ICOSAHEDRAL GROUP ACTIONS ON R3 Slawomir Kwasik and Reinhard Schultz Department of Mathematics Department of Mathematics Tulane University Purdue University New Orleans, Louisiana 70118 West Lafayette, Indiana 47907 In the past two decades the geometrization theorems and conjectures of W. T* *hurston (see [Th1]) have played a crucial role in the study of 3-manifolds. One of the firs* *t advances in this direction was the solution of the Smith Conjecture, which states that an orient* *ation-preserving periodic diffeomorphism of the 3-sphere is equivalent to an orthogonal transfor* *mation (cf. [Mor1]). There are natural analogs of the Smith Conjecture for other symmetry phenomena * *in 3-dimensional topology. One of these is the following linearity question: Problem. Is every smooth action of a compact Lie group G on the sphere S3, the * *disk D3 , or Euclidean space R3 equivalent to an orthogonal action? In [MY] W. Meeks and S.-T. Yau proved that a smooth orientation-preserving * *action of a finite group G on R3 is equivalent to an orthogonal action provided G is not is* *omorphic to the alternating group A5 (which is isomorphic to the group of orientation-preservin* *g symmetries of a regular icosahedron). The results of [MY] also yield orthogonality theorems for* * smooth actions of the same groups on D3 and also smooth actions of these groups on S3 with nonemp* *ty fixed sets. The purpose of this paper is to prove the analog of the Meeks-Yau result for A5: Theorem A. Let be a smooth action of A5 on R3, D3 , or S3; in the latter case,* * assume the fixed set is nonempty. Then is differentiably equivalent to a linear action. Since the analogous results for smooth actions of positive-dimensional comp* *act Lie groups on R3, D3 , and S3 are well known (cf. [Ra]), we have the following general statem* *ent. Theorem B. Every smooth action of a compact Lie group G on R3 or D3 is differe* *ntiably equivalent to a linear action. A similar conclusion holds for smooth actions on* * S3 with nonempty fixed point sets. Remark. Examples of Bing [Bi] show that the analog of Theorem B for topological* * actions is false. A preprint of W. Thurston from 1982 announces many strong results about smo* *oth group actions on 3-manifolds ([Th2]; cf. the footnote on p. 55 in [Mor2]), and the or* *thogonality of smooth A5-actionson R3, D3, and S3 is included in the results stated there. Our approa* *ch involves some developments that have taken place since the appearance of [Th2] - specifically* *, the formulation of invariant gauge theory for certain 4-manifolds with smooth group actions (cf* *. [BKS]) and the _________________________ The authors were partially supported by NSF Grants DMS 8901583 and DM 89025* *43, respec- tively. The second named author would like to thank Northwestern University fo* *r access to its facilities during portions of this work. Version 1.0,* * July, 1991 2 results of [KL] on equivariant smoothings of 3-manifolds - and thus it seems sa* *fe to say that the argument in this paper is different from the ideas underlying [Th2]. This paper has five sections. The first contains statements of several ver* *sions of the main results and indicates the logical relationships between the different versions.* * Section 2 contains the crucial insights in the proofs of the main results: A smooth A5-actionon S3 wit* *h a nonempty fixed point set has an equivariant Heegaard splitting of genus 60 (= orderA5), and if* * one fixed point is removed from S3 (with such an action) the induced action on the complement i* *s equivariantly diffeomorphic to an invariant open neighborhood of the origin in R3 with an ort* *hogonal action. In the third section we combine these observations with standard infinite cancella* *tion arguments to show that a smooth action of A5 on S3 with nonempty fixed point set is topologi* *cally equivalent to an orthogonal action; the analog of the Meeks-Yau result then follows from t* *he methods of [MY] and some standard considerations in equivariant differential topology. Section * *4 contains a proof of the uniqueness of equivariant smoothings of 3-manifolds. The paper ends with re* *marks concerning orientation reversing actions on R3, D3 and S3. Although a large part of the di* *scussion in [MY] applies to smooth A5-actionon R3, at certain points [MY] uses in a crucial way * *the solvability of all proper closed subgroups of SO3 except A5 to draw important conclusions. It * *is also worthwhile to point out that the troubles with handling A5-actionson S3 (or R3) are also p* *resent in [DM] and [CS], where techniques different from those of [MY] were employed (also see* * [Fe]). The results of [BKS] overcome some of the difficulties caused by the nonsolvability of A5, * *and of course the arguments in Sections 2 and 4 are designed to deal with the remaining problems.* * In fact, this paper could be viewed as a complement to [BKS], strengthening some of the concl* *usions of [BKS] obtained by gauge theoretic techniques. The most important idea in Section 2 ma* *y be described as follows: The orthogonality theorem for A5-actions on R3 would follow quickly* * if one knew that a smooth A5-action on S2 x [0; 1] was equivalent to the product of a smoot* *h (orthogonal) action on S2 with the unit interval. Results of W. Meeks and P. Scott [MS, Thm.* * 8.1, pp. 40-341] yield a "stable" result along these lines; namely, if we form an equivariant co* *bordism connected sum of a smooth A5-action on S2 x [0; 1] with A5 x T 2x [0; 1] by suitably remo* *ving copies of A5x IntD2 x [0; 1] from each and identifying their frontier sets A5x S1 x [0; 1* *], then we obtain a smooth action of A5 on S60x[0; 1], where S60is the orientable surface of genus * *60, and the results of [MS] imply that this action is equivalent to the product of a standard A5-ac* *tion on S60 with the unit interval. Acknowledgment. Visits by the second named author to Tulane University play* *ed a crucial role in the completion of this work. Both authors are grateful to the Tulane Un* *iversity Mathematics Department for providing these opportunities. 