Proper holomorphic self-maps of symmetric powers of balls
aa r X i v : . [ m a t h . C V ] J un PROPER HOLOMORPHIC SELF-MAPS OF SYMMETRIC POWERSOF BALLS
DEBRAJ CHAKRABARTI AND CHRISTOPHER GROW
Abstract.
We show that each proper holomorphic self map of a symmetric power ofthe unit ball is an automorphism naturally induced by an automorphism of the unit ball,provided the ball is of dimension at least two. Introduction
Let D m denote the m -dimensional polydisc in C m , and for 1 ď k ď m , denote by σ k the k -th elementary symmetric polynomial in m variables. The subset of C m given byΣ m D “ tp σ p z q , σ p z q , . . . , σ m p z qq P C m | z P D m u (1.1)is known as the symmetrized polydisc of m -dimensions. It turns out that Σ m D is a pseu-doconvex domain C n with remarkable function theoretic properties, and applications toengineering (see [1] and the work inspired by it.) Of particular interest are the symme-tries and mapping properties of these domains. In [11], Jarnicki and Pflug determined thebiholomorphic automorphisms of Σ m D . More generally, we have the following result ofEdigarian and Zwonek on proper self-maps of Σ m D : Theorem 1 (See [7, 8]) . Let f : Σ m D Ñ Σ m D be a proper holomorphic map, and let σ “ p σ , . . . , σ m q : C m Ñ C m be as in (1.1) . Then, there exists a proper holomorphic map B : D Ñ D such that f p σ p z , . . . , z m q “ σ p B p z q , . . . , B p z m qq . Recall that a proper holomorphic map B : D Ñ D is represented by a finite Blaschkeproduct, so the above result gives a complete characterization of proper holomorphic self-maps of Σ m D , so that each such map is induced by a proper holomorphic self-map of thedisc.In this note we prove an analogous result for proper self-maps of complex analytic spacesanalogous to the symmetrized polydisc, where the disc is replaced by a higher dimensionalball. To define these spaces, let B s Ă C s denote the unit ball in C s , and for some positiveinteger m , let p B s q m Sym denote the m -fold symmetric power of B s , i.e., the collection of all unordered m -tuples x z , z , . . . , z m y , where each z j P B s . Note that the construction of thesymmetric power is functorial: given any map g : B s Ñ B s , there is a naturally definedmap g m Sym : p B s q m Sym
Ñ p B s q m Sym given by g m Sym px z , z , . . . , z m yq “ x g p z q , g p z q , . . . , f p z m qy , Debraj Chakrabarti was partially supported by a grant from the NSF ( which is easily seen to be well-defined. See below in Section 2 for a discussion of this notion,and further details. Then, p B s q m Sym is a complex analytic space, and Σ m D is biholomorphicto D m Sym . The main result of this paper is:
Theorem 2.
Let s ě , m ě , and let f : p B s q m Sym
Ñ p B s q m Sym be a proper holomorphicmap. Then, there exists a holomorphic automorphism g : B s Ñ B s such that f “ g m Sym , that is f is the m -th symmetric power of g . It follows in particular that each proper self-map of p B s q m Sym is in fact an automorphism.Recall also (see [13, p. 25] that an automorphism of B s is of the form ϕ a p z q “ a ´ P a z ´ s a Q a z ´ x z, a y , (1.2)where a P B n , P a is the orthogonal projection from C n onto the one dimensional complexlinear subspace spanned by a , Q a “ id ´ P a is the orthogonal projection from C n onto theorthogonal complement of the one dimensional complex linear subspace spanned by a , and s a “ p ´ | a | q .Note also that the domain B s may be replaced in Theorem 2 by any strongly pseudo-convex domain, without any change in the proof. We prefer to state it in this special casefor simplicity. 2. Symmetric Powers
Complex Symmetric powers.
