Part III Differential Geometry Notes on Example Sheet 1
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Tight Immersions of Simplicial Surfaces in Three Space
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon Topology Vol. 35, No. 4, pp. 863-873, 19% Copyright Q 1996 Elsevier Sctence Ltd Printed in Great Britain. All tights reserved CO4&9383/96 $15.00 + 0.00 0040-9383(95)00047-X TIGHT IMMERSIONS OF SIMPLICIAL SURFACES IN THREE SPACE DAVIDE P. CERVONE (Received for publication 13 July 1995) 1. INTRODUCTION IN 1960,Kuiper [12,13] initiated the study of tight immersions of surfaces in three space, and showed that every compact surface admits a tight immersion except for the Klein bottle, the real projective plane, and possibly the real projective plane with one handle. The fate of the latter surface went unresolved for 30 years until Haab recently proved [9] that there is no smooth tight immersion of the projective plane with one handle. In light of this, a recently described simplicial tight immersion of this surface [6] came as quite a surprise, and its discovery completes Kuiper’s initial survey. Pinkall [17] broadened the question in 1985 by looking for tight immersions in each equivalence class of immersed surfaces under image homotopy. He first produced a complete description of these classes, and proceeded to generate tight examples in all but finitely many of them. In this paper, we improve Pinkall’s result by giving tight simplicial examples in all but two of the immersion classes under image homotopy for which tight immersions are possible, and in particular, we point out and resolve an error in one of his examples that would have left an infinite number of immersion classes with no tight examples. -
MA4E0 Lie Groups
MA4E0 Lie Groups December 5, 2012 Contents 0 Topological Groups 1 1 Manifolds 2 2 Lie Groups 3 3 The Lie Algebra 6 4 The Tangent space 8 5 The Lie Algebra of a Lie Group 10 6 Integrating Vector fields and the exponential map 13 7 Lie Subgroups 17 8 Continuous homomorphisms are smooth 22 9 Distributions and Frobenius Theorem 23 10 Lie Group topics 24 These notes are based on the 2012 MA4E0 Lie Groups course, taught by John Rawnsley, typeset by Matthew Egginton. No guarantee is given that they are accurate or applicable, but hopefully they will assist your study. Please report any errors, factual or typographical, to [email protected] i MA4E0 Lie Groups Lecture Notes Autumn 2012 0 Topological Groups Let G be a group and then write the group structure in terms of maps; multiplication becomes m : G × G ! G −1 defined by m(g1; g2) = g1g2 and inversion becomes i : G ! G defined by i(g) = g . If we suppose that there is a topology on G as a set given by a subset T ⊂ P (G) with the usual rules. We give G × G the product topology. Then we require that m and i are continuous maps. Then G with this topology is a topological group Examples include 1. G any group with the discrete topology 2. Rn with the Euclidean topology and m(x; y) = x + y and i(x) = −x 1 2 2 2 3. S = f(x; y) 2 R : x + y = 1g = fz 2 C : jzj = 1g with m(z1; z2) = z1z2 and i(z) =z ¯. -
Monomorphism - Wikipedia, the Free Encyclopedia
Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G. -
Induced Homology Homomorphisms for Set-Valued Maps
Pacific Journal of Mathematics INDUCED HOMOLOGY HOMOMORPHISMS FOR SET-VALUED MAPS BARRETT O’NEILL Vol. 7, No. 2 February 1957 INDUCED HOMOLOGY HOMOMORPHISMS FOR SET-VALUED MAPS BARRETT O'NEILL § 1 • If X and Y are topological spaces, a set-valued function F: X->Y assigns to each point a; of la closed nonempty subset F(x) of Y. Let H denote Cech homology theory with coefficients in a field. If X and Y are compact metric spaces, we shall define for each such function F a vector space of homomorphisms from H(X) to H{Y) which deserve to be called the induced homomorphisms of F. Using this notion we prove two fixed point theorems of the Lefschetz type. All spaces we deal with are assumed to be compact metric. Thus the group H{X) can be based on a group C(X) of protective chains [4]. Define the support of a coordinate ct of c e C(X) to be the union of the closures of the kernels of the simplexes appearing in ct. Then the intersection of the supports of the coordinates of c is defined to be the support \c\ of c. If F: X-+Y is a set-valued function, let F'1: Y~->Xbe the function such that x e F~\y) if and only if y e F(x). Then F is upper (lower) semi- continuous provided F'1 is closed {open). If both conditions hold, F is continuous. If ε^>0 is a real number, we shall also denote by ε : X->X the set-valued function such that ε(x)={x'\d(x, x')<Lε} for each xeX. -
1 Whitehead's Theorem
