DOCSLIB.ORG

A Radius Sphere Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Invent. math. 112, 577-583 (1993) Inventiones mathematicae Springer-Verlag1993 A radius sphere theorem Karsten Grove 1'* and Peter Petersen 2'** 1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA 2 Department of Mathematics, University of California, Los Angeles, CA 90024-1555, USA Oblatum 9-X-1992 Introduction The purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter Sphere Theorem in Riemannian geometry. There has lately been considerable interest in studying spaces which are more singular than Riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff limits of Riemannian manifolds are almost never Riemannian manifolds, but usually only inner metric spaces with various nice properties. The kind of spaces we wish to study here are the so-called Alexandrov spaces. Alexan- drov spaces are finite Hausdorff dimensional inner metric spaces with a lower curvature bound in the distance comparison sense. The structure of Alexandrov spaces was studied in [BGP], [P1] and [P]. We point out that the curvature assumption implies that the Hausdorff dimension is equal to the topological dimension. Moreover, if X is an Alexandrov space and peX then the space of directions Zp at p is an Alexandrov space of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is homeomorphic to the linear cone over Zv- One of the important implications of this is that the local structure of n-dimensional Alexandrov spaces is determined by the structure of (n - 1)-dimen- sional Alexandrov spaces with curvature > 1. Sphere theorems in this context seem to be particularly interesting. For if one can give geometric characterizations of spheres, then one can also characterize manifold points in Alexandrov spaces (see however, Example 2.1). In [P] there is a generalization of the Diameter Sphere Theorem (see [GS]): If X is an Alexandrov space with curvature > 1 and diameter > g/2 then X is * Supported in part by the NSF ** Supported in part by the NSF, the Alfred P. Sloan Foundation and an NSF Young Investigator Award 578 K. Grove and P. Petersen a suspension. Closed hemispheres or suspensions over projective spaces, however, are examples of suspensions that have curvature > 1 and diameter n, but which are not spheres. So to get a sphere theorem for Alexandrov spaces we therefore need a stronger invariant than the diameter. The radius of a metric space X is defined as: rad X = minp~x maxq~x d(p, q), where d(, ) denotes the distance function on X (see [GP1], [GM] and [SY] for other results using the radius concept). If therefore radX > r then for every point p~X there is qeX such that d(p, q) > r. With this behind us we can now state our main result as. Main theorem. Let X be an n-dimensional Alexandrov space with curvature > 1 and radius > n/2 then X is homeomorphic to the n-sphere S". This theorem is optimal in the sense that the radius condition cannot be relaxed to a condition on diameter or to the condition that radius > n/2. To see this just note that the above mentioned examples have radius = n/2 (see also Sect. 2 for more examples). In [GW] two different optimal sphere theorems for n-dimensional Alexandrov spaces with curvature > 1 are proved. In the first one it is assumed that Pack,+2(X) > n/4. This condition is easily seen to imply our radius condition, so in this case our result is more general. In the second result it is assumed that X has no boundary and that Pack,(X)> n/4. These conditions are, however, neither stronger nor weaker than our assumption. The Main Theorem implies another result about Alexandrov spaces with large 1-systole. The 1-systole sysl (X) for a compact connected Alexandrov space X is the length of the shortest closed non-contractible curve. The existence of such a curve follows from the fact that X is a compact ANR. The 1-systole of real projective space for instance is n, while other space forms have smaller 1-systole. Corollary. Let X be an Alexandrov space with curvature > 1 and 1-systole > n/2. Then the universal cover of X is homeomorphic to a sphere and the fundamental group of X is cyclic of order 2 or 3. Note that if a cyclic group of order k acts freely by isometries on the standard sphere then the 1-systole of the quotient space is 2n/k. The corollary is therefore optimal, although it still leaves open the question of what exactly X can be (see [W] for related results). The rest of the paper is divided into two sections. In the first section we prove the Main Theorem and the corollary. In the second section we give some general procedures for constructing Alexandrov spaces with curvature ~ 1 and big radius. The abundance of examples we construct will show that it is hard to get a grip on those spaces which have diameter or radius equal to hi2. This is somewhat in contrast with the manifold situation, where all Riemannian manifolds with sec- tional curvature > 1 and diameter = n/2 have been classified (see [GG1]). 