1. Equivalent statements For some time it has been understood that the orthogonality results for act* *ions on R3, D3 , and S3 are logically equivalent. We shall begin with a formal statement of the * *logical relationships. Theorem 1.0. Each of the statements (1:1)-(1:4) in the list below implies the n* *ext one; and (1:4) implies (1:1): (1.1) Every smooth A5-action on S3 with nonempty fixed point set is topological* *ly equivalent to an orthogonal action. (1.2) Every smooth A5-action on S3 with nonempty fixed point set is differentia* *bly equivalent to 3 an orthogonal action. (1.3) Every smooth A5-action on the closed disk D3 is differentiably equivalent* * to an orthogonal action. (1.4) Every smooth A5-action on R3 is differentiably equivalent to an orthogona* *l action. We shall prove the implications in the cyclic order 1 =) 2 =) 3 =) 4 =) 1. * * To avoid degenerate cases, we assume the actions are nontrivial. Proof that (1.1) implies (1.2). This is a special case of a more general resul* *t: If G is a finite group that acts smoothly and orientation preservingly on the compact 3-m* *anifolds M3 and N3 , and h : M3 ! N3 is a G-equivariant homeomorphism, then h is equivariantl* *y isotopic to an equivariant diffeomorphism. Although the validity of this result seems to ha* *ve been generally accepted, the proof turns out to be less trivial than one might expect. For the* * sake of completeness, we shall include a proof in Section 4. Proof that (1.2) implies (1.3). Given a smooth action on D3 we know that the ac* *tion on the boundary is orthogonal. If we attach a linear disk 0 along the boundary, we obt* *ain a smooth action on S3 with a nonempty fixed point set, and thus the conclusion of (1.3) * *implies that the constructed action on S3 is equivariantly diffeomorphic to a linear action. By * *the closed Equivariant Tubular Neighborhood Theorem we can choose the equivariant diffeomorphism so th* *at it takes the disk 0 to a standardly embedded linear disk about a fixed point of the linear a* *ctions. If we remove the interior of 0 and its image we obtain an equivariant diffeomorphism * *from the original A5-action on D3 to a linear disk. Proof that (1.3) implies (1.4). This is formally identical to the argument in [* *MY] for smooth actions of all other compact Lie groups on R3 modulo a single change: Theorem 3* * in [MY], which states that a smooth G-actionon D3 is orthogonal if G 6~=A5 , must be replaced * *with (1.4). Most of the argument in [MY] goes through for arbitrary compact Lie groups. The only* * point at which A5 must be excluded is the application of [MY, Thm. 3] at the top of [MY, p. 17* *8]. Proof that (1.4) implies (1.1). Let x0 2 S3 be a fixed point, and consider the* * induced A5-action on S3 - {x0} ~=R3. By (1.4) this action is equivalent to an orthogon* *al action; it follows that the one point compactifications of these actions are equivariantly* * homeomorphic. But the one point compactification of the action on S3- {x0} is merely the original* * action on S3, and the one point compactification of an orthogonal action on R3 is an orthogonal a* *ction on S3, and therefore the original smooth A5-actionon S3 is topologically equivalent to an * *orthogonal action. 2. Stabilization techniques The results of [BKS] show that a nontrivial smooth A5-action on S3 has exac* *tly two fixed points. If we remove the interiors of closed linear disks about these points w* *e obtain a smooth action of A5 on S2 x [0; 1]. The restriction of this group action to either bou* *ndary component is linear by the geometrization results for finite group actions on surfaces, and * *as indicated in Section 1 the proofs of the main results essentially amount to showing that the induced* * action on S2x(0; 1) is equivalent to the product of the action on either boundary component with th* *e trivial action on (0; 1). We shall begin by showing that the action on S2 x [0; 1] becomes a prod* *uct if one stabilizes 4 by taking suitable cobordism connected sums with copies of the product cobordis* *m T 2x[0; 1] This is formally parallel to a well known principle in 4-dimensional topology: Thing* *s often simplify if one stabilizes by taking connected sums with copies of S2 x S2. However, the an* *alogy is limited because the underlying techniques have little in common. The basic idea behind the cobordism connected sum is simple; one takes prop* *erly embedded curves in each cobordism starting at one end and terminating at another, forms * *nice closed tubular neighborhoods of the curves, and identifies the complements of the interiors of* * the tubular neigh- borhoods along the unit sphere bundles. In order to avoid questions about the d* *ependence of this construction on various choices, we shall only work with the case that is neede* *d in this paper. Let V be a smooth A5-manifold that is nonequivariantly diffeomorphic to S2 * *x [0; 1]. The results of Section 1 imply that the nonfree part of the action is a finite set * *of properly embedded smooth curves, each starting in S2 x {0} and ending in S2 x {1}. Let X S2 be * *the image of the nonfree part of the A5-action under the composite V ~= S2 x [0; 1] proj-* *!S2. Since X is the smooth image of a finite set of smooth curves, it follows that X has measur* *e zero. Therefore we can find a vertical curve in S2 x [0; 1] ~=V that lies in the free part of t* *he A5-action on V . Furthermore, we can take a small tubular neighborhood of the form : D2x[0; 1] * * S2x[0; 1] ~=V that also lies in the free part of the action and is disjoint from all its tran* *slates under the action of A5 . Let T 2be a smoothly embedded closed disk. Consider the A5-manifold U fo* *rmed by gluing V - A5. (IntD2 x [0; 1]) to A5 x (T 2- Int) x [0; 1] by identifying A5 .* * (S1 x [0; 1]) to A5 x@x[0; 1]. Since we can choose a diffeomorphism from V to S2x[0; 1] that is * *compatible with boundary collars, we can perform this construction so that U becomes a smooth A* *5-manifold. By construction the boundary components of U are surfaces of genus 60 (= order * *ofA5) and the action of A5 is given by a connected sum of a linear action on S2 with 60 copi* *es of T 2over the free part of . Proposition 2.1. The smooth A5-manifold U is equivariantly diffeomorphic to the* * product of S2()#A5(A5 x T 2) with the trivial A5-action on [0; 1]. The notation means that S2() is a linear A5-actionon S2 and #A5(A5 x T 2) m* *eans that one takes connected sums with 60 copies of T 2equivariantly over the free part * *of S2(). Proof . By construction, U is nonequivariantly equivalent to S60x [0; 1] where * *S60 is an oriented surface of genus 60. Therefore a result of Meeks and Scott [MS, Thm. 8.1, pp. 3* *40-341] shows that the action on U is equivalent to the product of an A5-actionon S60 with th* *e trivial action on [0; 1]. The action on S60 is clearly equivalent to the restriction of the A* *5-action on either boundary component of U , and therefore the action on S60 has the dorm describe* *d above. Proposition 2.1 implies that a smooth A5-actionon S3 with two fixed points * *has a very well behaved equivariant Heegaard splitting: Proposition 2.2. Let M3 be a smooth A5-manifoldwith two fixed points such that * *M is nonequi- variantly diffeomorphic to S3. Then M is equivalent to the A5-manifold formed f* *rom D3()#A5(A5 x S1 x D2) and D3()#A5(A5 x D2 x S1) by gluing the boundaries together via an equivariant diffeomorphism. 