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\centerline{\bf PART III}
\bigskip

\centerline{\bf FINITE GROUP ACTIONS ON HOMOLOGY 3-SPHERES}
\bigskip
\bigskip
\bigskip
\centerline{\it Background references}
\bigskip
In contrast to Parts I and II, the emphasis in this final part will be
on specific geometric problems involving transformation groups on 
3-manifolds.  The following references contain most of the background 
material we shall need:
\bigskip
\itemitem{(1)} Article by M. Davis and J. Morgan in Bass--Morgan (eds.), 
{\it The Smith Conjecture} (:= [DaMo]).
\bigskip
\itemitem{(2)} Article by Edmonds in Contemp. Math. Vol.
36 (1985) (:= [Ed]).
\bigskip
\itemitem{(3)} Article by Raymond in Transactions Amer. Math. Soc. 
Vol. 131 (1968) (:= [Ra]).
\bigskip
\itemitem{(4)} Notes from lectures of Thurston on the geometry and 
topology of 3-manifolds (:= [Th1]; the main points are summarized
in [Th2]).
\bigskip
During the past fifteen years the concepts of {\sl orbifold} and
{\sl orbifold fundamental group} have become fairly standard in
3-dimensional topology.  Since both arose from considerations 
involving transformation groups, we shall use these concepts as
needed.  The basic definitions and examples can be found in 
Thurston's notes [Th1, Ch. 13, especially p. 13.5] or the first 
few sections of [DaMo].
\bigskip\bigskip\bigskip
\centerline{\bf 1. A survey of known results}
\bigskip\bigskip
For several decades topologists have known that dimensions 3 and 4
form a transitional range from the geometric rigidity of line and 
surface topology to the freedom of movement one has in dimensions
$\geq 5$.  References on this topic are numerous and include an old
article of Siebenmann [Si1], work of A. Casson and C. Gordon
[CG], and books of M. Freedman and F. Quinn [FQ], S. Donaldson and
P. Kronheimer [DK], and S. Akbulut and J. McCarthy [AM].  Our 
interest in this article lies with transitional properties of 
group actions on spheres and manifolds closely resembling spheres
({\it e.g.}, manifolds homeomorphic but not necessarily diffeomorphic
to the standard sphere).  As indicated by the title above, we shall
concentrate on the 3-dimensional case; some references for the 
4-dimensional case include [Ed] for work done through the early 
nineteen eighties, [BKS] and [DM] for results about the fixed 
point sets of smooth actions on $S^4$, [KwS2--4] for
results on topological actions on $S^4$ with isolated singular 
points, and [KwS1] and [KwS5] for results concerning topological
circle actions on $S^4$ and other 4-manifolds.

In 1- and 2-dimensional topology all group actions on manifolds 
are equivalent to smooth actions that preserve nontrivial 
geometric structures ({\it cf.}, [Ed, p. 341]).  For example, all 
compact Lie group actions on $S^1$ and $S^2$ are equivalent to
orthogonal actions.  On the other hand, it is well known that a
smooth action of a finite cyclic group on $S^4$ need not be 
orthogonal because the fixed point set can be a nontrivially 
knotted 2-sphere (see Giffen [Gi] and Gordon [Go1]).  The
known results for dimension 3 lie somewhere between these two
adjacent cases.

\medskip
\proclaim{Fact 1.1}  There is a continuum of pairwise 
inequivalent topological  ${\Bbb Z}_k$ actions on $S^3$ for
every prime $k$.
\endproclaim

As noted in [Ed], this is due to Bing [Bi] and Alford 
[{\smc Al}].$\blacksquare$
\medskip
\proclaim{Fact 1.2}  All smooth actions of compact Lie groups
on $S^3$ with nonempty fixed point sets are orthogonal.
Modulo some possibly exceptional cases, all actions of compact
Lie groups on $S^3$ with positive-dimensional singular sets are 
orthogonal.
\endproclaim

If $G=S^1$ this result is contained in [Ra], and for larger Lie
groups the result is a straightforward exercise.  When $G$ is finite
cyclic and the fixed point set is a circle this was conjectured by
P. A. Smith and solved in the late nineteen seventies by the combined
efforts of several mathematicians (see the book containing 
[DaMo]).  Results for other finite groups appear in several different 
places, including papers of M. Davis and J. Morgan [DaMo], M. Feighn 
[Fn], and S. Kwasik and the author [KwS6].  About ten years ago W. 