1 The antipodal map For the rest of this section we will assume that X is an n-dimensional Alexandrov space with curvature > 1 and radius > ~/2. If we fix a point peX then our radius condition implies that the complement to the open ~/2-ball around p, C(p)= X - B(p, ~/2) is a closed set with non empty interior. Toponogov's Distance (or Angle) Comparison Theorem (see IT, P1, BGP]) then shows that C(p) is convex A radius sphere theorem 579 and that the distance function d(p,. ) is concave on C(p) and strictly concave on the interior of C(p). Thus there is a unique point A(p) at maximal distance from p. Note that the map p ~ A(p) is clearly continuous. We call this map the antipodal map for X, since on the standard sphere A is the antipodal map. It follows from [P] that X is the suspension over 2~a~p) for all p. All we need to do, to show that X is a sphere, is to make sure that A(p) is a manifold point (see also Example 2.1). Because then X itself becomes a manifold, since it is a suspension and spheres are the only manifolds which are also suspensions. We prove that A(p) is a manifold point for some p by contradiction. Denote by S c X the set of non manifold points in X, thus we assume that A : X ---, S. It follows from the topological stratification of X described in [P] that S is a closed subset of dimension < n - 1. Furthermore the dimension can only be n - 1 if X has boundary. Now any Alexandrov space with curvature > 1 and non empty boundary must have radius < n/2 (see e.g. [P-l). Thus our radius condition implies that dim S < n - 2. We will show in Lemma 1 and 2 below that the Alexander-Spanier cohomology group H"(X, 7/2) = 7/2 and that A 2= A.A is homotopic to the identity map id:X--,X. Thus A 2 induces a map on/4"(X, 7/z), which on one hand is the identity map, and on the other hand is zero since A 2 factors through S which has dimension < n. Hence we get a contradiction. Lemma 1 Let X be a compact Alexandrov space without boundary. Then X has a fundamental class in Alexander-Spanier cohomology with 7Z2 coefficients i.e. l~"(X, 7/2) = 7/2. Proof. We use Alexander-Spanier cohomology as it is described in IS]. Denote again by S the set of non manifold points. Since X doesn't have any boundary it follows from the topological stratification results in [P] that dim S < n - 2. Then X - S is a connected n-dimensional manifold. Therefore the top cohomology class with compact support satisfies/~2(X - S, 7/2) = 7/2. Now the advantage of using Alexander-Spanier cohomology is that/~"(X, S, 7/2) =/42(X - S, 712) as long as X and S are compact. Using the long exact sequence for the pair (X, S) now yields: 0 =/~"- I(S, 7/2) -~ ~q"(x, S, 7/2) --'/~"(X, Z2) --, a"(S, 7/2) = 0, because S has dimension < n - 2. Thus we get the desired conclusion. [] Lemma 2 Let X be as in the Main Theorem and A the antipodal map described above. Then A 2 is homotopic to the identity map on X. Proof All of the results on ANR's and decompositions we are going to need for the proof can be found in [D]. The sets C(p)= X- B(p, g/2) are compact and vary continuously in the Hausdorff metric. Furthermore it is proved in [P] that they are Alexandrov spaces which are contractible. Now any Alexandrov space is an ANR since it is finite dimensional and locally contractible. Thus the sets C(p) are cell-like. Consider now the decomposition G on X x X consisting of the sets {(x, y)}, where y(~C(A(x)), and {(x, C(A (x)))}. This is an upper semicontinuous cell-like decomposition, since the sets C(A(x)) are cell-like and vary continuously with x. Let p: X xX~ X x X/G be the natural projection and P2: X x X ~ X the projection onto the second factor.
Recommended publications
  • The Suspension X ∧S and the Fake Suspension ΣX of a Spectrum X