Recall that, informally, a complex analytic space ismade of local analytic subsets of C n glued together analytically, just as a complex manifoldof dimension s is made of open sets of C s analytically glued together (see [6, 14, 10] formore information). Recall also that an analytic subset of C n is given near each of pointof C n by the vanishing of a family of analytic functions, and a local analytic subset of C n is an open subset of an analytic subset of C n .Let X be a complex manifold, or more generally, a complex analytic space. let X m denote the m -th Cartesian power of X , which is by definition the collection of orderedtuples tp x , . . . , x m q , x j P X, j “ , . . . , m u .X m is then a complex manifold in an obvious way. The symmetric group S m of bijectivemappings of the set t , . . . , m u acts on X m in as biholomorphic automorphisms: for σ P S m , and x “ p x , . . . , x m q P X m we set σ ¨ x “ p x σ p q , . . . , x σ p m q q . The m -th symmetric Power of X , denoted by X m Sym is the quotient of X m under the actionof S m defined above, i.e, points of X m Sym are orbits of the action of S m on X m . We denoteby π : X m Ñ X m Sym the natural quotient map. It follows from the general theory of complex analytic spacesthat X m Sym has a canonical structure of an analytic space, i.e., the quotient analytic spaceof X m under the action of S m as biholomorphic automorphisms (see [3, 4] and [10]). When X m Sym is given this complex structure, the map π becomes a proper holomorphic map. YMMETRIC POWERS OF BALLS 3
In our application, we are interested in the case when X “ B s , the unit ball in C s . Thesymmetric power p B s q m Sym is then biholomorphic to a local analytic set, in fact to an opensubset of a certain affine algebraic variety in C N for some large N depending on m and s .Though logically not needed for the proof of Theorem 2, we give a short account of thisconstruction in order to explain the relation of Theorem 2 with Theorem 1, as well as toemphasize the elementary nature of the constructions.2.2. Embedding of symmetric powers of projective spaces.
Recall that the m -fold symmetric tensor product of a (complex) vector space V , denoted V d m is defined as V d m “ V b m { „ , where V b m “ V b V b ¨ ¨ ¨ b V is the m -fold tensor product of V withitself, and the equivalence relation „ is defined by: v σ p q b v σ p q b ¨ ¨ ¨ b v σ p m q „ v b v b ¨ ¨ ¨ b v m (2.1)for all v , v , . . . , v m P V and all σ P S m . We denote the equivalence class of v b v b¨ ¨ ¨b v m under this equivalence relation (which is an element of V d m ) by v d v d ¨ ¨ ¨ d v m . Just asthe tensor product of vector spaces is itself a vector space, the symmetric tensor product V d m of a vector space is again a vector space, with a natural linear structure as a quotientvector space of V b m . Suppose V is finite-dimensional of dimension s `
1, and let t e , . . . , e s u be a basis of V . Let µ “ p µ , µ , . . . , µ s q P N s ` be a multi-index with | µ | “ ř sj “ µ j “ m ,and let e µ denote the element of V d m given by e µ “ e i d e i d ¨ ¨ ¨ d e i m (2.2)where exactly µ j of the e i k are equal to e j . It is not difficult to see that t e µ u | µ |“ m gives abasis for V d m . Therefore the dimension of V d m is the number of solutions of ř sj “ µ j “ m ,i.e. dim V d m “ ˆ m ` sm ˙ . (2.3)Given a vector space V , there is a natural mapping from the symmetric power, V m Sym (which is just a set), to the symmetric tensor product V d m (which is a vector space),given by x v , v , . . . , v m y ÞÑ v d v d ¨ ¨ ¨ d v m , (2.4)which is easily seen to be well-defined.Denote by P p V q the projectivization of the vector space V , and for v P V , denote by r v s its equivalence class in the projectivization P p V q . There is a natural map ψ : p P p V qq m Sym Ñ P p V d m q induced by the map (2.4) given by ψ pxr v s , r v s , . . . , r v m syq “ “ v d v d ¨ ¨ ¨ d v m ‰ P P p V d m q , (2.5)with v j P V . We call this the Segre-Whitney map (see Appendix V of [14]). It is easilyseen to be well-defined and injective, and is a symmetric version of the well-known classical