Thurston announced that the second assertion in 1.2 holds without 
exception [Th3]; although workers
in the area have few doubts about the correctness of this statement, 
it is not clear when a complete written proof will be 
available.$\smallsquare$  
\medskip
\proclaim{Fact 1.3}  There are many smooth group actions on integral
homology $3$-spheres that are analogous to important examples of
smooth actions on higher-dimensional spheres.
\endproclaim

\def\bvecz{\text{\bf z}}
A similar---and related---phenomenon occurs in the theory of isolated
singularities of complex hypersurfaces ({\it cf.} Mumford [Mum]):  
Given a complex polynomial $f({\bvecz})$ in $n+1$ variables such that
the origin is an isolated singularity, let $\Sigma_f$ be the intersection
of the zero set $\{ {\bvecz}\in {\Bbb C}^{n+1}| f({\bvecz})=0\}$ with a 
sphere of 
sufficiently small radius; it follows that $\Sigma_f$ is a closed 
smooth $(2n-1)$-manifold.  Mumford's result deals with the case $n=2$ 
and states that a 3-dimensional manifold of the form $\Sigma_f$ 
is diffeomorphic to $S^3$ if it is simply connected; on the other 
hand, there are many examples where $\Sigma_f$ is a nonsimply connected
integral homology sphere.  The analogs of the latter in higher dimensions
are homeomorphic but not necessarily diffeomorphic to the standard 
$(2n-1)$-sphere (see Milnor's book [{\smc Mln3}] for further information 
on this topic).  
\bigskip
\noindent{\bf Examples for Fact 1.3. }  (1)\qquad Perhaps the most 
basic of these are the {\sl pseudofree} 
smooth circle actions on Seifert homology 
3-spheres.  These actions are free on the complement of a finite 
set of pairwise disjoint circles, and the isotropy subgroups for points
on these circles are pairwise relatively prime integers $d_i > 1$.  
The family of such actions includes all fixed point free orthogonal
circle actions on $S^3$; in fact, the orthogonal actions are precisely
those for which there are at most two exceptional circles, and in 
all remaining cases the underlying homology spheres are not simply 
connected.  In contrast to the orthogonal case, the number of exceptional
orbits for an arbitrary action on a Seifert homology 3-sphere can be 
an arbitrary positive integer.  There are several excellent descriptions 
of these manifolds
in the literature, including Raymond's article [Ra], the book by 
P. Orlik [Or], and lecture notes by M. Jankins and W. Neumann [JN].  In 
higher dimensions one also has orthogonal pseudofree circle actions
on $S^{2n-1}$: If $(d_1, \cdots\ , d_n)$ is a sequence of pairwise
relatively prime positive integers, then the unit sphere in ${\Bbb C}^n$
for the linear action
$$t\cdot(z_1, \cdots\ ,z_n) := (t^{d_1}z_1, \cdots\ , t^{d_n}z_n)$$
is free on the complement of $m$ circles, where $m$ is the number
of integers $d_i$ that are greater than one.  Of course, it follows
that the number of exceptional circles for an orthogonal pseudofree
circle action on $S^{2n-1}$ is at most $n$, and it is natural to 
ask if it is possible to construct pseudofree smooth circle actions 
on $S^{2n-1}$ with larger numbers of exceptional circles for larger
values of $n$.  In [MnY1--2] D. Montgomery and C.-T. Yang succeeded 
in producing such examples when $n=3$; a survey of this and analogous
work in dimensions $(2n+1)\geq 9$ appears in [DPS, Sec. 2] (also see
[Pet1]).  To complete the picture, we note that the case $n = 2$ ({\it 
i.e.}, the 5-dimensional case)
reflects the basic difficulties of 4-dimensional topology.  One can use
M.~Freedman's work on 4-dimensional topological surgery [FQ] to construct
topological 
pseudofree circle actions on $S^5$ that are locally linear,
%and work of R. Fintushel and R. Stern [FS] shows that 
%gauge-theoretic methods  can be applied effectively to questions about
%smooth pseudofree circle actions on $S^5$.
and applications of gauge theory to 
smooth pseudofree circle actions on $S^5$ have also been considered
({\it cf}. [FS]; see also {\it Math. Reviews} {\bf 88e:}57032).
