classes of such equivalences are given by $[N^G,
F_G]$.  Finally, an analysis of the obstructions in the third step
shows that one can kill the isovariance obstruction by choosing a 
(possibly) different equivariant fiber homotopy equivalence on the
equivariant sphere bundles.$\blacksquare$
\bigskip\bigskip\bigskip
\centerline{\bf 3. Homotopy linear actions on spheres}
\bigskip\bigskip
As indicated in Section 1, the original interest in classifying smooth
manifolds in a given isovariant homotopy type involved certain smooth
group actions on homotopy spheres.  In this section we shall discuss
some basic questions in this area that can be analyzed, at least to some
extent, by the methods of the preceding sections.
\medskip
{}From a purely formal viewpoint we are interested in smooth
$G$-manifolds that are isovariantly homotopy equivalent to linear
actions on spheres.  However, for historical and practical reasons it is
more useful to deal with actions satisfying apparently weaker
assumptions and to prove that all such actions are isovariantly homotopy
equivalent to the appropriate linear example.

The basic concepts and constructions for homotopy linear actions are
summarized in [Sc10, Secs. 5--6].  We shall begin with a modified
version of the definition in [Sc10, Sec. 5, p. 274].
\bigskip
\noindent{\bf Definition. }  Let $\varphi_0$ be a linear representation
of $G$ on ${\Bbb R}^n+1$ that splits as $\varphi\oplus{\Bbb R}$ (with
trivial action on the second summand).  If $H$ is a subgroup of $G$ let
$n(H)+1$ denote the dimension of the real vector space $\varphi_0^H$
(hence $n(H)\geq 0$).  A smooth $G$-action $\gamma$ on a smooth manifold
$\Sigma^n$ is said to be {\sl strongly} $\varphi$-{\sl homotopy linear} 
($\equiv$ {\sl strongly} $\varphi$-{\sl homotopically linear} or {\sl
strongly} $\varphi$-{\sl semilinear}) if the following hold:
\medskip
\itemitem{(1)}  For each $H\subset G$ the fixed point set of $H$ is 
homeomorphic (but not necessarily diffeomorphic) to $S^{n(H)}$.

\itemitem{(2)}  If $H\subset K\subset G$ and $n(H)-n(K)=2$ then 
$\Sigma^H-\Sigma^K$ is homotopy equivalent to $S^1$.

\itemitem{(3)}  The induced $G$ representations at the tangent spaces of
points in $\Sigma^G$ are all equivalent to $\varphi$.
\bigskip
It is fairly elementary to show that each such action is $G$-homotopy
equivalent to the unit sphere $S(\varphi_0)$, or equivalently to the one
point compactification of $\varphi$ ({\it cf}. [Sc10, Prop. 5.1]); 
in fact, $\Sigma^n$ is usually $G$-homeomorphic to this linear sphere 
(see the remarks on [Sc10,p. 274] following Proposition 5.1), and in
the remaining cases the results of [DuS, Sec. 4] show that $\Sigma$ is
isovariantly homotopy equivalent to the linear action.  
\bigskip
{\sl Connected sums. }  If $G=\{1\}$ then a strongly homotopy  linear 
$G$-manifold is a manifold homeomorphic to $S^n$ ({\it i.e.}, an {\sl
exotic sphere}) by the Generalized Poincar\'e Conjecture [{\smc Mln1},
p. 109];  a diffeomorphism classification of such objects was developed 
in the previously mentioned work of Kervaire and Milnor during
the late nineteen fifties and early nineteen sixties [KM].  An
elementary but highly useful step in their program was the use of
objects with orientations and the introduction of an abelian group
structure on the oriented diffeomorphism classes of exotic spheres by
means of connected sums (see [Sc10, p. 275] and the references cited
there).  One can proceed similarly with strongly homotopy linear
$\varphi$-spheres:  Given two such $G$-manifolds $\Sigma_1$ and
$\Sigma_2$, let $D_i\subset\Sigma_i$ be $G$-diffeomorphic to the disk 
$D(\varphi)$  and glue $\Sigma_1-{\text{Int}}(D_1)$ to 
$\Sigma_2-{\text{Int}}(D_2)$ equivariantly along the common boundary;
once again one needs a suitable concept of orientation to ensure this
construction is well defined, and this can be done as in [Sc9, Sec.