Segre embedding of the product of two or more projective spaces as a projective variety ina higher dimensional projective space.Thanks to classical results in complex algebraic geometry, the image ψ ´ P p V qq m Sym ¯ isa projective algebraic variety in the projective space P p V d m q , which, if V “ C s ` is of DEBRAJ CHAKRABARTI AND CHRISTOPHER GROW dimension N p m, s q “ ˆ m ` sm ˙ ´ “ dim C ` p C s ` q d m ˘ ´ . (2.6)2.3. Embedding of p B s q m Sym . Let E be a subset of CP s “ P p C s ` q , and let i : E Ñ CP s be the inclusion map. Then we have a natural inclusion i m Sym : E m Sym
Ñ p CP s q m Sym , so that we can think of E m Sym as a subset of p CP s q m Sym . Composing with the Segre-Whitneyembedding ψ of p CP s q m Sym in CP N p m,s q , where N p m, s q is defined as in (2.6), we obtain anembedding ψ ˝ i m Sym : E m Sym Ñ CP N p m,s q which we will again call the Segre-Whitney embedding . In this way we can think of sym-metric powers as sitting in some projective space. We will denote this embedded versionof the m -th symmetric power of E by Σ m E , i.e.,Σ m E “ ψ ˝ i m Sym p E m Sym q Ă CP N p m,s q . (2.7)Two special cases of this construction are relevant here. The first is when E is an affinepiece of CP s , so that E can be identified with C s . Then, explicitly working through thecomputations, one can verify that Σ m E is an affine algebraic variety in an affine piece of CP N p m,s q . So we can think of Σ m C s as an affine algebraic variety in C N p m,s q . For detailsof this construction, see [14, Appendix V].Now, if E “ B s is the unit ball in an affine piece of CP s , then it is easy to see thatΣ m E “ Σ m B s is an open subset of the affine algebraic variety Σ m C s . In this way, p B s q m Sym is realized as a local analytic set, i.e., an open set of an analytic subset of C N p m,a q (actuallyan open set of an affine algebraic variety).2.4. The case s “ . When s “
1, we have N p m, s q “ m , and it is not difficult to seethat the Segre-Whitney map C m Sym Ñ C m is actually a biholomorphism, given by z ÞÑ p σ p z q , . . . , σ m p z qq , where σ k px z , . . . , z m yq is the k -th elementary symmetric polynomial in the variables p z , . . . , z m q . Then the image of D m Sym is a pseudoconvex domain Σ m D in C m , called the symmetrized polydisc . See [5] for more details. Consequently, the symmetric power D m Sym is biholomorphically identified with the domain Σ m D in C m , which shows that Theorem 2is indeed an extension of Theorem 1.3. Proper mappings of Cartesian to Symmetric powers
The first step in the proof of Theorem 2 is the following result, which is interesting inits own right:
Theorem 3.
Let s ě , m ě , and let f : p B s q m Ñ p B s q m Sym be a proper holomorphicmap. Then, there exists a proper holomorphic map r f : p B s q m Ñ p B s q m such that f “ π ˝ r f . In other words, the map f can be lifted to a proper holomorphic map r f such that thefollowing diagram commutes: YMMETRIC POWERS OF BALLS 5 p B s q m p B s q m Sym p B s q m f π r f Note however, that the map π is not a covering map, so that the classical theory oflifting maps into a covering space is not directly applicable. However, as we will see, wecan reduce this problem to a problem involving covering maps by removing the ramificationlocus and the branching locus of the map π from p B s q m and p B s q m Sym respectively.3.1.
Fundamental group of complements of analytic sets.
The proof of Theorem 3will involve the computation of some fundamental groups, for which we need the followingfact. We include a proof for completeness.
Proposition 3.1.