\medskip
(2)\qquad Let $p$ be an odd prime, and let $D_{2p}$ be the dihedral 
group of order $2p$.  If we are given a fixed point free, linear
action of $D_{2p}$ on $S(V)\approx S^{2n-1}$. then the fixed point
sets of the order two subgroups are all $(n-1)$-spheres, and if 
$H$ and $K$ are two distinct subgroups of order two then $S(V)^H$
and $S(V)^K$ have linking number $\pm 1$.  If we are given an arbitrary
fixed point free $D_{2p}$ action on $M\approx S^{2n-1}$ such that the 
fixed point set of every (equivalently, of some) order two subgroup
is an $(n-1)$-sphere, then Smith theory implies that the linking 
numbers of $M^H$ and $M^K$ are congruent to $\pm 1$ mod $p$; results
of J.~Davis and T. tom Dieck [DtD, tD3] show that one can realize 
exotic linking numbers by smooth $D_{2p}$ actions on the $(2n-1)$-sphere
if $n\geq 3$.  In contrast, special cases of Fact 1.2 imply that 
no such examples can exist if $n=2$ and $p>5$ ({\it e.g.}, see [DaMo]).
Despite this, C. Livingston has shown that fixed point free, smooth 
$D_{2p}$-actions with exotic linking numbers exist on integral homology
3-spheres [Liv]. 

\def\psph{\Sigma(2,3,5)}
\medskip
(3)\qquad If $A_5$ denotes the alternating group on 5 letters,
then $A_5$ can be viewed as the group of isometries of a 
regular dodecahedron or icosahedron, and consequently there are
natural realizations of $A_5$ as a subgroup of $SO_3$.  The 
homogeneous space $SO_3/A_5$ is the well-known {\sl Poincar\'e
homology $3$-sphere} that we shall denote by $\psph$; summaries
of the properties of this manifold appear in [Bre3, Sec. I.8]
and [KiSc].  As noted in [Bre3, pp. 55--56], the action of $A_5$ 
on $\psph$ obtained by restricting the transitive action of
$SO_3$ has exactly one fixed point.  Although smooth actions
with one fixed point cannot exist on a smooth homotopy 3-sphere
[BKS], the methods of geometric topology have produced numerous
examples of smooth actions on higher dimensional spheres with one 
fixed point during
the past two decades; the first examples were due to
E. V. Stein [Stn], with additional families of examples due to 
T. Petrie [Pet1--2] appearing shortly afterwards.  We shall not 
attempt to summarize subsequent work here, but many further references
appear in [BKS], applications to algebraic group actions are discussed
in [DMP], and an article by E. Laitinen, M. Morimoto,
and K. Pawa{\l}owski [LMP] provides the most recent information
available at this time.
\bigskip
In the remaining sections of Part III we shall concentrate on 
the following question:
\bigskip
\itemitem{($\ddag$)}{\sl What is the role of
the standard one fixed point action on $\psph$ in the family of all smooth 
one fixed point actions on homology $3$-spheres?}
\bigskip
Standard considerations involving P. A. Smith cohomological fixed point
theory, the local linearity of smooth actions near fixed points, and the 
subgroups of $SO_3$ show that $A_5$ is the only group that can act on a
closed integral homology 3-sphere with exactly one fixed point.
\bigskip
One motivation for studying such actions is that symmetry considerations 
have led to interesting classes of 3-manifolds such as
Seifert manifolds. On the other hand, 
smooth group actions with one fixed point have been studied extensively 
over the past two decades, both as test cases for the sorts of exotic smooth 
group actions that can exist on spheres 
and in connection with questions from algebraic 
transformation groups.  In particular, recent work has shown that smooth one 
fixed point actions on nonsimply connected homology 3-spheres are rather 
exceptional low-dimensional examples.  Standard geometrization
results ({\it cf.} [Ed]) imply that finite group actions with a single fixed 
point do not exist on (homology) spheres of dimension 1 or 2, and more recent 
results of [BKS], [DM], and [Mto2] show the 
nonexistence of smooth actions with exactly one fixed point on 
homology 4- and 5-spheres. In contrast, such smooth actions exist on genuine
spheres in all dimensions $\geq 6$ ({\it e.g.}, see [Stn], [Pet2], 
[Mto2], [BaMo], and related examples of [BKS]).  
