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Chapter 1. Preface In 1963, John Milnor put forward a list of problems in geometric topology. 3 3 1. Let M be a homology 3-sphere with π1 =0. Is the double suspension of M homeomorphic to S5? 2. Is simple homotopy type a topological invariant? 3. Can rational Pontrjagin classes be defined as topological invariants? 4. (Hauptvermutung) If two PL manifolds are homeomorphic, does it follow that they are PL homeomorphic? 5. Can topological manifolds be triangulated? 6. The Poincar´e hypothesis in dimensions 3, 4. 7. (The annulus conjecture) Is the region bounded by two locally flat n-spheres in (n + 1)-space necessarily homeomorphic to Sn × [0, 1]? These were presented at the 1963 conference on differential and algebraic topology in Seattle, Washington. A much larger problem set from the conference is published in Ann. Math.81 (1965) pp. 565–591. In the last 30 years, much progress has been made on these problems. Problems 1, 2, 3, and 7 were solved affirmatively by Edwards-Cannon, Chapman, Novikov, and Kirby in the late 1960’s and early 1970’s, while problems 4 and 5 were solved negatively by Kirby- Siebenmann in 1967. Freedman solved the 4-dimensional (TOP) Poincar´e Conjecture in 1980. The 3-dimensional Poincar´e Conjecture and the 4-dimensional PL/DIFF Poincar´e Conjecture remain open as of this writing. This book introduces high-dimensional PL and TOP topology by providing solutions – or at least useful information pointing towards solutions – of problems 1, 2, 3, 4, 5, and 7. This sort of geometric topology has recently been applied to Gromov-style differential geometry, index theory, and algebraic geometry. One of the author’s goals in writing this book is to help workers in other areas to understand what the machinery of geometric topology can do. In a sense, this is intended as a book for people who haven’t decided, yet, whether they want to make the (substantial) investment of learning geometric topology! Accordingly, the focus here is on examples and techniques of proof, rather than on developing any one theory or technique to its logical limits. 1 Chapter 2. Some TOP topology We begin with a study of embeddings of Sn−1 into Sn. We know from algebraic topology that if i : Sn−1 → Sn is an embedding, then i(Sn−1) separates Sn into two parts. We remind the student of the statement of Alexander Duality: n n ∼ n−−1 If K is a polyhedron and i : K → S is an embedding, then H¯(S −K) = H (K). n−1 n n−1 ∼ n−1 n−1 If K = S and =0, this says that H¯0(S − S ) = H (S )=Z.Thus, Sn − Sn−1 has two path components. Further applications of Alexander duality show that each of these complementary domains has the homology of a point. The naive conjecture is that each complementary domain is homeomorphic to Dn and that all embeddings Sn−1 → Sn which preserve orientation should be topologically equivalent – that if i : Sn−1 → Sn is an orientation-preserving embedding, then there should be a homeomorphism h : Sn → Sn so that h ◦ i is the standard embedding of Sn−1 onto the equator of Sn. A classical example, the Alexander Horned sphere, shows that this conjecture is false for n =3. In Example 2.22, we provide a slightly different example of an embedded S2 in S3 which fails to bound a disk in S3. To achieve positive results in the face of such counterexamples, we must impose hy- potheses on the embedding i. The classical condition is that i should be either locally or globally collared in Sn. Accordingly, we begin the section with a proof of Morton Brown’s collaring theorem. In the case of Sn−1 ⊂ Sn, Brown’s theorem says that if either complementary domain of Sn−1 in Sn is a manifold, then boundary of this com- plementary domain has a neighborhood (in the complementary domain) homeomorphic to Sn−1 × [0, 1). Here is the general definition of a local collaring: Definition 2.1. Let X be a topological space and let B be a subset of X.ThenB is collared in X if there is an open embedding h : B × [0, 1) → X with h|B ×{0} = id.If B can be covered by a collection of open subsets, each of which is collared in X,thenB is said to be locally collared in X. Theorem 2.2 (M. Brown [B3], [Con]). If X and B