5 In analogy with previous notation, if @A, @B 6= ; the equivariant boundary * *connected sum A#A5(A5 x B) is given by taking the connected sum of A with 60 copies of B along the boundar* *y equivariantly and over the free part of the action on A. If is a 3-dimensional representati* *on, then D3() refers to its unit disk. Of course the equivariant gluing construction merely y* *ields S3( x R) if one identifies the boundaries by the identity (in this case one is taking an equiva* *riant connected sum of S3( x R) with 60 copies of S3 over the free part of the action). Proof. As noted at the beginning of this section, M has an equivariant decompos* *ition D3+(1) [ V [ D3(2) where D3 (i) is a linear disk and V is nonequivariantly diffeomorphic* * to S2 x [0; 1]. In fact, since 1 and 2 are determined by their restrictions to cyclic groups an* *d the fixed point sets of cyclic subgroups and connected, we know that 1 and 2 are equivalent rep* *resentations; henceforth write = i. Let U be the equivariant cobordism connected sum of V an* *d 60 copies of T 2x [0; 1] constructed before the proof of Proposition 2.1. Consider the cu* *rves Ca (indexed by a 2 A5) in V that arise in the construction of U . These curves can be exten* *ded slightly into the interiors of D () so that their intersections with @D () are transverse, * *and in fact one can extend the tubular neighborhoods of the Ca to closed tubular neighborhoods * *of the extended curves C0a. Of course all this can be done equivariantly so that each piece is * *disjoint from all of its translates. The extended tubular neighborhoods N(C0a) meet D () in sets N (C0* *a) diffeomorphic to D3 \ R3+, where R3+is all points with nonnegative third coordinate. Along ea* *ch of the disks N+(C0a), form a boundary connected sum with S1 x D2 and along each disk N- (Ca)* *, form a boundary connected sum with D2xS1. It follows that the A5-manifoldM0 formed by * *identifying D3+()#A5(A5 x S1 x D2) ; D3-()#A5(A5 x D2 x S1) along the consecutive boundary pieces is equivariantly diffeomorphic to M#A5(A5* * x S3); since the latter is A5-diffeomorphicto M , it follows that M0 is A5-diffeomorphicto M* * . Since U is equivariantly diffeomorphic to S2()#A5(A5 x T 2x[0; 1] by Proposition 2.1, it * *follows that M0~= M is diffeomorphic to an A5-manifold formed by identifying the boundary of D3()* *#A5(A5 x S1 x D2) to that of D3()#A5(A5 x D2 x S1). Remark. Of course the two pieces in the preceding sentence are equivariantly di* *ffeomorphic to each other. The reason for writing the factors in both orders is to make S3( x * *R) the manifold formed by gluing along the identity. The techniques of Proposition 2.1 and 2.2 also yield another result that wi* *ll figure importantly in Section 3. Theorem 2.3. Let M3 be a smooth A5-manifold with two fixed points such that M3 * *is nonequiv- ariantly diffeomorphic to S3. Let x0 and x1 be the fixed points, and let V 3be * *the local represen- tation of A5 at the fixed point x0. Then M3 - {x1} is equivariantly diffeomorph* *ic to an invariant open neighborhood of the origin in V 3. Proof. Since M3-{x1} is equivariantly diffeomorphic to M3-D1 where D1 is an inv* *ariant linear disk about x1, it follows that we can work with M3 - D1 instead of M3 - {x1}. F* *rom the proof of Proposition 2.2, it follows that the interior of D3()#A5(A5 x D2x S1) contains * *an invariant open 6 subset of the form M3 - D1 - (A5 x E()), where is the image of a smooth proper* * embedding from [0; 1) to M3 - D1 such that lies in the free part of the A5-actionon M3 -* * D1, and the translates of satisfy g \ = ; for all g 2 A5 such that g 6= 1, and E() is a c* *losed tubular neighborhood of (so that A5 x is properly and equivariantly embedded in M3 - * *D1). If {j} is a (say) finite family of pairwise disjoint images of proper smooth embedding* *s [0; 1) ! Nk for some smooth manifold Nk (k 2), then it is well-known that Nk - [E(j) is diffe* *omorphic to Nk (compare [La, numbered paragraph 7, p. 143] or the top half of [Wh, p. 237])* *, where E(j) is a family of pairwise disjoint closed tubular neighborhoods for the curves j; the * *same considerations in our example show that (M3 - D1) - (A5 x ) is equivariantly diffeomorphic to * *M3 - D1. Therefore we conclude that M3 - D1 is equivariantly diffeomorphic to an invaria* *nt open subset of Int D3()#A5(A5 x D2 x S1) . On the other hand, if we apply Proposition 2.2 t* *o the linear action S3() we see that W = D3()#A5(A5 x D2 x S1) is equivariantly diffeomorp* *hic to an invariant open subset of S3() that contains exactly one fixed point of the latt* *er. By construction V is the complement of the remaining fixed points, and thus W is an invariant * *open subset of V . Since M3 - D1 is equivariantly diffeomorphic to an invariant subset of W ,* * it follows that M3 - D1 ~=M3 - {x1} is also equivariantly diffeomorphic to an invariant open su* *bset of V . 3. Infinite processes As noted in Section 2, a smooth A5-action on M3 ~=S3 with two fixed points * *determines a smooth action of A5 on M0 ~=S2 x [0; 1] by removing the interiors of disjoint* * closed linear disks about the fixed points. Of course an orientation on M3 determines an ori* *entation on the submanifold M0, and @M0 is equivalent to the disjoint union S2() t -S2() where * *one chooses some orientation ! for the representation that will remain fixed throughout th* *e discussion. One can define a set A(; !) of oriented equivariant diffeomorphism classes of o* *riented smooth A5-manifolds N3 that are nonequivariantly diffeomorphic to S2x[0; 1], and such * *that the bound- ary components are equivariantly diffeomorphic to S2(). The induced orientatio* *n !@N of @N determines the "right" orientation of S2() on one component that will be denote* *d by @+N , and !@N determines the "wrong" orientation on the other component, which will be de* *noted by @- N . The reason for dwelling on orientation questions is to construct a natural alge* *braic structure on A(; !). Proposition 3.1. If ff; fi 2 A(; !) are represented by (A; A) and (B; B ), wher* *e X is asso- ciated orientation, then the class of ffi ffi A t B @+A @- B ; A t B @+A @- B depends only on the classes of (A; A) and (B; B ) in A(; !), and thus the const* *ruction yields a well-defined element ff O fi 2 A(; !). This binary operation is associative,* * and the class of (S2() x [0; 1]; standard orientation) is a two-sided identity. The proof of 3.1 is standard. Of course, objects like A(; !) have arisen repeatedly in geometric topology* *. A well known infinite process argument for such constructions (cf. [St]) applies in our sett* *ing and immediately yields the following result: Corollary 3.2. If (A; A) determines an element in A(; !) with two-sided inverse* *, then Int(A)[ D() is equivariantly diffeomorphic to the associated orthogonal action to on R* *3. 