\medskip
A result of G. Bredon [Bre1] states that $\psph$ is the only integral
homology sphere that admits a transitive action of a compact Lie group
that is not equivalent to an orthogonal action on a standard sphere.
Since the one fixed point action of $A_5$ on $\psph$ is the restriction
of this exceptional transitive action, the evidence in this and the
preceding paragraph may suggest that all one fixed point actions on
homology 3-spheres are closely related to the standard examples in some
fashion ({\it e.g.}, perhaps there is an equivariant degree one map
into $\psph$). However, our main results 
show the existence of many smooth one fixed point actions on 
irreducible homology 3-spheres.  Some of these actions are clearly 
related to the standard actions on $\psph$, but others are quite different.
%\bigskip\bigskip\bigskip
\vfill\eject
\centerline{\bf 2. Equivariant surgery in three dimensions}
\bigskip\bigskip
Our objective is to construct exotic examples of smooth one fixed 
point actions on homology 3-spheres by means of equivariant surgery
and other techniques.  Of course, it is well known that many basic
results of surgery theory fail in dimension three; the purpose of this
section is to summarize some aspects of surgery theory that are both
valid and useful for 3-manifolds.
\bigskip
For many years geometric topologists have known that surgery
theory applies to dimension three if one is willing to settle for
homology equivalences rather than genuine homotopy equivalences;
informal discussions of this appear in the writeup by F. Quinn on page 
225 of the book containing [Brw2] and also in [FQ, p. 200, lines --3 to
--1].  Here is a more formal statement that ``homology surgery
works for 3-manifolds.''
\medskip
\proclaim{Folk Theorem 2.1} Suppose that
$f:(M,\partial M)\to (X,\partial X)$ and appropriate bundle data 
determine a degree one normal map of compact $3$-manifolds with boundary
such that $\partial f:\partial M\to\partial X$ induces a ${\Bbb Z} 
[\pi_1(X)]$-homology equivalence (with possibly twisted coefficients).
Then $f$ is normally cobordant rel boundaries to a simple ${\Bbb Z} 
[\pi_1(X)]$-homology equivalence if the ordinary Wall surgery
obstruction $\sigma(f)\in L^s_3({\Bbb Z}[\pi_1(X)],w)$ is trivial.
\endproclaim
\medskip
The proof of this follows directly from the methods of Wall's book
for (2n+1)-manifolds with $n\geq 2$ subject to one complication:  If one 
removes an embedded $n$-sphere from a $(2n+1)$-manifold, the fundamental
group does not change if $n\geq 2$ but it usually changes drastically if
$n=1$.  This means that one loses control of the fundamental group of 
the source manifold for the normal map, but the underlying homological
arguments remain valid if we we work with twisted coefficients in the
group ring ${\Bbb Z}[\pi_1(X)]$.$\blacksquare$
\medskip
In [BaMo] Bak and Morimoto formulate a version of this for equivariant
surgery on 3-manifolds with orientation-preserving actions.
\proclaim{Theorem 2.2}  Let $M$ and $X$ be closed smooth $G$-manifolds 
with orientation-preserving actions of a finite group $G$, let
$f:M\to X$ and suitable bundle data define 
a degree one equivariant surgery problem, where $X$ is simply connected
and $f$ maps ${\text{\rm Sing}}(M)$ to ${\text{\rm Sing}}(X)$ by an
equivariant homotopy equivalence.  Furthermore, assume that the
projective class group ${\widetilde K_0}({\Bbb Z}[G])$ is zero.  Then
$f$ is equivariantly normally bordant to a $G$-homotopy equivalence if
an obstruction in the Bak group
$${\text{\bf W}}^h_{3} ({\Bbb Z}[G], \varGamma G(X);1)$$
is trivial.$\blacksquare$
\endproclaim
\medskip
As noted in Section 2.4, the Bak group 
${\text{\bf W}}^h_{3} ({\Bbb Z}[G], \varGamma G(X);1)$
is defined in [Ba] as a quotient of the Wall group 
$L^h_3({\Bbb Z}[G],1)$.