1].  As noted in [Sc10, Prop. 5.2, p. 276], this yields a monoid
structure on the set of all (suitably equivariantly oriented) 
diffeomorphism classes of strongly $\varphi$-semilinear spheres, the
resulting monoid is abelian if the fixed point set dimension is at least
2, and if we factor out the submonoid of actions that bound
equivariantly contractible $G$-manifolds, then the resulting quotient is
a group (abelian if the fixed point sets are at least 2-dimensional).
Following [Sc10] we shall denote this group by $\Theta^G(\varphi)$.
\bigskip
\centerline{\it Digression---some motivation}
\medskip
Although one can certainly study the groups $\Theta^G(\varphi)$ for
their own sake, these groups also arise naturally in connection with
certain questions of independent interest.  Before proceeding with 
further results on such actions we shall describe some of these
contexts.
\medskip
\noindent{\bf Examples. } {\bf 1.} {\sl Smooth actions of arbitrary
$p$-groups on exotic spheres. }  In fact, as noted in [Sc6] 
this was one of the original motivations for studying the classification 
of smooth $G$-manifolds in a given isovariant homotopy type.  If one is
given an action of a finite abelian $p$-group on an exotic sphere, then
Smith theory shows that all the fixed point sets are mod $p$ homology
spheres.  This implies that an arbitrary such action admits an
isovariant map to a linear model with degree prime to $p$.  If the
dimension of the fixed point set of $G$ is at least 2, then one can use
these maps and the methods of Section 1 to define $p$-localized versions 
of the knot invariant with values in the abelian groups 
$$F/O_{G,iso}(S(\varphi^H\oplus{\Bbb R})\times [\varphi/\varphi^H] 
\ {\text{rel}}\ \{{\text{basept.}}\})_{(p)}$$
where $H$ is an arbitrary isotropy subgroup of the action.  
For the special case of cyclic $p$-groups where $H$ is the minimal
nontrivial isotropy subgroup, this was done previously
in [Sc4] and [Sc9, Sec. 4], where the invariants were used 
to obtain restrictions on the fixed point structure of smooth 
${\Bbb Z}_{p^r}$-actions on exotic spheres.  In subsequent work the more
general invariants will be used to study actions of other abelian
$p$-groups on exotic spheres in relatively low dimensions.
\bigskip
{\bf 2. } {\sl Fixed point sets of differentiable actions on spheres. }
Results of L. Jones [Jo], A. Assadi [As] and others show that certain 
variants of the
groups $\Theta^G(\varphi)$ carry the obstructions to realizing a mod $p$
homology $k$-sphere as the fixed point set of a smooth semifree 
${\Bbb Z}_{p^r}$-action on some homotopy $(k+2m)$-sphere, where $m\geq
2$ ({\it cf}.
[As]).  The basic idea is simple:  If $A$ is the homology sphere, let
$A_0$ denote $A$ with the interior of a closed disk removed.  Then one
can realize $A_0$ as the fixed point set of a smooth semifree 
${\Bbb Z}_{p^r}$-action on $D^{k+2m}$.  The induced action on the
boundary then determines an element of the appropriate variant group 
${\text{Var}}\Theta^G(\varphi)$, and one can extend the action to an
action on a homotopy sphere if and only if this element is zero.
Related ideas are used in [Sc14] to construct examples of smooth 
${\Bbb Z}_{pq}$-actions on spheres, where $p$ and $q$ are distinct odd
primes, such that the Pontrjagin numbers of the fixed point set 
are nontrivial.