7 We can combine this with Theorem 2.3 to obtain a result that yields (1.1), * *and hence also (1.2)-(1.4), almost immediately. Proposition 3.3. Every element A(; !) is invertible. Corollary 3.4. Statement (1:1) is true; specifically, every smooth A5-action on* * S3 with non- empty fixed point set is topologically equivalent to an orthogonal action. We shall first prove that 3.3 implies 3.4, and afterwards we shall prove 3.* *3. Proof that (3.3) implies (3.4). As noted earlier, the results of [BKS] imply th* *at the action has exactly two fixed points. Let D+ and D- be disjoint closed linear disks about t* *hese fixed points, and let A be the closure of the complement of D+ [ D- . Then A with its inherit* *ed orientation determines an element of A(; !) , and by Proposition 3.3 and Corollary 3.2, we * *know that the induced action on Int(A[@D()) is orthogonal. Since the one point compactificati* *on of the former is equivariantly homeomorphic to the original action on S3and the one point com* *pactification of the orthogonal action on R3 is orthogonal, it follows that the original action is t* *opologically equivalent to an orthogonal action. There is only one thing left to do. Proof of 3.3. Let A3 be a smooth A5-manifoldthat is nonequivariantly diffeomorp* *hic to S2x[0; 1] and such that @A3 = S2() t -S2(). Form a smooth action on S3 by attaching orth* *ogonal disks D3 () on each boundary component, and call the resulting A5-manifold M3. * * If x1 is the fixed point in D3-() and V 3is R3 with the orthogonal action , then Theorem* * 2.3 implies that M3 - {x1} is equivariantly diffeomorphic to an invariant open neighborhood* * of 0 in V 3. In fact, by the Equivariant Closed Tubular Neighborhood Theorem we can find an equ* *ivariant self- diffeomorphism of V 3such that the unit disk D3() is mapped to D3+() M3 - {x1}* *. If we remove these disks, we obtain an equivariant smooth embedding h : A ! S2() x [1* *; 1) such that @A \ S2() x {1} = @+A. Choose K 2 R so large that A is contained in S2() x [1; * *K]. Then if B = S2() x [1; K] - (Int(A) [ @+A) ; it follows that B and its associated orientation defines a class in A(; !) so t* *hat [A; A]O[B; B ] = 1. Repeating this procedure with (B; B ) in place of (A; A) we obtain a pair (C* *; C ) such that [B; B ][C; C ] = 1. By associativity, it follows (as usual) that [A; A] = [C; C* * ] and that [B; B ] is a two sided inverse to [A; A]. 4. Equivariant smoothings of 3-manifolds The purpose of this section is to prove the uniqueness of equivariant smoot* *hings of 3-manifolds. The main result is the following. Theorem 4.1. Let M; N be compact oriented 3-manifolds with smooth orientation * *preserving actions of a finite group G, and let h : M ! N be an equivariant homeomorphism.* * Then there exists an equivariant diffeomorphism d : M ! N equivariantly isotopic to h. In Section 5 we shall indicate how one can prove similar results for action* *s that do not neces- sarily preserve orientations. 8 The basic ideas in the proof of Theorem 4.1 are fairly predictable; one use* *s the existence and uniqueness theorems for nonequivariant smoothings in dimensions 3 together wit* *h an induction argument involving the orbit and slice types. For the sakes of brevity and comp* *leteness, we shall also use results of [MY], [SM], [KL], and this paper; all of these can probably* * be replaced by more elementary considerations (cf. the remarks following the statement of Propositi* *on 4.2 below), but such an argument would require some lengthy digressions and the tedious verific* *ation of many additional elementary facts. We shall begin with a result that is needed to start the inductive argument: Proposition 4.2. (Equivariant Annulus Conjecture) Let G be a finite group actin* *g orthogonally and orientation-preservingly on R3 with a zero-dimensional fixed point set, let* * S1 and S2 be dis- joint, invariant, G-locally flat submanifolds that are G-diffeomorphic to the u* *nit sphere, let A0 be the unique component of R3 - (S1 [ S2) whose closure contains S1 [ S2, and let * *A be the closure of A0. Then A is G-equivariantly homeomorphic to S1 x [0; 1]. Remark. One could probably establish this result by extending the ordinary 3-d* *imensional proof of the Annulus Conjecture to relative annuli, applying this extension to * *the orbit space A=G R3=G ~=R3, and lifting the homeomorphism A=G ~=S1=G x [0; 1] to an equivar* *iant home- omorphism from A to S1 x [0; 1]. However, the work needed to carry this out is * *more extensive than one might first expect. Proof. By the validity of the ordinary 3-dimensional Annulus Conjecture we know* * that A is home- omorphic to S2 x [0; 1]. Furthermore, by [KL] we know that A has an equivariant* * smoothing Aff, and by the uniqueness of (nonequivariant) smoothings of 3-manifolds, we know th* *at the equivari- ant smoothing of A is nonequivariantly diffeomorphic to S2 x [0; 1]. If G is no* *t isomorphic to the icosahedral group A5, then the results of [MY] imply that A is also equivariant* *ly diffeomorphic to S2 x [0; 1], and this completes the proof when G 6~=A5 . If G ~=A5, then the results of Sections 2 and 3 show that Affis equivariant* *ly diffeomorphic to an invariant submanifold Bffof R3 (with the given linear action) such that o* *ne component of R3 - Bffis the open unit disk and one component of @Bffis the unit sphere (1). * *We need to prove that Bffis equivariantly diffeomorphic to (1) x [0; 1]. The singular set of Bffconsists of finitely many properly embedded, locally* * flat arcs from points in @- Bff= (1) to points in the other component of @Bff; the latter comp* *onent will be denoted by @+Bffhenceforth. As in [Hi] we can assume these arcs are neatly embe* *dded and we can find a G-invariant system of neat tubular neighborhoods {Nj} of the singula* *r arcs. Let W be the closure of Bff- [Nj; by construction W is an invariant smooth submanifol* *d with corners (hence an invariant topological submanifold). We claim that it suffices to show that W is a handlebody. For if this is tr* *ue, then a result of C. Gordon [G, Thm. 20] (see also [Fr]) shows that