    The Suspension X ∧S and the Fake Suspension ΣX of a Spectrum X

    Contents 41 Suspensions and shift 1 42 The telescope construction 10 43 Fibrations and cofibrations 15 44 Cofibrant generation 22 41 Suspensions and shift The suspension X ^S1 and the fake suspension SX of a spectrum X were defined in Section 35 — the constructions differ by a non-trivial twist of bond- ing maps. The loop spectrum for X is the function complex object 1 hom∗(S ;X): There is a natural bijection 1 ∼ 1 hom(X ^ S ;Y) = hom(X;hom∗(S ;Y)) so that the suspension and loop functors are ad- joint. The fake loop spectrum WY for a spectrum Y con- sists of the pointed spaces WY n, n ≥ 0, with adjoint bonding maps n 2 n+1 Ws∗ : WY ! W Y : 1 There is a natural bijection hom(SX;Y) =∼ hom(X;WY); so the fake suspension functor is left adjoint to fake loops. n n+1 The adjoint bonding maps s∗ : Y ! WY define a natural map g : Y ! WY[1]: for spectra Y. The map w : Y ! W¥Y of the last section is the filtered colimit of the maps g Wg[1] W2g[2] Y −! WY[1] −−−! W2Y[2] −−−! ::: Recall the statement of the Freudenthal suspen- sion theorem (Theorem 34.2): Theorem 41.1. Suppose that a pointed space X is n-connected, where n ≥ 0. Then the homotopy fibre F of the canonical map h : X ! W(X ^ S1) is 2n-connected. 2 In particular, the suspension homomorphism 1 ∼ 1 piX ! pi(W(X ^ S )) = pi+1(X ^ S ) is an isomorphism for i ≤ 2n and is an epimor- phism for i = 2n+1, provided that X is connected.
  • Zuoqin Wang Time: March 25, 2021 the QUOTIENT TOPOLOGY 1. The

    Zuoqin Wang Time: March 25, 2021 the QUOTIENT TOPOLOGY 1. The

    Topology (H) Lecture 6 Lecturer: Zuoqin Wang Time: March 25, 2021 THE QUOTIENT TOPOLOGY 1. The quotient topology { The quotient topology. Last time we introduced several abstract methods to construct topologies on ab- stract spaces (which is widely used in point-set topology and analysis). Today we will introduce another way to construct topological spaces: the quotient topology. In fact the quotient topology is not a brand new method to construct topology. It is merely a simple special case of the co-induced topology that we introduced last time. However, since it is very concrete and \visible", it is widely used in geometry and algebraic topology. Here is the definition: Definition 1.1 (The quotient topology). (1) Let (X; TX ) be a topological space, Y be a set, and p : X ! Y be a surjective map. The co-induced topology on Y induced by the map p is called the quotient topology on Y . In other words, −1 a set V ⊂ Y is open if and only if p (V ) is open in (X; TX ). (2) A continuous surjective map p :(X; TX ) ! (Y; TY ) is called a quotient map, and Y is called the quotient space of X if TY coincides with the quotient topology on Y induced by p. (3) Given a quotient map p, we call p−1(y) the fiber of p over the point y 2 Y . Note: by definition, the composition of two quotient maps is again a quotient map. Here is a typical way to construct quotient maps/quotient topology: Start with a topological space (X; TX ), and define an equivalent relation ∼ on X.
  • Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension

    Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension

    LECTURE 2: SPACES OF MAPS, LOOP SPACES AND REDUCED SUSPENSION In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there will be a short digression on spaces of maps between (pointed) spaces and the relevant topologies. To be a bit more specific, one aim is to see that given a pointed space (X; x0), then there is an entire pointed space of loops in X. In order to obtain such a loop space Ω(X; x0) 2 Top∗; we have to specify an underlying set, choose a base point, and construct a topology on it. The underlying set of Ω(X; x0) is just given by the set of maps 1 Top∗((S ; ∗); (X; x0)): A base point is also easily found by considering the constant loop κx0 at x0 defined by: 1 κx0 :(S ; ∗) ! (X; x0): t 7! x0 The topology which we will consider on this set is a special case of the so-called compact-open topology. We begin by introducing this topology in a more general context. 1. Function spaces Let K be a compact Hausdorff space, and let X be an arbitrary space. The set Top(K; X) of continuous maps K ! X carries a natural topology, called the compact-open topology. It has a subbasis formed by the sets of the form B(T;U) = ff : K ! X j f(T ) ⊆ Ug where T ⊆ K is compact and U ⊆ X is open. Thus, for a map f : K ! X, one can form a typical basis open neighborhood by choosing compact subsets T1;:::;Tn ⊆ K and small open sets Ui ⊆ X with f(Ti) ⊆ Ui to get a neighborhood Of of f, Of = B(T1;U1) \ ::: \ B(Tn;Un): One can even choose the Ti to cover K, so as to `control' the behavior of functions g 2 Of on all of K.
  • The Double Suspension of the Mazur Homology 3-Sphere

    The Double Suspension of the Mazur Homology 3-Sphere

    THE DOUBLE SUSPENSION OF THE MAZUR HOMOLOGY SPHERE FADI MEZHER 1 Introduction The main objects of this text are homology spheres, which are defined below. Definition 1.1. A manifold M of dimension n is called a homology n-sphere if it has the same homology groups as Sn; that is, ( Z if k ∈ {0, n} Hk(M) = 0 otherwise A result of J.W. Cannon in [Can79] establishes the following theorem Theorem 1.2 (Double Suspension Theorem). The double suspension of any homology n-sphere is homeomorphic to Sn+2. This, however, is beyond the scope of this text. We will, instead, construct a homology sphere, the Mazur homology 3-sphere, and show the double suspension theorem for this particular manifold. However, before beginning with the proper content, let us study the following famous example of a nontrivial homology sphere. Example 1.3. Let I be the group of (orientation preserving) symmetries of the icosahedron, which we recall is a regular polyhedron with twenty faces, twelve vertices, and thirty edges. This group, called the icosahedral group, is finite, with sixty elements, and is naturally a subgroup of SO(3). It is a well-known fact that we have a 2-fold covering ξ : SU(2) → SO(3), where ∼ 3 ∼ 3 SU(2) = S , and SO(3) = RP . We then consider the following pullback diagram S0 S0 Ie := ι∗(SU(2)) SU(2) ι∗ξ ξ I SO(3) Then, Ie is also a group, where the multiplication is given by the lift of the map µ ◦ (ι∗ξ × ι∗ξ), where µ is the multiplication in I.
  • L22 Adams Spectral Sequence

    L22 Adams Spectral Sequence

    Lecture 22: Adams Spectral Sequence for HZ=p 4/10/15-4/17/15 1 A brief recall on Ext We review the definition of Ext from homological algebra. Ext is the derived functor of Hom. Let R be an (ordinary) ring. There is a model structure on the category of chain complexes of modules over R where the weak equivalences are maps of chain complexes which are isomorphisms on homology. Given a short exact sequence 0 ! M1 ! M2 ! M3 ! 0 of R-modules, applying Hom(P; −) produces a left exact sequence 0 ! Hom(P; M1) ! Hom(P; M2) ! Hom(P; M3): A module P is projective if this sequence is always exact. Equivalently, P is projective if and only if if it is a retract of a free module. Let M∗ be a chain complex of R-modules. A projective resolution is a chain complex P∗ of projec- tive modules and a map P∗ ! M∗ which is an isomorphism on homology. This is a cofibrant replacement, i.e. a factorization of 0 ! M∗ as a cofibration fol- lowed by a trivial fibration is 0 ! P∗ ! M∗. There is a functor Hom that takes two chain complexes M and N and produces a chain complex Hom(M∗;N∗) de- Q fined as follows. The degree n module, are the ∗ Hom(M∗;N∗+n). To give the differential, we should fix conventions about degrees. Say that our differentials have degree −1. Then there are two maps Y Y Hom(M∗;N∗+n) ! Hom(M∗;N∗+n−1): ∗ ∗ Q Q The first takes fn to dN fn.
  • Arxiv:1812.07604V1 [Math.AT] 18 Dec 2018 Ons Hsvledosa H Ieo H Oe of Model the of Size the As Drops Value This Points

