\bigskip\bigskip\bigskip
\centerline{\bf 4. Borderline cases of the Gap Hypothesis}
\bigskip\bigskip
To simplify the discussion, in this section we shall only consider
degree one equivariant normal maps $(f:M\to X,\ {\text{\rm bundle
data}})$ such that $f$ maps the singular set of $M$ to the singular set
of $X$ by an equivariant homotopy equivalence.  Since the key inductive
step in equivariant surgery involves situations of this type, our
hypothesis is basically a way of concentrating on a single inductive
step.  In any case, for such maps the appropriate Gap Hypothesis
assumption is that 
$$\dim X\geq 2\cdot\dim ({\text{\rm Sing}}(X))+2.$$
In this section we are interested in examples that lie just outside this
Gap Hypothesis range but have been studied effectively by the standard
techniques of equivariant surgery.  There are two reasons for our
interest in such cases.  First of all, some are needed in Part III.  
Second, special cases of these results have implications for equivariant
and isovariant homotopy theory that are not presently obtainable by
purely homotopy-theoretic methods; needless to say, it would be
enlightening to have intrinsically homotopy theoretic proofs for 
such results.
\medskip
Following [DoS2, Section III.2] we define the {\sl Gap Hypothesis balance} 
to be $\Delta(X):=\dim X - 2\cdot\dim ({\text{\rm Sing}}(X))$; with 
this terminology the appropriate version of the Gap Hypothesis is $\Delta
(X)\geq 2$.  The cases of interest here are $\Delta (X)=0$ or $1$; as
one might expect, the similarities with the Gap Hypothesis range
decrease as $\Delta(X)$ gets smaller.  In particular, 
examples of M. Rothenberg and S. Weinberger (described in [DoS2, Sec. 
I.6]) indicate that the situation becomes even more complicated if
$\Delta(X)$ is negative.  We begin by discussing the situation when the
Gap Hypothesis holds.
\bigskip
\centerline{\it The cases $\Delta(X)\geq 2$}
\medskip
In these cases the methods of equivariant surgery yield the following
conclusion (compare [DR] or [DoS2, Sec. I.5]):
\proclaim{Theorem 4.1}  Suppose that $f:M\to X$ and appropriate bundle data 
determine an equivariant degree one normal map of closed smooth 
$1$-connected $n$-manifolds $(n\geq 5)$ 
such that the associated map of singular sets is an
equivariant homotopy equivalence and $\Delta(X)\geq 2$.
Then $f$ is normally cobordant to an equivariant simple 
homotopy equivalence if an ordinary Wall surgery
obstruction $\sigma(f)\in L^s_n({\Bbb Z}[G],w)$ is trivial.
\endproclaim
\medskip
As indicated in [DoS2, Sec. I.4], the crucial idea in the proof is that
the pair $(M,M-{\text{\rm Sing}}(M)\,)$ is highly connected, and this
allows one to deform all surgical constructions into the complement of
the singular set.  It follows that the equivariant surgery problem
essentially reduces to an ordinary surgery problem on the orbit space of
the free part of the action.$\blacksquare$
\bigskip
\centerline{\it The case $\Delta(X)=1$}
\medskip
The crucial new insights in this case are due to M. Morimoto.  In 
[Ba] A. Bak defines a quotient of the Wall group $L^c_{2k+1}({\Bbb Z}
[G],w)$ that is denoted by 
${\text{\bf W}}^c_{2k+1} ({\Bbb Z}[G], \varGamma G(X);w)$; the set 
$\varGamma G(X)$ is a set of order two elements $g\in G$ such that 
$\dim X^g=k$ ({\it cf.} [Mto1, p. 467]) and $w$ denotes the first
Stiefel-Whitney class.  Following Morimoto, we shall 
call the group ${\text{\bf W}}^c_*(-)$ the {\sl Bak group} associated
to the given data.  Frequently the Bak group is isomorphic to the 
corresponding Wall group.  In
particular, this is true if $G$ has odd order or $k$ is even
and the orientation homomorphism is trivial.  However, the
example in [Mto1, Corollary C, page 468] shows that 
the projection from the Wall group to the Bak group has a nontrivial
kernel in some cases, and the results of [BaMo] yield additional
examples. 