the embedding of the arcs* * in the singular set in Bffis nonequivariantly equivalent to a vertically embedded family of arc* *s. One can then use the methods of [MY] (see Remark 2, p.172) to show that W is equivariantly d* *iffeomorphic to D# x [0; 1], where D# is the disk with holes given by removing disks about the * *(isolated) singular points of (1). Therefore, if T is the singular set of the induced action on (1)* * and D(T ) is a closed tubular neighborhood, the Bffis given by gluing W = D# x [0; 1] to D(T )* * x [0; 1] via an equivariant diffeomorphism of @D(T ) x [0; 1]. But the group G acts freely on @* *D(T ) x [0; 1] with 9 orbit space equal to three copies of S1x[0; 1], and standard considerations inv* *olving transformation groups and surface automorphisms imply that every choice of equivariant gluing * *diffeomorphisms yields the same smooth G-manifold. Since (1) x [0; 1] is obtainable in this way* *, it follows that Bffmust be equivariantly diffeomorphic to (1) x [0; 1]. It remains to prove that W is a handlebody; more precisely, we shall show t* *hat W is home- omorphic to D# x [0; 1] (nonequivariantly!), where D# is defined as in the prec* *eding paragraph. The first step is to notice that W and D# x [0; 1] are homotopy equivalent; in * *fact, Proposition 3.2 implies that the interiors of W and D# x [0; 1] are homeomorphic. The secon* *d step is to notice that W and D# x [0; 1] are both Haken manifolds; it is well known that D# x [0;* * 1] is Haken for homological reasons, and the homeomorphism Int(W ) ~=Int(D# x [0; 1]) shows tha* *t W has the same homological properties as D# x [0; 1]. The third step is to notice that tw* *o Haken manifolds W1 and W2 are homeomorphic if there is a homotopy equivalence from W1 to W2 tha* *t sends @W1 to @W2 by a homotopy equivalence (compare [J, Cor. X.8, p. 203]). The final ste* *p is to construct such a map from W to D# [0; 1]. Let be the unit disk in R3 (so that (1) = @),* * let * be a disk of large radius such that Int* contains the invariant submanifold Bff* *and , and let = *- Int(). By adding invariant tubular neighborhoods of the singular arcs in * * to Bffand rounding corners we can find a new submanifold Cff Bffsuch that (Cff; @Cff) is * *G-diffeomorphic to (Bff; @Bff), the submanifold Cffis contained in and contains the latter's s* *ingular set, and @*\ Cffis G-diffeomorphic to D(T ), where D(T ) is defined as before. Now remov* *e the interiors of neat closed tubular neighborhoods Vi about the singular arcs (where neatness* * is again defined as in [Hi]), and let C0 be the closure of Cff- [ Vi. It follows that C0 is diff* *eomorphic to W and the inclusion W C0 is a homotopy equivalence. Let C1 be the closure of -[ Vi; * *it then follows that the inclusion W C0 C1 is also a homotopy equivalence (in fact, the inclu* *sion of W \ (1) in each of these submanifolds is a homotopy equivalence). We only need to show * *that the inclusion @C0 C0 C1 can be deformed so that @C0 maps to @C1 by a homotopy equivalence. * *But a map from @C0 to @C1 can be defined by the identity off @C0\ @+Cffand on the lat* *ter set by the composite @C0 \ @+Cff C1 ~=D# x [0; 1] proj-!D# ~=@* \ : By construction this map's composite with the inclusion @C1 C1 is homotopic to* * the inclusion of @C0 into C1. Furthermore, the map also sends Closure(@C0- @+Cff) to Closure(* *@C1- @*) homeomorphically such that the intersection of the first set with @C0 \ @+Cffco* *rresponds to the intersection of the second set with C1 \ @*, and the map sends @C0 \ @+Cfft* *o C1 \ @* by a homotopy equivalence. Therefore by the gluing theorem for homotopy equiva* *lences [Bro, Thm. 7.4.3, pp. 270-271], it follows that the map @C0 ! @C1 described above i* *s a homotopy equivalence. We shall also need the following uniqueness result for topological tubular * *neighborhoods: Proposition 4.3. Let G be a finite cyclic group acting smoothly and semifreely * *on an unbounded manifold Mn with fixed set Nn-2 , let 'i : D(i) ! M be equivariant closed topol* *ogical tubular neighborhoods of N in M for i = 0 or 1, and assume there is a compact subset K * * N and an invariant open neighborhood K U N such that '0 = '1f , where F is a smooth or* *thogonal disk bundle isomorphism over U . Then there is an ambient equivariant isotopy H that* * is fixed near K , a positive real number ", and an equivariant (smooth orthogonal) disk bundl* *e isomorphism F : "D(0) ! "D(1) such that F = f over an invariant subneighborhood V such tha* *t K V 10 __ V U and H1'0 = '1F . Notation. If D(ff) refers to the unit disk bundle of a vector bundle ff an* *d " > 0, then "D(ff) will denote the corresponding disk bundle of radius "; usually we can assume " * *< 1, and in this case we view "D(ff) as a subspace of D(ff). Proof. There are three main steps. First, the orbit spaces of all the relevant * *actions are manifolds with canonical smooth structures (see the first part of [Sch2]). Passage to the* * orbit space sends the disk bundles D(i) to (probably different) disk bundles D(ji), and the maps 'i=G* * are topological tubular neighborhoods of N in M=G. Furthermore, the induced map f=G induces a * *smooth orthogonal disk bundle automorphism over U . The second step is to use the uniq* *ueness results of [KiS1, Thm. A, p. 310] for topological tubular neighborhoods of the codimension* * two locally flat embedding N M=G; more accurately, one also needs the results of [FQ, Section 9* *.4] to cover the case n = 4 (also see [FQ, Section 9.4]). These results yield an ambient isotopy* * ht that is fixed near K , sends N to itself, and satisfies h1('0=G) = '1F *, where F *is a smooth ort* *hogonal bundle isomorphism from "D(j0|V ) to_"D(j1|V ) for some " > 0, where V is an invariant* * neighborhood of K in N such that K V V U . The third step is to construct liftings of h* * and F * to an ambient isotopy H on M and an orthogonal bundle isomorphism F : "D(0) ! "* *D(1). Such liftings can be constructed in a direct elementary fashion; in particular,* * we can use the isovariant covering homotopy property for stratum preserving maps of orbit spac* *es (cf. the second chapter of [Bre]). Perhaps the most subtle point is that the lifting of a (smoo* *th) orthogonal bundle isomorphism is a map of the same type, but this can be verified by looking at s* *uitable product neighborhoods. Proof of Theorem 4.1. Let h : M ! N be an equivariant homeomorphism between com* *pact smooth G-manifolds (G finite) with orientation-preserving group actions. Since * *the proof can be reduced to the special case of connected manifolds by elementary considerations* * involving connected