Let M be a connected complex manifold without boundary, A Ă M bean analytic subset, x a point in M z A , and i : M z A Ñ M the inclusion map. Then if thecomplex codimension of A is at least 2, then the homomorphism of the fundamental groups i ˚ : π p M z A, x q Ñ π p M, x q is an isomorphism.Proof. We begin by recalling the following fact from Differential Topology:
Let X be aconnected C differentiable manifold without boundary, Y Ă M be closed submanifold, x a point in X z Y , and i : X z Y Ñ X the inclusion map. Then if the codimension of Y is atleast 3, then the group homomorphism i ˚ : π p X z Y, x q Ñ π p X, x q is an isomorphism. For a proof, see [9, Th´eor`eme 2.3, page 146]. Essentially, this is a reflection of the factthat thanks to the low codimension of Y , by a standard transversality argument, thereis no problem in homotopically deforming a loop based at x to a loop based at x andnot intersecting Y , and further, given two loops based at x homotopic in X , there is noproblem in homotopically deforming them to each other in X z Y .Now, since A is an analytic subset of a complex manifold M , there is a stratificationof A by local analytic subsets (see [6] for details). More precisely, there exist pairwisedisjoint local analytic subsets A j of A such that A “ n ď i “ A i , where the set B k “ Ť ki “ A i is an analytic subset of M and A k is a closed submanifold of M z B k ´ for each 1 ď k ď n . Note that, since A has codimension at least 2 in M , then A k also has codimension at least 2 in M z B k ´ for each 1 ď k ď n . Thus, assuming A ‰ M , M k “ M z B k is a open submanifold of M for each 1 ď k ď n . Hence, since M k “ M k ´ z A k , DEBRAJ CHAKRABARTI AND CHRISTOPHER GROW each inclusion in the following chain M z A “ M n M n ´ . . . M M “ Mi i i i is an isomorphism of groups by the fact from Differential Topology quoted in the firstparagraph, since we may take X “ M k and Y “ A k and the conditions are satisfied. Theconclusion follows. (cid:3) Branching behavior of π . Since π is proper holomorphic map of equidimensionalcomplex analytic sets, it follows that π must be a covering map when the analytic setsover which it is branched are removed from the source and the target. However, giventhe elementary nature of the considerations here, one can be much more explicit in thisspecial case. Let m , . . . , m k be a partition of m , i.e., m j be positive integers such that ř kj “ m j “ m . We denote by x x : m , x : m , . . . , x k : m k y (3.1)the element of p B s q m Sym in which x j is repeated m j times. Let V p m , . . . , m k q be the setof points α in B sm Sym such that there are distinct x , . . . , x k P B s such that α “ x x : m , . . . , x k : m k y , where we use the notation (3.1), that is x j is repeated m j times. Also let r V p m , . . . , m k q “ π ´ p V p m , . . . , m k qq Ă B sm . Proposition 3.2.
The restricted map π : r V p m , . . . , m k q Ñ V p m , . . . , m k q is a holomorphic covering map of degree m ! m ! m ! ¨ ¨ ¨ m k ! . Proof.
Let α P V p m , . . . , m k q . We will show that there is an open set U α Ă V p m , . . . , m k q ,containing α , such that π ´ p U α q is a disjoint union of open subsets r U iα Ă r V p m , . . . , m k q ,with the restriction of π to each r U iα a homeomorphism onto its image π p r U iα q .We can write α “ x x : m , . . . , x k : m k y for some distinct x , . . . , x k P B s . Since B s is Hausdorff, there exist disjoint open subsets U i Ă B s with x i P U j if and only if i “ j .Now, let U α “ tx α : m , . . . , α k : m k y P V p m , . . . , m k q | α i P U i u . Note that U α is open in in the subspace topology defined on V p m , . . . , m k q , since U α “ π p U m ˆ ¨ ¨ ¨ ˆ U m k k q X V p m , . . . , m k q , where U m i i denotes the m i -fold Cartesian power of U i . Let σ p y , . . . , y m q denote p y σ p q , . . . , y σ p m q q for σ P S m and p y , . . . , y m q P B sm . Then, π ´ px y , . . . , y m yq “ t σ p y , . . . , y m q | σ P S m u , YMMETRIC POWERS OF BALLS 7 and we have π ´ p U α q “ ď σ P S m σ ´ p U m ˆ ¨ ¨ ¨ ˆ U m k k q X r V p m , . . . , m k q ¯ “ ď σ P S m ´ σ p U m ˆ ¨ ¨ ¨ ˆ U m k k q X r V p m , . . . , m k q ¯ “ ˜ ď σ P S m σ p U m ˆ ¨ ¨ ¨ ˆ U m k k q ¸ X r V p m , . . . , m k q . Since the sets σ p U m ˆ ¨ ¨ ¨ ˆ U m k k q are open in B sm , it follows that π ´ p U α q is open in r V p m a , . . . , m k q with the subspace topology. Also, since U i X U j “ H for i ‰ j , the sets σ p U m ˆ ¨ ¨ ¨ ˆ U m k k q and τ p U m ˆ ¨ ¨ ¨ ˆ U m k k q are either identical or disjoint for each σ, τ P S m , and the number of distinct sets σ p U m ˆ ¨ ¨ ¨ ˆ U m k k q is equal to the number ofdistinct preimages π ´ px α : m , . . . , α k : m k yq , which is exactly m ! m ! m ! ¨ ¨ ¨ m k ! . Now, the restricted map π : σ p U m ˆ¨ ¨ ¨ˆ U m k k q Ñ π p U m ˆ¨ ¨ ¨ˆ U m k k q is one-to-one, andhence a homeomorphism, as π is a quotient map. Thus, by restricting π to the subspace σ p U m ˆ¨ ¨ ¨ˆ U m k k qX r V p m , . . . , m k q , we have π : σ p U m ˆ¨ ¨ ¨ˆ U m k k qX r V p m , . . . , m k q Ñ U α is a homeomorphism as well. (cid:3) Proposition 3.3.