\medskip
We shall need a slight extension of the preceding result:
\proclaim{Complement 2.3} The preceding result remains valid if one
merely assumes that $X$ is obtained from a compact smooth 3-manifold by
attaching $G$-free equivariant cells on the free part of $X$.$\blacksquare$
\endproclaim
\medskip
We shall also need some computational results for Bak groups and certain
other algebraic $K$-theoretic groups.  These are all established in 
[BaMo].
\proclaim{Theorem 2.4}  Let $G$ be the alternating group $A_5$.  Then
the projective class group of ${\Bbb Z}[G]$, the Whitehead group of $G$,
and the Bak group ${\text{\bf W}}^h_{3} ({\Bbb Z}[G], \varGamma 
G(X);1)$ are all trivial.$\blacksquare$
\endproclaim
\bigskip
In contrast to the preceding result, the Wall group $L^h_3({\Bbb Z} 
[A_5],1)$ is nontrivial, for $L^h_3({\Bbb Z} [{\Bbb Z}_2],1) \approx 
{\Bbb Z}_2$ implies that $L^h_3({\Bbb Z} [{\Bbb Z}_2\times
{\Bbb Z}_2],1) \supset {\Bbb Z}_2\times {\Bbb Z}_2$, and by transfer 
considerations it follows that $L^h_3({\Bbb Z} [A_5],1)$ also contains 
a copy of ${\Bbb Z}_2\times {\Bbb Z}_2$.$\blacksquare$
\bigskip\bigskip\bigskip
\centerline{\bf 3. Construction of exotic examples}
\bigskip\bigskip
Elementary considerations show that
the one fixed point action on $\psph$ is not the only smooth 
action on a homology 3-sphere with exactly one fixed point.
For example, one can construct many 
examples by taking an equivariant connected sum of 60 = $|A_5|$
copies of some homotopy sphere $P^3$ over the free part of the
action on $\psph$; this yields an infinite family of pairwise
inequivalent one fixed point $A_5$-actions on homology 3-spheres.  
More generally, if $H$ is an isotropy subgroup
of the action on $\psph$ and $P^3$ has a smooth action of $H$,
then one can often form a {\it stratumwise equivariant connected sum} of
$\psph$ with $|G/H|$ copies of $P$ along the fixed point sets of the
conjugates of $H$; constructions of this type are used extensively 
by Meeks and Yau in [MeY1, Sec. 9].
In particular, one can take $H=A_4$ and obtain
a one fixed point action on a connected sum of 6 copies of $\psph$; 
we mention this because it appears to be the simplest example of 
an integral homology 3-sphere that supports a smooth one fixed 
point action and has a vanishing Rochlin invariant.
\bigskip
The preceding examples are all obtainable from $\psph$  by familiar
sorts of constructions.  In fact, there have been some informal
regularity conjectures that all one smooth fixed point actions on 
homology 3-spheres are somehow modeled after $\psph$.  Perhaps
the weakest of these conjectures is that the singular set of such
an action is always equivalent to Sing$\,(\,\psph\,)$.  The main
result of this section provides a negative answer to this particular
question and describes all possible fixed point sets.

\proclaim{Theorem 3.1} There are exactly four equivariant homeomorphism
classes of singular sets for smooth $A_5$-actions on ${\Bbb Z}$-homology 
$3$-spheres with exactly one fixed point.
\endproclaim
\bigskip
The possibilities for the singular set may be described as 
follows:  Suppose we are 
given a smooth action of $A_5$ on the homology 3-sphere $\Sigma^3$ 
with exactly one fixed point.  For each subgroup $C$ of order 2 
the fixed set of $C$ is the union of two semicircles with two 
endpoints in common; one of the endpoints is the fixed point of
the $A_5$-action, and the other is fixed under the unique subgroup
of $A_5$ that contains $C$ and is isomorphic to the alternating 
group on four letters.  Each semicircle also contains a point that 
is fixed under a subgroup of order 6 containing $C$ and a point 
that is fixed under a subgroup of order 10 containing $C$.  The union of 
all fixed sets of order 2 subgroups consists of 30 semicircles, and
$A_5$ acts transitively on this set of semicircles.  On the other 
hand, a direct analysis shows that there are exactly four ways of
constructing an $A_5$-orbit of data $(\Gamma_C; x_6, H_6, x_{10},
H_{10})$, where $C$ is a subgroup of order 2, $\Gamma_C$ is 
homeomorphic to [0,1], the points $x_6$ and $x_{10}$ belong to
$\{ {1\over 3},{2\over 3} \}$, and $H_6$ and $H_{10}$ are subgroups
of order 6 and 10 respectively containing $C$.  Furthermore,
Smith theory implies that for each class of semicircles 
there is a unique 1-complex with cell-preserving $A_5$-action 
that is a potential singular set for a smooth action on a homology
3-sphere with one fixed point.