components, we shall assume M and N are connected. Our proof that h is equivari* *antly isotopic to an equivariant diffeomorphism will use the uniqueness theorem for smooth struct* *ures on manifolds of dimension 3 and an induction with respect to the orbit types. We shall need some standard terminology for the strata of locally linear gr* *oup actions as introduced in [Bre] and [LR, pp. 229-230]. The partially ordered set of G-compo* *nents of M(H), where H G, will be denoted by {M(H)i}, and the corresponding set for N(H) will* * be denoted by {N(H)i}. Since we are dealing with orientation-preserving actions, there are* * no 2-dimensional G-components. By construction h maps each G-component M(H)ionto a G-component * *of the form N(H)i; it follows that the slice representations for these G-components ar* *e equivariantly homeomorphic (compare [Sch1]), and since topologically equivalent representatio* *ns of dimension 3 are linearly equivalent, it follows that the slice representations for N(H)ia* *nd M(H)iare linearly equivalent. We shall use this without further comment in the argument below. Let @h : @M ! @N be the induced equivariant homeomorphism on the boundary. * * By the geometrization results for group actions on surfaces (see [E]), the equivariant* * isotopy extension theorem [LR], and the uniqueness of equivariant collar neighborhoods there is a* *n isotopy from h to an equivariant homeomorphism h0that is an equivariant diffeomorphism near the b* *oundary. Since isotopy is a transitive concept, we shall assume that h already has this proper* *ty to simplify the discussion. 11 Smoothing an equivariant homeomorphism near 0-dimensional G-components. L* *et M(H)i be a 0-dimensional G-component of M ; then M(H)i=G lies in the interior of M , * *and we may as well assume the neighborhoods in the preceding paragraph are so small that they* * do not contain M(H)ior N(H)i. As noted above, h induces a homeomorphism h(H)i=G from M(H)i=G t* *o N(H)i=G. We claim there is an invariant neighborhood U(H)iof M(H)isuch that h|U(H)iis is* *otopic to an equivariant map that is a diffeomorphism onto its image. To see this, let D(H)i* *be a closed tubular neighborhood of N(H)i; of course, D(H)iis just a collection of linear disks. Ch* *oose a closed tubular neighborhood E(H)iof M(H)isuch that h(E(H)i) IntD(H)i. Then Proposition 4.2 i* *mplies that D(H)i- h(IntE(H)i) ~=@E(H)ix [0; 1], and one can use this and the Alexander tri* *ck to define an isotopy from h|E(H)ito a map that is a conical extension of an equivariant home* *omorphism from @E(H)ito @D(H)i. Now use the geometrization results for group actions on surfac* *es (cf. [E]) to isotop the boundary map to an orthogonal equivariant homeomorphism, and take cones to * *construct an isotopy from the original conical extension to a linear map from E(H)ito D(H)i. Combining the claim with the equivariant isotopy extension theorem for loca* *lly linear G- manifolds [LR], we can equivariantly isotop h to an equivariant homeomorphism t* *hat is an equiv- ariant diffeomorphism near M(H)i. Since the 0-dimensional G-components in {M(H)* *i} are closed and pairwise disjoint, we can proceed similarly by induction to obtain an equiv* *ariant homeomor- phism that is equivariantly isotopic to h and is an equivariant diffeomorphism * *on an invariant neighborhood of the boundary and all the 0-dimensional G-components M(H)i. In * *fact, by the closed equivariant tubular neighborhood theorem we can construct the new equiva* *riant homeomor- phism h0 and invariant neighborhood U so that for each 1-dimensional G-componen* *t M(H)ithe map h0 is a smooth disk bundle isomorphism on the intersection of an invariant * *tubular neighbor- hood of M(H)iwith U . As before, we shall assume that h itself has all the spec* *ial properties of h0 in order to simplify the rest of the discussion. Smoothing an equivariant homeomorphism near 1-dimensional G-components. * *Now let M(H)ibe a 1-dimensional G-component. We can use Proposition 4.3 to isotopy h s* *o that h is unchanged on a subneighborhood of U but h maps an equivariant tubular neighborh* *ood of M(H)i=G to an equivariant tubular neighborhood of N(H)i=G by an equivariant disk bundle* * isomorphism. By our original hypotheses and the additional properties assumed thus far, the * *map h induces a homeomorphism h(H)i=G from M(H)i=G to N(H)i=G that is a diffeomorphism on a c* *ompact subset K contained in a neighborhood of @M and the 0-dimensional G-components. * *Therefore the relative versions of the uniqueness theorems for smooth structures in dimen* *sions 3 imply that h(H)i=G is isotopic to a diffeomorphism by an isotopy that is fixed near K . Si* *nce the projections M(H)i! M(H)i=G and N(H)i! N(H)i=G are smooth covering projections, the isotopy * *can be lifted to an isotopy from h(H)ito a diffeomorphism, and by construction the lifting is* * equivariant. One can now use the pullback construction, the covering homotopy property, and cutt* *ing and pasting to construct an isotopy on a tubular neighborhood of MHi such that the isotopy * *is fixed near @M and the 0-dimensional G-components and maps the tubular neighborhood by a map t* *hat is an equivariant diffeomorphism on the zero section and an equivariant disk bundle i* *somorphism along the fibers. Another application of the equivariant isotopy extension theorem y* *ields an isotopy from h to an equivariant homeomorphism with the same properties near MHi. Fina* *lly, we can use standard smooth approximation theorems to deform the equivariant disk bundl* *e isomorphism to a smooth equivariant disk bundle isomorphism, and one more application of th* *e equivariant 12 isotopy extension theorem implies that h is equivariantly isotopic to an equiva* *riant homeomorphism such that the map is a diffeomorphism on the zero section and a smooth equivari* *ant disk bundle isomorphism along the fibers. Since a map with the preceding two properties is* * automatically an equivariant diffeomorphism, one more application of the equivariant isotopy * *extension theorem implies that h is equivariantly isotopic to a map that is an equivariant diffeo* *morphism near the boundary, the 0-dimensional G-components, and the set MHi. Since the 1-dimensional G-components are pairwise disjoint and the equivari* *ant isotopies in the paragraph above can be constructed to be constant off some decent neighborhood * *of the given G- component, a similar argument shows that that h is equivariantly isotopic to an* * equivariant homeo- morphism that is an equivariant diffeomorphism near all 0- and 1-dimensional G-* *components. But we have