    Arxiv:1812.07604V1 [Math.AT] 18 Dec 2018 Ons Hsvledosa H Ieo H Oe of Model the of Size the As Drops Value This Points

    THE TOPOLOGICAL COMPLEXITY OF FINITE MODELS OF SPHERES SHELLEY KANDOLA Abstract. In [2], Farber defined topological complexity (TC) to be the mini- mal number of continuous motion planning rules required to navigate between any two points in a topological space. Papers by [4] and [3] define notions of topological complexity for simplicial complexes. In [9], Tanaka defines a notion of topological complexity, called combinatorial complexity, for finite topologi- cal spaces. As is common with papers discussing topological complexity, each includes a computation of the TC of some sort of circle. In this paper, we compare the TC of models of S1 across each definition, exhibiting some of the nuances of TC that become apparent in the finite setting. In particular, we show that the TC of finite models of S1 can be 3 or 4 and that the TC of the minimal finite model of any n-sphere is equal to 4. Furthermore, we exhibit spaces weakly homotopy equivalent to a wedge of circles with arbitrarily high TC. 1. Introduction Farber introduced the notion of topological complexity in [2] as it relates to motion planning in robotics. Informally, the topological complexity of a robot’s space of configurations represents the minimal number of continuous motion plan- ning rules required to instruct that robot to move from one position into another position. Although topological complexity was originally defined for robots with a smooth, infinite range of motion (e.g. products of spheres or real projective space), it makes sense to consider the topological complexity of finite topological spaces. For example, one could determine the topological complexity of a finite state machine or a robot powered by stepper motors.
  • A Suspension Theorem for Continuous Trace C*-Algebras

    A Suspension Theorem for Continuous Trace C*-Algebras

    proceedings of the american mathematical society Volume 120, Number 3, March 1994 A SUSPENSION THEOREM FOR CONTINUOUS TRACE C*-ALGEBRAS MARIUS DADARLAT (Communicated by Jonathan M. Rosenberg) Abstract. Let 3 be a stable continuous trace C*-algebra with spectrum Y . We prove that the natural suspension map S,: [Cq{X), 5§\ -* [Cq{X) ® Co(R), 3 ® Co(R)] is a bijection, provided that both X and Y are locally compact connected spaces whose one-point compactifications have the homo- topy type of a finite CW-complex and X is noncompact. This is used to compute the second homotopy group of 31 in terms of AMheory. That is, [C0(R2), 3] = K0{30) i where 3fa is a maximal ideal of 3 if Y is com- pact, and 3o = 3! if Y is noncompact. 1. Introduction For C*-algebras A, B let Hom(^4, 5) denote the space of (not necessarily unital) *-homomorphisms from A to 5 endowed with the topology of point- wise norm convergence. By definition <p, w e Hom(^, 5) are called homotopic if they lie in the same pathwise connected component of Horn (A, 5). The set of homotopy classes of the *-homomorphisms in Hom(^, 5) is denoted by [A, B]. For a locally compact space X let Cq(X) denote the C*-algebra of complex-valued continuous functions on X vanishing at infinity. The suspen- sion functor S for C*-algebras is defined as follows: it takes a C*-algebra A to A ® C0(R) and q>e Hom(^, 5) to <P0 idem € Hom(,4 ® C0(R), 5 ® C0(R)).
  • Regular Cell Complexes in Total Positivity

