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On Embeddings of Spheres ON EMBEDDINGS OF SPHERES BY B. C. MAZUR Princeton, U.S.A. (1) Introduction If we embed an (n-1)-sphere in an n-sphere, the complement consists of two components. Our problem is to describe the components more exactly. For n = 2, there is a classical theorem of SchSenflies which says that an arbitrary simple closed curve in the two-dimensional sphere S 2 separates S 2 into two components whose closures are both topologically equivalent to a disk. The Riemann mapping theorem yields, moreover, a conformal equivalence between the interior of a simple closed curve and the open disk. The reasonable conjecture to make would be that some analogous result holds for all dimensions; more precisely, that the complementary components of an (n- 1)- sphere embedded in n-space are topologically equivalent to n-cells. A classical counter-example (in dimension n = 3) to this unrestricted analogue of the "two-dimensional SchSenflies theorem is a wild embedding of S 2 in S 3 known as the Alexander Horned Sphere [1]. One of the complementary components of this embedding is not homeomorphie with the n-cell, and, in fact, not simply connected. One's intuition shrugs at this counter-example, attributes its existence to the 'pa- thology of the non-differentiable', or whatever, and persists in believing the statement true--at least for nice imbeddings. In particular, the Alexander Horned Sphere embedding can be made neither differentiable nor polyhedral. Under the assumption that the two-sphere S 2 is embedded polyhedrally in S 3, Alexander [1], and later, Moise [4], proved that the closures of the complementary components of S 2 were topological 3-cells. (1) A research announcement has already appeared in Bull. Amer. Math. Soc. 65, 1959. 1- 61173047. Acta mathematica 105. Imprim6 le 11 mars 1961 2 B.C. MAZUR My aim is to prove that if S n-1 is embedded nicely in S ~, the closure of the complementary components are n-cells. The word "nicely" is defined in section 1, and it includes the class of differentiable embeddings as a special case. However, it is yet unknown whether the class of polyhedral embeddings is also included as a subclass of nice embeddings. The main theorem is proved in section 2. A corollary, the Open Star Theorem, is given in the last section. It states that the open st~r of a vertex in a triangulated mani- fold is homeomorphic with Euclidean n-space. This is a weak partial result towards what is known as the Sphere Problem. The Sphere Problem asks whether or not the closed star of a vertex in a triangulated manifold is combinatorially equivalent to a closed n-cell, I n. It is not known yet whether the closed star is even topologically equivalent to IL I am thankful to Professor R. H. Bing, Professor Ralph Fox, and John Stallings for suggestions yielding improvements in the presentation of the proof of the main theorem. 1. Section of terminology The Euclidean n-space, or the Cartesian product of n copies of the real line R, will be denoted R n. A point x E R ~ is thus an n-tuple of real numbers (x1 ..... x,) and I1 11 = max Ix, I. i = 1 ..... n We use the following notations. The standard n-cube I n = (x e Rnl [I x I[ ~< 1}, 11 = I. The standard n-sphere S n = (x e Rn+l I II x II = 1). The standard n-annulus A m = ( x ~ Rn [ 1 ~< II x II < 2). The standard n-stock St n is obtained by attaching two copies (An)l (An)2 of A n via the identification: {x e (A~)I IXl = e) ~ {x e (A n)21Xl = -- e). The n-annulus has two boundary components, each homeomorphic with S n. I shall refer to {x e R~l [Ixll = 1) as the internal boundary component, denoted in aA n, and (xeRnl Ilxll=2) as the external boundary component, denoted ex ~A n. I shall also need names for standard homeomorphisms of S n-1 onto each boundary compo- nent. Denote by i: sn-1--->in ~A n the identity homeomorphism, and by i*: sn-1---> ex ~A ~ the radial projection homeomorphism. Similarly, I shall need a name for the external boundary component of St n. Call it ex ~St ~. Call the two internal boundary components w 1 and w 2. Let T1, T~ be standard identity homeomorphisms of S n-1 into each internal boundary component. o~ ~MBwDD~S OF SPHeReS Let ~ be some fixed homeomorphism of ex ~St n onto S ~-1, for