already noted that M has no 2-dimensional G-components, and therefore b* *y the Principal Orbit Theorem we have found an equivariant isotopy from h to an equivariant hom* *eomorphism that is an equivariant diffeomorphism on a neighborhood UM of the singular set. Let WM be the complement of the singular set and boundary of M , let VM = W* *M \UM , and let WN ; VN ; UN be defined similarly with respect to N . Then the preceding paragr* *aphs show that the original equivariant homeomorphism h is equivariantly isotopic to an equivarian* *t homeomorphism h0that maps WM to WN and is an equivariant diffeomorphism from VM to VN . The* * induced map of orbit spaces k : WM =G ! WN =G is a homeomorphism of 3-manifolds that is a d* *iffeomorphism on VM =G; by construction the set (WM - VM )=G is compact. Therefore a relativ* *e version of the uniqueness of smoothings of 3-manifolds implies that k is isotopic to a dif* *feomorphism by an isotopy that is fixed off a compact set (the relative smoothing result is impli* *cit in [Moi]; the results of [KiS2, Essay V] for 3-manifolds contain explicit theorems on relative smooth* *ings). If we lift the isotopy to WM and extend it to M by the identity on the singular set, we obtai* *n an equivariant isotopy from h0 to an equivariant diffeomorphism. Since h0 is isotopic to h by* * construction, it follows that h is isotopic to an equivariant diffeomorphism. 5. The non-orientation-preserving case In this section we shall discuss the modifications needed to prove the main* * results of this paper for actions that are not orientation preserving. We begin with an orthogonality* * theorem for smooth actions on S3. Theorem 5.1. Every smooth action of a finite group on S3 with nonempty fixed po* *int set is topologically equivalent to an orthogonal action. Proof. The results of Sections 1-3 establish this for orientation-preserving a* *ctions, so it suffices to consider the case of actions that do not preserve orientations. There are th* *ree cases depending on the dimension of the set of singular orbits. Case I. The set of singular orbits is 0-dimensional. In this case the group must be Z2 and the action is an involution with two fixe* *d points, and a result of G. R. Livesay [Lv,Ru] implies that the action is differentiably equiv* *alent to an orthogonal action on S3. Case II. The set of singular orbits is 1-dimensional. In this case the subgroup SG of orientation-preserving elements is a nontrivial* * subgroup of index 2. By the results of Sections 1-3 the restriction of the action to SG is orthog* *onal, and therefore 13 the fixed point set of SG is either two points or a circle. In either subcase * *the fixed set of the entire group consists of two points; furthermore, the action's singular set is * *a finite union of circles such that each pair's intersection is the fixed point set. This closely resembl* *es the description of an A5-action's singular set in Section 1, and the techniques of Sections 1-3 for A* *5-actions generalize directly to this case. Case III. The set of singular orbits is 2-dimensional. In this case G must contain elements of order 2 whose fixed point sets are 2-sp* *heres. As before, the results of [BKS] imply the existence of at least two fixed points. Let D0 * *and D1 be closed orthogonal G-disks containing these two points in their interiors, and let A be* * the closure of the complement of D0 [ D1. It will suffice to prove that A is equivariantly hom* *eomorphic to a cylinder @D0x [0; 1], for this implies that the action is given by gluing D0 to* * D1 by an equivariant homeomorphism of the boundary; since each Di is a cone on its boundary, such a * *homeomorphism extends to either Di by taking cones, and this implies that the action in quest* *ion is equivalent to an orthogonal action. Let R(G) G be the reflection subgroup generated by elements of order 2 wit* *h 2-dimensional fixed point sets (i.e., the reflections in G). Let SingR(G)and SingG be the si* *ngular sets of the respective R(G)- and G-actions on A. It follows that the restriction of SingR(G* *)to either boundary component is the union of all circles fixed by reflections, and by Smith theory* * and the classification of surfaces the entire set SingR(G)is equivariantly isomorphic to the product i j SingR(G)\ @Di x [0; 1]: Since the set of reflections is closed under conjugation, this shows that the s* *et SingR(G)is G- invariant. Furthermore, it follows that SingG is a disjoint union of SingR(G)wi* *th finitely many pairwise disjoint 1-disks, where the isotropy subgroups for points on the 1-dis* *ks are all cyclic. Let N0 be a piecewise smooth invariant regular neighborhood of @- A [ SingR* *(G)constructed from a collar neighborhood of @- A and a union of neat tubular neighborhoods of* * the cylinders in A that are fixed under reflections. It follows that Closure(A - N0) is a union * *of 3-disks, and the intersection of each 3-disk Ej with N0 is a nicely embedded 2-disk in @N0\ @Ej.* * By construction the singular set of the induced G-action on [jEj is a disjoint union of 1-disks* *, and the isotropy subgroups are constant over each connected component. This forces the G-action * *on [jEj to be equivalent to a disjoint union of balanced product actions of the form a B := G xH(`)F` ` where each H(`) is a cyclic subgroup of G and F` is a 3-disk with a smooth acti* *on of H(`) whose fixed point set is a 1-disk. Combining these observations`we see that A i* *s an equivariant connected sum of N0 with B along a union of linear 2-disks G xH(`)`, where ` * * @F`. By the validity of the Smith Conjecture, for all ` the action of H(`) on F` is * *orthogonal, and therefore A must be equivariantly homeomorphic to N0. Finally, since SingR(G)is* * equivariantly homeomorphic to the product of SingR(G)\ @Di with the unit interval, we also kn* *ow that N0 is equivariantly homeomorphic to @- A x [0; 1]; combining this with the previous c* *onclusion we see that A is equivariantly homeomorphic to @- A x [0; 1] as required. 