\medskip
\noindent{\bf Sketch of proof of Theorem 3.1. } Each of the 
1-complexes $K$ in the preceding paragraph can be realized as 
the singular set of a smooth action on {\sl some} closed oriented 
3-manifold by an elementary ``rewiring'' construction on $\psph$
with its standard one fixed point action.  Let $A_K$ be 
the manifold so obtained; it follows from the construction that
$H_1(A_K;{\Bbb Q})\neq 0$.  One can then use machinery developed by 
R. Oliver [{\smc Ol}] to add equivariant cells along the free part of
$A_K$ and obtain an equivariant CW complex $B_K$ with the same 
structure on and near the singular set and such that $B_K$ is 
homotopy equivalent to $S^3$; if we split $A_K$ equivariantly
as $D\cup_\partial E$ where $D$ is a linear disk about a fixed
point, the equivariant cells can all be added over $E$ and one
obtains a corresponding splitting $D\cup_\partial E'$ where 
$E'$ is contractible.  Therefore the inclusion of $A_K$ in
$B_K$ can be viewed as a map of triads, and from this it follows
that the inclusion is an isomorphism on $H_3(-;{\Bbb Z})$ and
can be viewed as a map of degree one.  In order to make this into
an equivariant surgery problem, it is necessary to introduce some
equivariant bundle data; the details of the rewiring construction
imply that the equivariant tangent bundle is stably isomorphic to
a product bundle on the complement of a finite invariant subset 
$F$, and it follows that equivariant bundle data can be given by 
crossing the map $A_K-F\to B_K$ with the identity on $\Omega$ for 
a suitable $A_5$-representation $\Omega$.  This suffices for 
surgery-theoretic purposes because the latter involve maps from
positive-codimensional manifolds into $A_K$ and such maps can
always be deformed to avoid a finite subset (similar considerations
arise in [DR], where bundle data with deficiencies are discussed 
in greater detail).  The results of Bak and Morimoto [BaMo] (cited 
in Section 2) now show that 

\itemitem{$(i)$} one can do equivariant surgery away from the singular
set of $A_K$ to convert the map $A_K\to B_K$ into a 
${\Bbb Z}$-homology equivalence if an obstruction in some quotient
group of the Wall group $L^h_3(A_5;1)$ -- specifically, the 
associated Bak group $W_3^h(A_5,\varGamma G(X);1)$ -- is zero,

\itemitem{$(ii)$} the Bak group in $(i)$ is equal to zero.

\noindent
Therefore one can modify $A_K$ by equivariant surgery away from the 
singular set to obtain a homology sphere with a smooth one fixed 
point action.$\blacksquare$
\bigskip
\centerline{\it Fixed point free actions on homology $3$-spheres}
\medskip
The methods of this section also yield results about fixed point free
actions of finite groups on homology 3-spheres.  Some of this is 
work in progress, but the results for $G=A_5$ can be stated fairly 
simply.  There is
a unique linear action of this type; namely, let $V$ be the orthogonal
complement of the diagonal in ${\Bbb R}^5$ where $A_5$ acts on the
latter by permuting the coordinates, and take the induced action on the
unit sphere $S(V)$.  
It is immediately clear that one can find examples with exotic orbit
structures; in particular, this can be done by taking an
equivariant connected sum of two one fixed point actions on homology 
3-spheres at the fixed points.  The methods of this section
show that {\sl the singular set of a fixed point free $A_5$-action on
an integral homology $3$-sphere can be an 
arbitrary $1$-dimensional $A_5$-complex
that satisfies the necessary conditions imposed by Smith theory}; the
list of all such possibilities is fairly short, but it does contain more
than the singular set of the linear action and the singular
sets obtained from the equivariant connected sums described above.