Let X and Y be analytic subsets of Ω Ă C n and Ω Ă C m , respectively,and let f : X Ñ Ω be a proper holomorphic map such that f p X q Ă Y . Then,(1) If f p X q “ Y , then dim X “ dim Y .(2) If Y is irreducible and dim X “ dim Y , then f p X q “ Y .Proof. By Remmert’s Theorem, f p X q is an analytic subset of Ω , with dim f p X q “ dim f .Since f is proper, we cannot have dim f ă dim X . Otherwise, the Rank Theorem wouldimply that for some y P f p X q , f ´ p y q is a compact analytic subset of X with positivedimension, which is impossible. Hence, we conclude that dim f p X q “ dim X .Suppose first f p X q “ Y . Then, evidently, dim X “ dim f p X q “ dim Y , establishing(1).Now suppose dim X “ dim Y and suppose that f p X q ‰ Y . Then, by well-knownproperties of analytic sets, Y z f p X q is an analytic set, and since Y is closed in Ω , Y z f p X q is contained in Y . Additionally, since dim f p X q “ dim Y , Y z f p X q cannot be all of Y . Now,since Y “ f p X q Y Y z f p X q , Y is reducible. Hence, if Y is irreducible and dim X “ dim Y ,then f p X q “ Y , completing the proof of (2). (cid:3) We now prove the following lemma:
Lemma 3.4.
Let A “ tp z , . . . , z m q P p B s q m : z i “ z j for some i ‰ j u . Then A is ananalytic subset of p B s q m of codimension s and π p A q is an analytic subset of p B s q m Sym ofcodimension s . The restricted map π : p B s q m z A Ñ p B s q m Sym z π p A q is a holomorphic covering map of complex manifolds. DEBRAJ CHAKRABARTI AND CHRISTOPHER GROW
Proof.
Let A ij be the linear subspace of p C s q m – C sm given by A ij “ p z , . . . , z m q| z i “ z j ( . Then A ij is defined by the vanishing of s linearly independent linear functionals z ÞÑ z ik ´ z jk where 1 ď k ď s , and consequently A ij is of codimension s in p C s q m . Since A “ ď i ă j p A ij X p B s q m q , it now follows that A is an analytic subset of codimension s in p B s q m . Since the finite-to-one quotient map π : p B s q m Ñ p B s q m Sym is proper and holomorphic, by Remmert’sTheorem, π p A q is an analytic subset of p B s q m Sym . Since π ´ p π p A qq “ A , it follows that π | A is a proper map A Ñ π p A q , and since π is also surjective, it is easy to see that we musthave dim A “ dim π p A q . Since dim pp B s q m q “ dim pp B s q m Sym q , it follows that π p A q has thesame codimension in p B s q m Sym as A has in p B s q m , which is s . (cid:3) Recall that, for a partition m , . . . , m k of m , V p m , . . . , m k q is the set of points α in p B s q m Sym such that there are distinct z , . . . , z k P B s such that α “ x z : m , . . . , z k : m k y , where this notation is as in (3.1), i.e., z j is repeated m j times. Let us also set r V p m , . . . , m k q “ π ´ p V p m , . . . , m k qq Ă p B s q m . Then, p B s q m z A is precisely the set r V p m , . . . , m k q and p B s q m Sym z π p A q is precisely the set V p m , . . . , m k q .We are now ready to prove Theorem 3. Proof of Theorem 3.