14 We can now proceed in analogy with Section 4. The following result is the * *counterpart to Proposition 4.2 (the Equivariant Annulus Conjecture). Proposition 5.2. Let G be a finite group acting orthogonally on R3 with a zero-* *dimensional fixed point set, let S1 and S2 be disjoint, invariant, G-locally flat submanifolds th* *at are G-diffeomorphic to the unit sphere, let A0 be the unique component of R3 -(S1[S2) whose closure* * contains S1[S2, and let A be the closure of A0. Then A is G-equivariantly homeomorphic to S1 x * *[0; 1]. Proof. Once again there are three cases depending upon the dimension of the si* *ngular set. If the singular set is 0-dimensional, then G Z2 and the action on A is fre* *e; of course the action is also smoothable. Construct a smooth involution by attaching linear d* *isks to the two boundary components of the annulus. It follows that the manifold constructed in* * this way is S3 and the involution has two fixed points. Therefore the result of Livesay shows * *that this involution is orthogonal, and the Closed Equivariant Tubular Neighborhood Theorem then imp* *lies that the action on A is a product action. If the singular set is 1-dimensional, then its intersection with A is a fin* *ite union of properly embedded 1-disks, and the result follows in this case by an argument that is fo* *rmally identical to the proof of Proposition 4.2. Finally, if the singular set is 2-dimensional, one can use the argument in * *Case III of Theorem 5.1 to prove that the action on A is a product action; the argument goes through wi* *th no changes. There is also an analog of the uniqueness result (Proposition 4.3) for topo* *logical tubular neighborhoods of codimension 2 G-components: Proposition 5.3. Let G be a finite dihedral group acting smoothly on an unbound* *ed manifold Mn with fixed set Nn-2 , let 'i: D(i) ! M be equivariant closed topological tubula* *r neighborhoods of N in M for i = 0 or 1 (where i is a 2-dimensional real G-vector bundle), and as* *sume there is a compact subset K N and an invariant open neighborhood K U N such that '0 = '* *1f , where F is a smooth orthogonal disk bundle isomorphism over U . Then there is an ambi* *ent equivariant isotopy H that is fixed near K , a positive real number ", and an equivariant (* *smooth orthogonal) disk bundle isomorphism_F : "D(0) ! "D(1) such that F = f over an invariant sub* *neighborhood V such that K V V U and H1'0 = '1F . Proof. The argument resembles the proof of Proposition 4.3, the main differe* *nces arising from the structure of the orbit spaces D(i)=G. If G is dihedral, this orbit sp* *ace has the form D(i) x [0; 1] where i is a real line bundle over the trivial G-space N . Furthe* *rmore, the points of (D(i) - 0 section)x {0} are orbits whose isotropy groups have order 2 and ar* *e given by reflec- tions in G, and if IntD(-) denotes the open disk bundle, then IntD()=G is a 3-m* *anifold whose boundary is IntD(i) x {0}. Since this differs from the description of D()=G in* * the proof of Proposition 4.3, the uniqueness theorem of [KiS1] (and [FQ]) is of no use here.* * Instead, one needs the uniqueness theorems for codimension 1 topological tubular neighborhoods (se* *e [RoS, Thm. 4.9, p. 408]) together with the uniqueness theorems for collar neighborhoods (see [K* *iS2, Appendix A to Essay I]). With these substitutions the argument for Proposition 4.3 extends in* * a straightforward formal manner. We also need a uniqueness theorem for tubular neighborhoods of 2-dimensiona* *l G-components 15 if we work with actions that are not orientation-preserving. The following anal* *og of 4.3 and 5.3 will suffice: Proposition 5.4. Let Z2 act smoothly on an unbounded manifold Mn with fixed set* * Nn-1 , let 'i : D(i) ! M be equivariant closed topological tubular neighborhoods of N in M* * for i = 0 or 1 (where i is a 1-dimensional real G-vector bundle), and assume there is a c* *ompact subset K N and an invariant open neighborhood K U N such that '0 = '1f , where F is* * a smooth orthogonal disk bundle isomorphism over U . Then there is an ambient equ* *ivariant isotopy H that is fixed near K , a positive real number ", and an equivariant (smooth o* *rthogonal) disk bundle isomorphism_F : "D(0) ! "D(1) such that F = f over an invariant subneigh* *borhood V such that K V V U and H1'0 = '1F . Proof. As in the previous argument, the strategy is to modify the proof of Pro* *position 4.3. In this case D()=G is merely a closed boundary collar N x [0; 1] and IntD()=G is an ope* *n boundary collar N x[0; 1). Therefore, as in the proof of 5.3, the proof of Proposition 4* *.3 will go through if we replace the uniqueness theorem of [Kis1] and [FQ] with the uniqueness theorem o* *f [KiS2, Appendix A to Essay I] for collar neighborhoods. The preceding results provide everything that is needed to extend Theorem 4* *.1 to actions that do not preserve orientations. Theorem 5.5. Let M; N be compact 3-manifolds with smooth actions of a finite gr* *oup G, and let h : M ! N be an equivariant homeomorphism. Then there exists an equivariant dif* *feomorphism d : M ! N equivariantly isotopic to h. Proof. As in the proof of 4.1 we may as well assume that M and N are connecte* *d, and since group actions lift to oriented double coverings (cf. [Bre, Chapter I]) we also * *may as well assume that M and N are orientable. Once again the first step is to deform the equivariant homeomorphism to an * *equivariant dif- feomorphism near the boundary. The geometrization results for surfaces do not r* *equire that the underlying groups actions be orientation-preserving, so this step goes through * *with no changes. The second step is to deform the equivariant homeomorphism further so that it is al* *so an equivariant diffeomorphism near the 0-dimensional G-components. The key step in doing this * *for orientation- preserving actions was the equivariant annulus theorem of Proposition 4.2, and * *Proposition 5.2 establishes the corresponding result for linear representations that do not pre* *serve orientation. The third step is to deform the equivariant homeomorphism still further so that it * *is also an equivariant diffeomorphism near the 1-dimensional G-components. Here the argument in the o* *rientation- preserving case depends upon Proposition 4.3, and Proposition 5.3 provides the * *extra information needed to handle the general case. The next step has no counterpart in the orie* *ntation-preserving case; we need to deform the equivariant homeomorphism again so that it is also * *an equivariant diffeomorphism near the 2-dimensional G-components. This can be done by combini* *ng the argu- ment for 1-dimensional G-components with Proposition 5.4. At this stage we have* * an equivariant homeomorphism that is an equivariant diffeomorphism near the singular set, and * *one can deform this map to be an equivariant diffeomorphism everywhere exactly as in the orien* *tation-preserving case. 16 With all this at our disposal the extension of (1.2)-(1.4) to general smoot* *h actions proceeds by the same formal steps as in the orientation-preserving case. For the sake of* * completeness here is an explicit statement: Theorem 5.6. Let G be a finite group. 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