One reason for interest in such actions involves the linearity question
for smooth actions of finite groups on $S^3$ ({\it cf.} [DaMo], [Fn],
[KwS6]); fixed point free 
$A_5$-actions represent one basic type that is included in Thurston's 
announcement [Th3] but has not yet been verified elsewhere.
\bigskip\bigskip\bigskip
\centerline{\bf 4. Actions on hyperbolic homology spheres}
\bigskip\bigskip
One obvious drawback of Theorem 3.1 is that the argument does
not yield explicit examples of actions with exotic singular sets.  In
particular, it is natural to ask if such examples can be found on
homology 3-spheres that are {\sl irreducible} and geometric in the sense
of Thurston [Th2].  More specifically, Thurston's hyperbolization
theorem [Th4] implies that one can often find hyperbolic 3-manifolds 
with certain topological or geometric properties by applying suitable 
conversion procedures to general 3-manifolds with such properties,
and in this connection one
would like to know if smooth one fixed point actions can be found on
hyperbolic homology 3-spheres.  According to the main result of this
section (Theorem 4.3), such examples can be found.  The results of
this section generate a variety of questions; some examples are presented
after the proof of Theorem 4.3.  
\bigskip
One can interpret Theorem 3.1 as a negative answer to questions about
the existence of a single basic model for one fixed point actions on
homology 3-spheres.  However, the following result shows that the one
fixed point actions on irreducible homology 3-spheres form a family of
models for all such actions.
\medskip
\proclaim{Theorem 4.1} If $\Sigma^3$ is a closed integral homology
$3$-sphere with a smooth one fixed point action of $A_5$, then $\Sigma^3$
is equivariantly diffeomorphic to an iterated equivariant connected 
sum along strata of the form 
$\Sigma_0\, {\#}\, |G/H_1|\Sigma_1\, \cdots\, {\#}
\,|G/H_r|\Sigma_r$, where $\Sigma_i$ is an irreducible homology 
$3$-sphere with a smooth action of $H_i$ if $i>1$ and $\Sigma_0$
is an irreducible homology $3$-sphere with a smooth one fixed 
point action of $A_5$.
\endproclaim
\bigskip
In particular, it follows that every one fixed point action on
a homology 3-sphere has an {\it irreducible nucleus} given by 
a smooth one fixed point action on some connected summand of $\Sigma$.
\bigskip
The proof of Theorem 4.11 follows from the standard equivariant 
analogs of the Papakyriakopoulos Sphere Theorem [MeY2, JR] and 
an analysis of the ways in which the invariant separating spheres
in $\Sigma$ can meet the singular set of the action;
this set turns out to be a 1-dimensional
finite cell complex with a cell-preserving group action (that can
be equivariantly subdivided to yield an equivariant regular
simplicial action in the sense of, say, [Bre3, Ch. III]).$\blacksquare$
\medskip
In view of Theorem 4.1 and Thurston's Geometrization Conjecture
[Th1--2], it is natural to ask next about one fixed point actions
on irreducible geometric homology spheres.  These manifolds fall
into three distinct classes; namely, {\it Seifert fibered}, 
{\it non-simple Haken} (in other words, containing an incompressible
torus), and {\it hyperbolic}.  Since the Poincar\'e homology $3$-sphere 
$\psph$ is an example of a Seifert
fibered $3$-manifold (in fact, it is a Brieskorn variety), the most 
immediate question is whether other Seifert fibered
homology $3$-spheres support smooth one fixed point actions.  This
question has a simple negative answer:
\bigskip
\proclaim{Proposition 4.2}  Let $\Sigma^3$ be a Seifert fibered 
homology $3$-sphere.
Then $\Sigma^3$ admits a smooth action of a finite group with one fixed
point if and only if $\Sigma^3$ is diffeomorphic to the Poincar\'e homology
$3$-sphere $\psph$.
\endproclaim

\bigskip
This follows from considerations involving the fundamental 
group of $\Sigma$ and the orbifold fundamental group associated
to the group action.$\blacksquare$
\bigskip
In contrast to the preceding result, situation is completely different
for hyperbolic and Haken homology $3$-spheres:
\medskip
\proclaim{Theorem 4.3}  $(i)$  There exist infinitely many irreducible 
non-simple Haken homology $3$-spheres that support smooth actions of
$A_5$ with exactly one fixed point.