    Regular Cell Complexes in Total Positivity

    REGULAR CELL COMPLEXES IN TOTAL POSITIVITY PATRICIA HERSH Abstract. Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over R is a reg- ular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) in- tervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology. 1. Introduction In this paper, the following conjecture of Sergey Fomin and Michael Shapiro from [12] is proven. Conjecture 1.1. Let Y be the link of the identity in the totally non- negative real part of the unipotent radical of a Borel subgroup B in a semisimple, simply connected algebraic group defined and split over R. − − Let Bu = B uB for u in the Weyl group W . Then the stratification of Y into Bruhat cells Y \ Bu is a regular CW decomposition. More- over, for each w 2 W , Yw = [u≤wY \ Bu is a regular CW complex homeomorphic to a ball, as is the link of each of its cells.
  • Proving N-Dimensional Linking in Complete N-Complexes in (2N+1)-Dimensional Space

    Proving N-Dimensional Linking in Complete N-Complexes in (2N+1)-Dimensional Space

    Rose-Hulman Undergraduate Mathematics Journal Volume 12 Issue 1 Article 2 Proving n-Dimensional Linking in Complete n-Complexes in (2n+1)-Dimensional Space Megan Gregory Massey University, Palmerston North, New Zealand, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Gregory, Megan (2011) "Proving n-Dimensional Linking in Complete n-Complexes in (2n+1)-Dimensional Space," Rose-Hulman Undergraduate Mathematics Journal: Vol. 12 : Iss. 1 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol12/iss1/2 Rose- Hulman Undergraduate Mathematics Journal Proving n-Dimensional Linking in Complete n-Complexes in (2n + 1)-Dimensional Space Megan Gregorya Volume 12, no. 1, Spring 2011 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] http://www.rose-hulman.edu/mathjournal aMassey University, Palmerston North, New Zealand, [email protected] Rose-Hulman Undergraduate Mathematics Journal Volume 12, no. 1, Spring 2011 Proving n-Dimensional Linking in Complete n-Complexes in (2n + 1)-Dimensional Space Megan Gregory Abstract. This paper proves that there is an intrinsic link in complete n-complexes on (2n + 4)-vertices for n = 1, 2, 3 using the method of Conway and Gordon from their 1983 paper. The argument uses the sum of the linking number mod 2 of each pair of disjoint n-spheres contained in the n-complex as an invariant. We show that crossing changes do not affect the value of this invariant. We assert that ambient isotopies and crossing changes suffice to change any specific embedding to any other specific embedding.
  • HOMOTOPY THEORY for BEGINNERS Contents 1. Notation

    HOMOTOPY THEORY for BEGINNERS Contents 1. Notation

    HOMOTOPY THEORY FOR BEGINNERS JESPER M. MØLLER Abstract. This note contains comments to Chapter 0 in Allan Hatcher's book [5]. Contents 1. Notation and some standard spaces and constructions1 1.1. Standard topological spaces1 1.2. The quotient topology 2 1.3. The category of topological spaces and continuous maps3 2. Homotopy 4 2.1. Relative homotopy 5 2.2. Retracts and deformation retracts5 3. Constructions on topological spaces6 4. CW-complexes 9 4.1. Topological properties of CW-complexes 11 4.2. Subcomplexes 12 4.3. Products of CW-complexes 12 5. The Homotopy Extension Property 14 5.1. What is the HEP good for? 14 5.2. Are there any pairs of spaces that have the HEP? 16 References 21 1. Notation and some standard spaces and constructions In this section we fix some notation and recollect some standard facts from general topology. 1.1. Standard topological spaces. We will often refer to these standard spaces: • R is the real line and Rn = R × · · · × R is the n-dimensional real vector space • C is the field of complex numbers and Cn = C × · · · × C is the n-dimensional complex vector space • H is the (skew-)field of quaternions and Hn = H × · · · × H is the n-dimensional quaternion vector space • Sn = fx 2 Rn+1 j jxj = 1g is the unit n-sphere in Rn+1 • Dn = fx 2 Rn j jxj ≤ 1g is the unit n-disc in Rn • I = [0; 1] ⊂ R is the unit interval • RP n, CP n, HP n is the topological space of 1-dimensional linear subspaces of Rn+1, Cn+1, Hn+1.
  • SEPARATION of the «-SPHERE by an (N - 1)-SPHERE by JAMES C