\medskip
\proclaim{Theorem 4.2}  Suppose that $f:M\to X$ and appropriate bundle data 
determine an equivariant degree one normal map of closed smooth 
$1$-connected $(2k+1)$-manifolds $(k\geq 2)$ 
such that the associated map of singular sets is an
equivariant homotopy equivalence and $\Delta(X) = 1$ (hence the 
singular set is $k$-dimensional).
Then $f$ is normally cobordant to an equivariant simple 
homotopy equivalence if the image of an ordinary Wall surgery
obstruction $\sigma(f)\in L^s_n({\Bbb Z}[G],w)$ 
in the Bak group
${\text{\bf W}}^s_{2k+1} ({\Bbb Z}[G], \varGamma G(X);w)$
is trivial.

An analogous result holds for equivariant homotopy equivalences if one 
replaces $L^s$ by $L^h$ and ${\text{\bf W}}^s$ by ${\text{\bf W}}^h$. 
\endproclaim
\medskip
Under the hypotheses of Theorem 4.2
the pair $(M,M-{\text{\rm Sing}}(M)\,)$ is not quite so
highly connected, and one can deform some but not all surgical
constructions back to the free part of the action on $M$.  The methods
employed in the proof of Theorem 4.1 still yield a surgery obstruction in
the appropriate Wall group, but this obstruction is not necessarily 
well-defined.  However, if one passes to the Bak group, 
then one does obtain a well-defined obstruction.$\blacksquare$
\medskip
Results of Dovermann [Do2] relate the preceding to questions involving
isovariance.  Namely, if $f$ is isovariant on the singular set then the 
given conditions allow one to surger $f$ into an isovariant map
that is an equivariant homotopy equivalence on the singular set.
In this setting the Wall group element represents the obstruction to
surgering $f$ into an isovariant homotopy equivalence.  Thus the kernels
of the maps from Wall groups to Bak groups carry obstructions for
transforming certain equivariant equivalences into isovariant 
equivalences.
\bigskip
%\vfill\eject
\centerline{\it The case $\Delta(X)=0$}
\medskip
In this case results are only known for the case $G\approx {\Bbb Z}_2$,
and the main results are due to Dovermann [Do1] (see also [DoS1] and
[Sc15, Thm. 2.5]).
\medskip
\proclaim{Theorem 4.3}
Let $f :X \rightarrow Y$ come from a suitably defined degree one
${\Bbb Z}_2$-normal map of smooth $1$-connected $2k$-dimensional 
${\Bbb Z}_2$-manifolds, where $f$ induces a homotopy equivalence of 
fixed point sets and the latter are $k$-dimensional.  Then $f$ is 
normally cobordant to a ${\Bbb Z}_2$-homotopy 
equivalence, relative to the fixed point sets, if and only if
the following hold:

${(i)}$ If $k$ is even, the ${\Bbb Z}_2$-signatures of $X$ and 
$Y$ are equal.

${(ii)}$ If $k$ is odd, the ordinary Kervaire invariant of 
$f$ is trivial and a mod $2$ rank invariant of the surgery kernel
of $f$ is also trivial.
\endproclaim
\bigskip
Unlike the preceding cases, one must now consider homotopy classes in 
$\pi_q(M)$ that cannot be deformed into the complement of singular set; 
the obstruction to doing this is measured by a homological intersection
number.  One needs a modified concept of Hermitian form (called {\sl 
quasi-quadratic} in [Do1]); the invariants of such forms turn out to
be the algebraic invariants described in the statement of the 
theorem.$\blacksquare$
\bigskip
Theorem 4.3 has some curious homotopy theoretic implications that are
not yet derivable from other techniques.  As indicated earlier, it would
be enlightening to have intrinsically homotopy theoretic arguments, both
for the sake of making everything more self contained and also in the
interests of proving further results along the same lines.