are compact metric and B is locally collared in X,thenB is collared in X. Proof: The proof is by induction on the number of elements in the cover of B.This reduces immediately to the case in which B = U ∪ V with both U and V collared in X. 2 2. Some TOP topology 3 Form a space X+ = X ∪ B × [0, ∞). Using the collars on U and V , we can find open subsets U + and V + of X+ homeomorphic to U × [−∞, ∞]andV × [−∞, ∞]sothat U + ∪V + is a neighborhood of B. Choose functions σ, τ : B → [0, 1] so that σ+τ =1, and so that σ and τ are supported on closed subsets (in B) of U and V , respectively. Define + + + + homeomorphisms hσ : U → U and hτ : V → V so that hσ(u, t)=(u, t + σ(u)) and hτ (v, t)=(v, t + τ(v)). If we choose the inner collars on U and V carefully, perhaps by reparameterizing U × [−1, 0] and V × [−1, 0] to be the new U × [−∞, 0] and V × [−∞, 0], + we can assume that hσ and hτ extend continuously to X using the identity outside of + + U and V . The composition hσ ◦ hτ throws X onto X ∪ B × [0, 1], exhibiting a collar on B. X UV + X Corollary 2.3. The boundary of a topological manifold is collared. It is amazing that this was a new result as late as 1962. Our main goal for this section is to prove the Generalized Schoenfliess Theorem, which says that if both complementary domains of Sn−1 in Sn are topological manifolds, then they are homeomorphic to balls. Definition 2.4. By an n-ball B in a manifold M n, we mean the homeomorphic image ◦ of a standard n-ball in Rn.Ann-ball is collared if ∂B is collared in M n − B.The homeomorphic image of a k-ballisalsoreferredtoasak-cell. Many useful homeomorphisms in the topological category are constructed as compo- sitions of pushes. The next lemma constructs one such push. Lemma 2.5. Let B be a collared n-ball in Rn with collar C and let D be a compact subset of Rn. Then there is a homeomorphism h : Rn → Rn with compact support such 4 2. Some TOP topology that h|(B)=id and h(B ∪ C) ⊃ B ∪ D. ◦ ◦ n Proof: Choose a point b ∈ B and a standard ball B1 ⊂ B centered at b.Letρ : R → n R be a homeomorphism with compact support in B ∪ C shrinking B into B1.This homeomorphism is easy to construct after parametrizing B ∪ C as the cone from b on ∂(B ∪ C). n n Let σ : R → R be a homeomorphism with compact support so that σ|B1 = id and σ(B ∪ C) ⊃ B ∪ C ∪ ρ(D). Such a homeomorphism is obtained by radially stretching a −1 small collar on B1. Setting h = ρ ◦ σ ◦ ρ completes the proof, since we have h|B = id and h(B ∪ C)=ρ−1 ◦ σ ◦ ρ(B ∪ C)=ρ−1 ◦ σ(B ∪ C) ⊃ ρ−1(ρ(D)) = D. We obtain a striking characterization of euclidean n-space which is valid for any n. Theorem 2.6 (M. Brown [B2]). Let M n be a TOP n-manifold such that every com- ∼ pact subset D of M iscontainedinsomeopensubsetP of M n with P = Rn.Then ∼ M n = Rn. Proof: ∪∞ ⊂ ⊂ Write M = i=1Di with D1 D2 ... and Di compact for all i.Choosea ◦ small standard ball B ⊂ M with collar C.WriteB ∪ C = ∪Bi with Bi a ball, Bi+1 = ◦ Bi ∪ Ci,Ci a collar on Bi,B1 ⊂ B2 ⊂ .... We define a sequence of homeomorphisms n n hi : M → M so that hi|Bi = hi+1|Bi and hi(Bi) ⊃ Di. This will suffice to prove the Theorem, since limi→∞ hi|B ∪ C is a homeomorphism from B ∪ C onto M. Such a sequence of homeomorphisms is easily obtained by repeated application of ∼ n Lemma 2.5. Given hi,chooseP = R so that P ⊃ hi(B ∪ C) ∪ Di+1. Applying Lemma 2.5, we obtain a homeomorphism h : P → P with compact support so that h |hi(Bi)=id ◦ n and h (hi(Bi+1)) ⊃ Di+1. Setting hi+1 = h ◦ hi and extending to M by the identity completes the construction of hi+1. Theorem 2.7. Let M n be a compact TOP n-manifold which is the union of two open sets U and V which are homeomorphic to Rn.ThenM n is homeomorphic to Sn. Proof: Replace V by a smaller copy of Rn and choose a point p ∈ U −V¯ . We will show that M −{p} is homeomorphic to Rn, which will complete the proof, since the one-point compactification of Rn is Sn. Let D be a compact subset of M −{p} and let B1 be a standard ball in U −D centered at p.Now,U − V is a compact subset of U, so there is a homeomorphism σ : U → U with compact support so that σ(B1) ⊃ U − V .Extendσ by the identity to all of M.We 2.
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