instance: (Z 1 2 X~..... for X E (A~)I ~ ex 8St ~, x= (x 1.... x~) (x) = If X and Y are topological spaces, X ~ Y will signify that there is a homeo- morphism between them. If X is a manifold, ~X will refer to its boundary, and int X will be the space X-~X. If X and Y are topological spaces, and /:A-+ Y a continuous map, X UI Y or Y (JfX will refer to the topological space X tJ Y with the equivalence relation x N/(x), equipped with the identification topology. There is no ambiguity arising from reversing the order of X and Y; where it is absolutely cIear which attaching map / is meant, X tJI Y may be referred to as X U Y. And further, in the course of the proof, iter- ated identifications X 1 UrX~ IJr~ ... (JfnX~ will be used. Where no confusion can arise I shall dispense with the parentheses necessary to indicate the precise order of iden- tifications. By CX, the cone over a space X, is meant the space X• with X• identified to a point. By a subcone CtX is meant the image in C X of X• [0, t] for 0~<t~l. A similarity trans]ormation S: C X-+ C X is a map S of the form S (X • t)= X • S (t), where S: [0, l]-~ [0, 1] is a monotonic continuous function such that ~(0)=0. Finally, a Euclidean similarity trans/ormation is a mapping T of kn--~R ~ which is of the form: T: x--~x+b, where bER ~ and ~ is a positive number, for xERL D~FI~ITION: An embedding 7~: sn-I--~R ~ is nice if one can extend 7~ to a homeomorphism ~*: I • S ~-1 -~ R ~ (i.e. ~* (0 • S ~-~) = ~ (S ~-1)) such tha~ (i) ~* ((- 1)• ~-1) is contained in the bounded complementary component of ~* (0• n-l) [this requirement is made for convenience only] and (ii) ~* is linear in the neighborhood of some point of (-1, 0)• n. (This requirement, also, is phrased in a manner which saves words in a future ap- plication. Manifestly, there is no loss of generality incurred by assuming that the 'linear point' lie in that restricted territory.) Linear is meant in the sense of a map of a subset of the vector space R ~ into itself. 4 B.C. MAZUR 2. The main theorem The main theorem to be proved is the following: THEOREM. Let S n-1 be nicely imbedded in S n. Then the closures o/ the comple- mentary components o/ S n-1 are homeomorphic to the n-cell. Outline o/ the proo/. (A) In the class of n-manifolds which bound (n-1)-spheres (in a nice way), a multiplication is defined. Intuitively, one takes two such manifolds M, N and attaches them to the interior boundaries of an n-stock, forming a new manifold, M. N, which again has an (n-1)-sphere boundary. Complications exist in the defini- tion since we must prove that the final space M. N is well defined. For, a priori, M. N depends strongly on the homeomorphisms used to attach M and N to the interior boundaries. (B) (Lemma 2) M.N=N.M. (C) If M cS n, one can construct a manifold N for which M.N~I n. (N is roughly the complementary manifold Sn-int M.) (D) Let B ~ = M-N.M-N ... U ~ where ~ is the one-point compactification of the rest, then (Lemma 3) there are two ways of viewing B ~r (a) B ~=(M.N).(M.N) ... (J o~ and since M. N ~- I n, B ~=I n.I n...U ~, from which one easily deduces that B ~ ~I n. (b) B r =M . (N -M) . (N . M) ... U o~ and since N.M=M'N by (B), B ~=M.I n-I ~ ... U from which one easily finds that B~ M. (E) M~I n for by (b) M~B ~, and by (a), B~r n. The proo]. The Semi-Group X. Let X be the collection of couples ]m = (M, ~), where (i) M is a compact n-manifold with boundary embeddable in Euclidean n- space in such a manner that it has an (n-1)-sphere boundary 0M nicely em- bedded in Rn; ON EMBEDDINGS OF SPHERES (ii) (I): ~M--~S ~-1 is a homeomorphism. If ~ E X, 7?/= (M, @), I will denote M by ] ~ [ and (I) by (P~ when it is helpful to do so. Call IX I the set of manifolds M satisfying condition (i) above. Two ele- ments ~, 7?/' E X will be called equivalent (denoted: ~') if there is a homeo- morphism h: ] ~] -> ] ~'1 and a commutative diagram: pl 'l v S n-1 <Td> S n-1 The object of real interest is the set of equivalence classes of X, under the relation defined above. Denote this set by X. A multiplication is defined in X as follows: If ~, Tl are representatives in X of equivalence classes of X, Notice that this is just a definition of what is commonly known as "addition of manifolds".
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