\medskip
The first implication involves equivariant function spaces.  Following
[BeS], if the finite group $G$ acts freely and orthogonally on the
unit sphere $S(V)$ in the 
Euclidean space $V$, let $F_G(V)$ be the space of $G$-equivariant self
maps of $S(V)$.  Of course, if $H$ is a subgroup of $G$ there is a 
natural forgetful map $\rho$ from $F_G(V)$ to $F_H(V)$.  Also, there
is a stabilization map from $F_G(V)$ to a space 
$F_G:= \lim F_G(V\oplus W)$ where $W$ runs through all isomorphism
classes of free $G$-representations.  The main results of [BeS] state
that $F_G$ is homotopy equivalent to the free infinite loop space 
$\Omega^\infty S^\infty (K(G,1)_+)$ and the forgetful map from $F_G$ 
to $F$ is induced by the transfer map in stable homotopy ({\it cf}. 
[BG]).  In particular, by the Kahn-Priddy
Theorem [KP] the map $F_G\to F$ induces a split surjection in positive
dimensional homotopy groups if $G\approx {\Bbb Z}_2$.  Dovermann's
results yield the following unstable analog of the Kahn-Priddy theorem.
\medskip
\proclaim{Proposition 4.4}  In the terminology above, suppose that
$G\approx {\Bbb Z}_2$ and $V={\Bbb R}^n$ with the antipodal involution.
If $j:F_G(V)\to F_G\to F$ is the composite of stabilization with the
forgetful map, then the image of $j_*: \pi_n(F_G(V))\to\pi_n(F)\cong
\pi_n$ contains all positive-dimensional elements whose Hopf invariants 
are even and whose Kervaire invariants are zero.
\endproclaim
\medskip
In particular, it follows that the image of $j_*$ has index at most two
in every dimension and the index is one except for a very sparse set of
dimensions (recall that an element can have an odd Hopf invariant only in
dimensions 1, 3, and 7, and an element can have a nonzero Kervaire
invariant only in dimensions of the form $2^m-2$).
\bigskip
Proposition 4.4 follows by combining the results of [Sc4] on knot
invariants with the results of [Sc8] on realizing exotic spheres as
fixed point sets of involutions on homotopy spheres (in this connection
also see [L\"o]); the role of Dovermann's work is that the
crucial results from [Sc8] and [L\"o] depend upon [Do1].
\bigskip
{\sl Questions: } Can one eliminate the condition on Kervaire invariants
in Proposition 4.4?  Basic results in homotopy theory show that the
condition on Hopf invariants {\smc cannot} be eliminated.  Also, can one
prove a result similar to 4.4 for the image of $\pi_n(F_G(V))
\to\pi_n(F)$
when $V={\Bbb R}^{n-1}$?  An answer to this question would have
implications for realizing exotic $n$-spheres as fixed point sets of
smooth involutions on homotopy $(2n-1)$-spheres.
\bigskip
Here is another consequence of Theorem 4.3 to a question of interest in
nonequivariant homotopy theory (compare [Str], [Sc15]):
\proclaim{Proposition 4.5}
Let $M$ and $N$ be closed, homotopically equivalent $1$-connected 
$n$-manifolds where $n\geq 3$.  Then the deleted symmetric squares 
$\{M \times M - \Delta_M\}/{\Bbb Z}_2$ and $\{N \times N - 
\Delta_N\}/{\Bbb Z}_2$ are homotopically equivalent.$\blacksquare$
\endproclaim
\bigskip
{\sl Question: } Can one eliminate the simple connectivity hypothesis in
this result?  Straus proves an analogous result for deleted 
reduced cyclic $p$-th powers where $p$ is an odd prime, and the 
latter result has no hypothesis on the fundamental group [Str].  Results 
of P. L\"offler and R. J.  Milgram [L\"oMi] suggest that the answer to 
the question is yes.  An obvious suggestion for approaching this is to 
prove a version of Dovermann's result for involutions on nonsimply 
connected manifolds with $\Delta(X)=0$.