    SEPARATION of the «-SPHERE by an (N - 1)-SPHERE by JAMES C

    SEPARATION OF THE «-SPHERE BY AN (n - 1)-SPHERE BY JAMES C. CANTRELL(i) 1. Introduction. Let A be the closed spherical ball in E" centered at the origin 0, and with radius one, B the closed ball centered at O with radius one-half, and C the closed ball centered at O with radius two. The Generalized Schoenflies Theorem states that, if h is a homeomorphism of Cl (C —B) into S", then h(BdA) is tame in S" (the closure of either component of S" —h(BdA) is a closed n-cell) [5]. One is naturally led to the following question : if h is a homeomorphism of C\(A—B) into S", is the closure of the component of S" — h(BdA) which contains n(BdB) a closed n-cell? This question is answered affirmatively by Theorem 1 and should be listed as a corollary to the Generalized Schoenflies Theorem. Let D be the closed ball in E", centered at (0,0,-",0, — 1) with radius two. Two other types of embeddings of Bd.4 in S", n > 3, are considered in §2, (1) the embedding homeomorphism h can be extended to a homeomorphism of Cl(£>— B) into S" such that the extension is semi-linear on each finite polyhedron in the open annulus Int (A — B), and (2) h can be extended to a homeomorphism of C1(B —A) into S" such that the extension is semi-linear in a deleted neigh- borhood of (0, 0, ■■■,(), 1) (see Definition 1). Theorem 4 strongly suggests that, for an embedding of type (1), h(BdA) is tame in S".
  • BOTT PERIODICITY Bott's 1957 Announcement of His Periodicity

    BOTT PERIODICITY Bott's 1957 Announcement of His Periodicity

    BOTT PERIODICITY PETER MAY Bott's 1957 announcement of his periodicity theorem transformed algebraic topol- ogy forever. To quote Atiyah in his obituary for Bott \This paper was a bombshell. The results were beautiful, far-reaching and totally unexpected." In briefest form, the theorem says 2 8 BU × Z ' Ω (BU × Z) and BO × Z ' Ω (BO × Z) When Bott started work on this, π10(U(n)) was claimed to be Z=3Z in one paper and to be cyclic of order a power of two in another, as yet unpublished. He showed the unpublished claim was correct. I'll begin with a brief sketch of his original proof, restricting to the complex case, next place it in his original context of differential geometry, and then give a sketch proof by the methods of algebraic topology, still restricting to the complex case. We have S2n−1 =∼ U(n)=U(n − 1) since U(n) acts transitively with isotropy group of a point U(n − 1). By the long exact sequence of the bundle U(n − 1) −! U(n) −! S2n−1 q and the fact that πk(S ) = 0 for k < q, πk(U(n)) = πk(U) for k=2 < n. Let 2n 2n 2n Grn(C ) be the Grassmannian manifold of complex n-planes in C and let Vn(C ) be the Stiefel manifold of n-frames in C2n. We have a principal U(n)-bundle 2n 2n p: Vn(C ) −! Grn(C ); which can also be seen via 2n ∼ 2n ∼ Vn(C ) = U(2n)=U(n) and Grn(C ) = U(2n)=U(n) × U(n): By the long exact sequence of homotopy groups, if 1 ≤ k << n, ∼ 2n πk−1(U(n)) = πk(Grn(C )): 2n In a sense I'll sketch, Bott showed that Grn(C ) is the \manifold of minimal geodesics" in ΩSU(2n).