$(ii)$ There exist infinitely many irreducible hyperbolic 
homology $3$-spheres that support smooth actions of
$A_5$ with exactly one fixed point.
\endproclaim
\bigskip
\noindent{\bf Sketch of proof of Theorem 4.3. } 
The constructions of examples depend heavily on Theorem 3.1, Theorem
4.1, and the nonexistence of smooth one fixed point actions on homotopy
3-spheres [BKS].  When combined, these imply that {\sl every possible
singular set is realized by a smooth one fixed point $A_5$-action on a
nonsimply connected, irreducible homology $3$-sphere.}
\medskip
To prove statement $(i)$, one first takes a simple 
closed curve in the free part of $\psph$ that represents a nonzero element
of $\pi_1(\psph)$, then forms a connected sum with a knot in some 
small coordinate 3-disk, and afterwards deforms it to be disjoint from 
all its translates under the action of $A_5$.  Next, one takes a closed
invariant tubular neighborhood $U$ of these $|A_5|$ pairwise disjoint 
curves and replaces the interior of $U$ with the $|A_5|$ copies of the
interior of some nontrivial knot complement. One can do this such
that the manifold in question becomes an irreducible homology sphere
and the components of the boundary of $U$ become incompressible tori.
\medskip
The proof of statement $(ii)$ is somewhat more delicate.  As in the
preceding discussion, one removes a suitably chosen union Int$\,U$ of 
invariant open solid tori from the free part of the action to obtain 
a bounded Haken manifold $V$ with a smooth one fixed point action and
$\partial V= A_5\times T^2$.  Next one uses the splitting theorems of W.
Jaco and J. H. Rubinstein [JR] to construct an 
equivariant splitting of $V$ along incompressible tori into Seifert
fibered and hyperbolic pieces; the hyperbolicity assertion uses
Thurston's recognition principle for hyperbolic 3-manifolds ([Th4]; also
see C. McMullen's article [McM] and the references cited there).  The 
equivariant geometrization results
of W. Meeks and G. P. Scott [MS] then imply that the fixed point of 
the induced action on $V$ lies
in a hyperbolic piece, say $V_0$, and an argument involving the dual
graph of the splitting implies that one can attach solid tori to
the boundary components of $V_0$ (nonequivariantly) to obtain a 
homology 3-sphere.  Thurston's results on Dehn fillings [Th1--2] then
imply that infinitely many such attachments yield a hyperbolic
manifold; furthermore, elementary considerations imply that these
Dehn fillings will yield integral homology spheres, and a more 
detailed analysis also shows that infinitely many of these 
constructions can be done equivariantly.$\blacksquare$
\bigskip
The preceding results generate a variety of questions.  Here are
two examples:
\medskip
\noindent{\bf (1)} If $\Sigma$ is an irreducible homology 3-sphere with
a smooth $A_5$-action with one fixed point, is the Rochlin invariant
always equal to 1?  This is true for all examples checked thus far.
Similarly, one can ask about the Casson invariant [AM] or other invariants
from topological quantum field theory.
\medskip
\noindent{\bf (2)} In [Th3] Thurston announced results implying that 
the one fixed point actions on the hyperbolic homology 3-spheres 
of Theorem 4.3 are hyperbolic structure preserving.  There are 
many examples of one fixed point actions on spheres in dimensions 
$\geq 6$.  Can one use equivariant surgery to convert such actions 
into one fixed point actions on hyperbolic
homology spheres that preserve a hyperbolic structure?  Results of M.
Davis and T. Januszkiewicz [DJ] provide a means for converting manifolds 
and orbifolds to objects that are hyperbolic in the sense of M. Gromov 
[Gr] (also see [Bow], [GH]), and these 
suggest that one can find at least some hyperbolic one fixed point 
actions on higher dimensional integral homology spheres.
More 
generally, it would be interesting to know which of the many
exotic smooth finite group actions on spheres are simply connected
analogs of hyperbolic actions on hyperbolic homology spheres.

\vfill\eject
%\input k24
\bye