\bigskip
{\bf 3. } {\sl Equivariant smoothings of topological $G$-manifolds. }
Results of Lashof and Rothenberg [LaR] show that the smoothability of a
$G$-manifolds and the classification of equivariant smoothings reduce to
equivariant bundle-theoretic questions, at least if there are no
4-dimensional components in the fixed point sets of the isotropy
subgroups.  This is formally parallel to ordinary smoothing theory for
topological manifolds ({\it cf.} [KiSb]).  However, in ordinary
smoothing theory the results of [KM] and [KiSb] translate the
bundle-theoretic problems into well known questions of homotopy theory,
but comparable insights into equivariant smoothing theory only exist in
special cases.  This is already evident in known results on the
topological classification of linear representations ({\it e.g.}, see
[CS1--3, CSSW, CSSWW]), which is the first step in analyzing the
bundle-theoretic problems in [LaR].  Partial results on the higher order
steps appear explicitly in [LaR] and [MR], and implicitly in [Sc7],
[KL], and [KwS6].  Standard techniques of engulfing theory ([Hud, Ch.
VII] or [RSa, Ch. 7]) imply that a strongly $\varphi$-homotopically
linear $G$-manifold is equivariantly homeomorphic to
$S(\varphi\oplus{\Bbb R})$ if the dimension of $\varphi^G$ is
sufficiently large [CMY, {\smc Il5}, Ro, Sc7], and thus information 
about the groups
$\Theta^G(\varphi)$ has immediate implications for equivariant smoothing
theory ({\it cf}. the results on rational characteristic classes in
[Sc7]).  It is conceivable that information on the groups
$\Theta^G(\varphi)$ can also shed light on equivariant smoothing theory
for more general $G$-manifolds; in particular, the results of [LaR,
pp. 215 and 264--265] suggest this.
\bigskip
{\bf 4. } {\sl Rational invariants for classifying smooth $G$-manifolds
up to finite ambiguity. }  In a sequence of papers culminating with
[RT], Rothenberg and Triantafillou described an equivariant analog of D.
Sullivan's rational invariants for diffeomorphism classification of 
certain smooth simply connected manifolds up to finite ambiguity [Su].  
However, their invariants only provide an equivariant {\sl almost
diffeomorphsim} classification up to finite ambiguity in many cases; in
other words,  to complete the picture one needs a smooth 
equivariant classification up to
finite ambiguity for all $G$-manifolds that are equivariant connected
sums $M_0 \# \Sigma$, where $M_0$ is fixed and $\Sigma$ is a homotopy 
linear $G$-sphere.  Questions of this type have been studied by
M.~Masuda [Ms] and will be considered further in joint work of Masuda
and the author [MSc].
\bigskip
\centerline{\it Exact sequences}
\medskip
Of course, the usefulness of the groups $\Theta^G(\varphi)$ depends on
the extent to which they can be computed. The original work of [KM]
can be summarized in a long exact sequence
$$\cdots \to P_{n+1}\to\Theta_n\to\pi_n(F/O)\to \cdots$$
where the groups $P_k$ are $0$ if $k$ is odd, infinite cyclic if $k$ is
divisible by 4, and cyclic of order two if $k\equiv 2\mod 4$ (compare
[Lev] or 
[Sc10, Thm. 6.1, p. 277]).  One particular consequence of this sequence
is the finiteness of the groups $\Theta_n$ if $n\geq 4$.  The subsequent
work of [BP] and [RSo] yielded somewhat different exact sequences for
$\Theta^G(\varphi)$ when $G$ acts semifreely on $\varphi$ (see [Sc3,
(1.1)]; also compare [Sc10, Thm. 6.3, p. 277]).  As indicated in
[Sc10, Sec. 6] one can use these exact sequences to obtain fairly
complete information on the rationalized group $\Theta^G(\varphi)
\otimes {\Bbb Q}$.  In particular, the following conclusion is an
elementary consequence of the exact sequences in [Sc10, Thm. 6.3]:
\proclaim{Proposition 3.1}  If $G$ acts semifreely on $\varphi$ such
that $\dim\varphi\geq 5$ and $\dim\varphi-\dim\varphi^G\geq 3$, then
the dimension of the rational vector space 
$\Theta^G(\varphi) \otimes {\Bbb Q}$ is at most $|G|+3$.$\blacksquare$
\endproclaim
This estimate is not really the best possible, but it shows that
the ranks of the rationalized groups have uniform bounds depending only
on the order of the group.