Let f : p B s q m Ñ p B s q m Sym be a proper holomorphic map and let A beas defined in Lemma 3.4. Since π p A q is an analytic subset in p B s q m Sym , p B s q m Sym z π p A q is anopen, connected set.Since p B s q m z A “ r V p , , . . . , q and p B s q m Sym z π p A q “ V p , , . . . , q , by Proposition 3.2, π | p B s q m z A is a holomorphic covering p B s q m z A Ñ p B s q m Sym z π p A q . Since a holomorphic cov-ering is a local biholomorphism, p B s q m Sym z π p A q is an sm -dimensional complex manifold.Moreover, since π p A q is an analytic subset of p B s q m Sym , we have that p B s q m Sym z π p A q is con-nected and dense in p B s q m Sym . Since reg pp B s q m Sym q is an open subset of p B s q m Sym containing p B s q m Sym z π p A q , reg pp B s q m Sym q must be connected, and hence p B s q m Sym is an irreducible ana-lytic set, with dim pp B s q m Sym q “ dim pp B s q m Sym z π p A qq . Since f is a proper holomorphic mapfrom p B s q m , which is a manifold of dimension sm , to p B s q m Sym , which is an irreducibleanalytic set of dimension sm , by Theorem 3.3, f is surjective.Clearly f | f ´ p π p A qq is a proper holomorphic map from f ´ p π p A qq , which is an analyticsubset of p B s q m , onto π p A q , and so by Proposition 3.3, dim p f ´ p π p A qq “ dim p π p A qq . Sincewe also have dim pp B s q m q “ dim pp B s q m Sym q “ sm , and we know from Lemma 3.4 that π p A q has codimension at least s , f ´ p π p A qq must have codimension at least s in p B s q m .Let U “ p B s q m z f ´ p π p A qq . Since A and f ´ p π p A qq have complex codimension at least s ě
2, by Proposition 3.1, both p B s q m z A and U are simply-connected. Hence, f | U has aholomorphic lift r f | U : U Ñ p B s q m z A , where f | U “ π ˝ r f | U . Since r f | U is bounded on U YMMETRIC POWERS OF BALLS 9 and f ´ p π p A qq is an analytic set, by Riemann’s Continuation Theorem r f | U extends to aholomorphic function r f : p B s q m Ñ p B s q m with f “ π ˝ r f . U p B s q m Sym z π p A qp B s q m z Af | U π r f | U p B s q m p B s q m Sym p B s q m f π r f It remains to show that r f : p B s q m Ñ p B s q m is a proper map. Note that f “ π ˝ r f isa proper map, and f is proper. If r f were not proper, one could find a sequence t z n u n “ with no limit points in p B s q m such that r f p z n q Ñ w P p B s q m . But this composing with π ,we see that f is not proper, which is a contradiction. Therefore r f is proper. (cid:3) Proof of Theorem 2
Let f : p B s q m Sym
Ñ p B s q m Sym be a proper holomorphic map. Then, since π is proper andholomorphic, h “ f ˝ π is a proper holomorphic map p B s q m Ñ p B s q m Sym . By Theorem 3, h lifts to a proper holomorphic map r h : p B s q m Ñ p B s q m with h “ π ˝ r h . By a classicalapplication of the methods of Remmert and Stein (see [12, page 76]), we conclude that thereexist proper holomorphic self-maps of the ball B s , r h i for i “ , . . . , m and a permutation σ of t , , . . . , m u such that r h has the structure r h p τ , τ , . . . , τ m q “ ´r h p τ σ p q q , r h p τ σ p q q , . . . , r h m p τ σ p m q q ¯ . We get the following commutative diagram: p B s q m Sym p B s q m Sym p B s q m p B s q m f ππ h r h Since f ˝ π “ π ˝ r h , and the left-hand side is invariant under the action of S m on p τ , τ , . . . , τ m q , we must have x r h p τ σ p q q , r h p τ σ p q q , . . . , r h m p τ σ p m q qy “ x r h p τ q , r h p τ q , . . . , r h m p τ m qy (4.1)for every p τ , τ , . . . , τ m q P p B s q m and every σ P S m . Such a relation cannot hold unlessthere is a self map g of B s such that for each j “ , . . . , m , we have r h j “ g, since otherwise we could choose a σ for which the two sides would be distinct. Now, since f ˝ π “ h , we have f px τ , τ , . . . , τ m yq “ x g p τ q , g p τ q , . . . , g p τ m qy“ g m Sym px τ , τ , . . . , τ m yq Since f is proper and f “ g m Sym , it follows that g is a proper holomorphic self-map of theball B s . Thanks to a classical result of Alexander (see [2], and also [13, page 316]), for s ě
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Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA
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