\bigskip\bigskip\bigskip
\centerline{\bf 5. Isovariance and nonsmoothable group actions}
\bigskip\bigskip
Many questions arising in Parts I and II are also meaningful and
interesting for group actions that are not smooth.  In this section we
shall describe some results along these lines.
\bigskip
Beginning in the mid nineteen eighties Cappell and Weinberger developed
a variety of 
surgery-theoretic techniques for constructing exotic topological or PL
group actions with a given isovariant homotopy type.  Some of their
results for circle actions are summarized in [CW1].  The following
{\sl replacement theorem} is a simple but basic example of their results 
for finite group actions:
\proclaim{Theorem 5.1} Let $G={\Bbb Z}_p$ where $p$ is an odd prime, and
let $W^n$ be a closed simply connected manifold with a locally linear
topological or PL $G$-action $\Phi$ such that $M+W^G$ is also simply
connected and $\dim M\geq 5$.  If $h:M'\to M$ is a homotopy equivalence,
then there is another locally linear topological or PL (resp.)
$G$-action $\Phi'$ on $W$ such that 

$(i)$ the two actions are equivalent on the complements of their fixed
point sets,

$(ii)$ the fixed point set of $\Phi'$ is $M'$,

$(iii)$ the $G$-manifolds $(W,\Phi)$ and $(W',\Phi')$ are
$G$-isovariantly homotopy equivalent (but the equivalence is not
necessarily isovariantly homotopic to a transverse linear map in the
sense of Browder and Quinn).$\blacksquare$
\endproclaim
\medskip
Cappell and Weinberger also obtain several extensions and refinements of
Theorem 5.1; details of this work appear in [CW2], and further results
on obstructions to replacement appear in [DW].  In [Daw] Dawson uses his
version of the results in Section 1 to study similar replacement
questions for smooth actions when the codimension of the fixed point set 
is small.  Dawson has also obtained results on {\sl replacement of
tangential representations} up to homotopy, where one 
replaces the $G$-representation $\Omega$ at the tangent space of a 
fixed point by an isovariantly homotopy equivalent representation 
$\Omega'$ and attempts to find an action $G$-homotopy equivalent to the
orignal one with the same fixed point set and the modified tangential data.
\bigskip
In a forthcoming book [Wb1] Weinberger develops powerful and fairly
general machinery for classifying certain topological actions up to
isovariant homotopy equivalence. Specifically, his results apply to
group actions for which the fixed point sets of subgroups define a weak
analog of a Thom-Mather stratification ({\it e.g.}, a CS stratification
in the sense of Siebenmann [Si2] or a homotopy stratification in the
sense of Quinn [Q2]).  Since this work involves several deep concepts
that are not needed in the smooth category (for example, results on ends
of maps [Q1]), we shall not attempt to explain the main ideas here.
This work has already produced some further developments and
applications due to Weinberger and M. Yan [Wb2, WY, Y], including
counterexamples to equivariant analogs of the Borel rigidity conjecture
for aspherical manifolds [Wb2].
\medskip
In view of [Wb1] it would be useful to have an extension of the results
of Part I to isovariant maps of $G$-manifolds with weak stratifications
of an appropriate type.  Perhaps the most obvious complication is that
fixed point sets need not have closed tubular neighborhoods ({\it e.g.},
see the examples near the beginning of [Q1]).  An extension of the
results in [DuS] to nonsmoothable actions will probably require diagrams
involving Quinn's notion of {\sl homotopy collar} [FQ, pp. 214--215] for 
the sets $M_{(H)}$ associated to a $G$-manifold $M$ (homotopy collars
are called {\sl homotopy completions} in [Q1, Sec. 7.8]) and {\sl
homotopy links of $\infty$ in the one point compactification} in [Q2]).
\medskip
Finally, we note that the Browder-Straus proof of Theorem 2.1 goes
through for certain classes of nonsmoothable actions; for example, the
proof applies to semifree PL actions on manifolds such that the fixed
point sets are also manifolds.  Of course, it would be enlightening to
have an alternate proof as in the smooth category, with little or no
input from surgery theory.