\medskip
In [Sc9] the approach for semifree actions is extended to a more 
general class of actions that are called {\sl ultrasemifree}; precise
descriptions of the basic exact sequences appear in [Sc9, (6.2), p.
275], and rational computations with these exact sequences are discussed
in [Sc9, Sec. 7].  For these cases one again obtains bounds for the
ranks of the groups $\Theta^G(\varphi)$ that only depend upon the order
of $G$.
\bigskip
The machinery of Section 1 allows one to extend everything to more
general actions in a straightforward manner:
\medskip
\proclaim{Theorem 3.2} Let $\varphi$ be a $G$-representation such that
all isotropy subgroups are normal ({\it e.g.}, suppose $G$ is abelian)
and $\dim\varphi^G\geq 2$.  Let $H$ be an isotropy subgroup for the
action on $\varphi$, and let $\varphi_H := \varphi/\varphi^H$.  Then there 
is the following long exact sequence of abelian groups:
$$
\CD
\boldI{\Cal S}^{h}_G(D(\varphi^H\oplus{\Bbb R})\times 
S(\varphi_H),\partial(-)) \\
@VVV\\
\Theta^G(\varphi)\\
@VVV\\
\phantom{xxxxxxxx}\Theta^{G/H}(\varphi^H)\oplus F/O_{G,iso}(S(\varphi^H
\oplus{\Bbb R})\times
\varphi_H, {\text{\rm basept.}})\\
@VVV\\
\boldI{\Cal S}^{h}_G(D(\varphi^H)\times 
S(\varphi_H),\partial(-)) \\
@VVV
\endCD
$$
\endproclaim
\bigskip
In particular, it seems likely that the preceding exact sequence should
imply that the abelian groups $\Theta^G(\varphi)$ are finitely
generated, but we have not verified this.
\bigskip
Although the exact sequence in Theorem 3.2 has not yet been used to do many
computations beyond those for semifree and ultrasemifree actions, the following
result from [Sc16] indicates the potential usefulness of Theorem 3.2.
\medskip
\proclaim{Proposition 3.3} Let $p$ and $q$ be distinct odd primes, and 
let $\omega$ be an orthogonal representation of ${\Bbb Z}_{pq}$ such that 
the following hold:

$(i)$ If $\omega_p$ and $\omega_q$ are the fixed sets of ${\Bbb Z}_p$ and
${\Bbb Z}_q$ respectively, then each has dimension at least $4$ and 
their intersection is the zero subspace.

$(ii)$ The dimension of $\omega_1:= \omega/(\omega_p+\omega_q)$ is 
at least 4.

\noindent
Then there are finitely generated subgroups $V_k\subset
\Theta^G ({\Bbb R}^k+\omega)$ such that if $v_k=\dim V_k\otimes{\Bbb Q}$ 
then for all positive integers $n$ 
the sequence $\{ v_k/k^n\}$ is unbounded.$\blacksquare$
\endproclaim
\medskip
This contrasts sharply with the results on $\dim\Theta^G(\varphi)$ in
the semifree and ultrasemifree cases (refer back to Proposition 3.1).
\bigskip
%\vfill\eject
\centerline{\it The groups $\Theta^G(\varphi)$ and the Gap Hypothesis}
\medskip
If the $G$-manifold $\varphi$ satisfies the Gap Hypothesis, then the
``rather long'' equivariant surgery sequence of Dovermann and 
Rothenberg [DR] provides another means for computing 
$\Theta^G(\varphi)$.  In particular, the methods and results of [DR] 
yield a canonical bound on the dimensions of the rational vector spaces 
$\Theta^G (\varphi)\otimes {\Bbb Q}$ in terms of $|G|$; consequently, 
Proposition 3.3 shows the existence of many new rational 
classification invariants for $G$-homotopically equivalent smooth 
$G$-manifolds beyond the usual invariants that appear when the standard 
Gap Hypothesis holds.
\vfill\eject
\input k24
\input k25
\bye