1 Whitehead's theorem. Statement: If f : X ! Y is a map of CW complexes inducing isomorphisms on all homotopy groups, then f is a homotopy equivalence. Moreover, if f is the inclusion of a subcomplex X in Y , then there is a deformation retract of Y onto X. For future reference, we make the following definition: Definition: f : X ! Y is a Weak Homotopy Equivalence (WHE) if it induces isomor- phisms on all homotopy groups πn. Notice that a homotopy equivalence is a weak homotopy equivalence. Using the definition of Weak Homotopic Equivalence, we paraphrase the statement of Whitehead's theorem as: If f : X ! Y is a weak homotopy equivalences on CW complexes then f is a homotopy equivalence. In order to prove Whitehead's theorem, we will first recall the homotopy extension prop- erty and state and prove the Compression lemma. Homotopy Extension Property (HEP): Given a pair (X; A) and maps F0 : X ! Y , a homotopy ft : A ! Y such that f0 = F0jA, we say that (X; A) has (HEP) if there is a homotopy Ft : X ! Y extending ft and F0. In other words, (X; A) has homotopy extension property if any map X × f0g [ A × I ! Y extends to a map X × I ! Y . 1 Question: Does the pair ([0; 1]; f g ) have the homotopic extension property? n n2N (Answer: No.) Compression Lemma: If (X; A) is a CW pair and (Y; B) is a pair with B 6= ; so that for each n for which XnA has n-cells, πn(Y; B; b0) = 0 for all b0 2 B, then any map 0 0 f :(X; A) ! (Y; B) is homotopic to f : X ! B fixing A. -
LECTURE 1. Differentiable Manifolds, Differentiable Maps
LECTURE 1. Differentiable manifolds, differentiable maps Def: topological m-manifold. Locally euclidean, Hausdorff, second-countable space. At each point we have a local chart. (U; h), where h : U ! Rm is a topological embedding (homeomorphism onto its image) and h(U) is open in Rm. Def: differentiable structure. An atlas of class Cr (r ≥ 1 or r = 1 ) for a topological m-manifold M is a collection U of local charts (U; h) satisfying: (i) The domains U of the charts in U define an open cover of M; (ii) If two domains of charts (U; h); (V; k) in U overlap (U \V 6= ;) , then the transition map: k ◦ h−1 : h(U \ V ) ! k(U \ V ) is a Cr diffeomorphism of open sets in Rm. (iii) The atlas U is maximal for property (ii). Def: differentiable map between differentiable manifolds. f : M m ! N n (continuous) is differentiable at p if for any local charts (U; h); (V; k) at p, resp. f(p) with f(U) ⊂ V the map k ◦ f ◦ h−1 = F : h(U) ! k(V ) is m differentiable at x0 = h(p) (as a map from an open subset of R to an open subset of Rn.) f : M ! N (continuous) is of class Cr if for any charts (U; h); (V; k) on M resp. N with f(U) ⊂ V , the composition k ◦ f ◦ h−1 : h(U) ! k(V ) is of class Cr (between open subsets of Rm, resp. Rn.) f : M ! N of class Cr is an immersion if, for any local charts (U; h); (V; k) as above (for M, resp. -
Immersion of Self-Intersecting Solids and Surfaces: Supplementary Material 1: Topology
Immersion of Self-Intersecting Solids and Surfaces: Supplementary Material 1: Topology Yijing Li and Jernej Barbiˇc,University of Southern California May 15, 2018 1 Topological definitions A mapping f ∶ X → Y is injective (also called \one-to-one") if for every x; y ∈ X such that x ≠ y; we have f(x) ≠ f(y): It is surjective (also called \onto") if for each y ∈ Y; there exists x ∈ X such that f(x) = y; i.e., f(X) = Y: A topological space is a set X equipped with a topology. A topology is a collection τ of subsets of X satisfying the following properties: (1) the empty set and X belong to τ; (2) any union of members of τ still belongs to τ; and (3) the intersection of any finite number of members of τ belongs to τ: A neighborhood of x ∈ X is any set that contains an open set that contains x: A homeomorphism between two topological spaces is a bijection that is continuous in both directions. A mapping between two topological spaces is continuous if the pre-image of each open set is continuous. Homeomorphic topological spaces are considered equivalent for topological purposes. An immersion between two topological spaces X and Y is a continuous mapping f that is locally injective, but not necessarily globally injective. I.e., for each x ∈ X; there exists a neighborhood N of x such that the restriction of f onto N is injective. Topological closure of a subset A of a topological set X is the set A of elements of x ∈ X with the property that every neighborhood of x contains a point of A: A topological space X is path-connected if, for each x; y ∈ X; there is a continuous mapping γ ∶ [0; 1] → X (a curve), such that γ(0) = x and γ(1) = y: A manifold of dimension d is a topological space with the property that every point x has a d d neighborhood that is homeomorphic to R (or, equivalently, to the open unit ball in R ), or a d d halfspace of R (or, equivalently, to the open unit ball in R intersected with the upper halfspace of d R ). -
Lecture 9: the Whitney Embedding Theorem
LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word \manifold" (Mannigfaltigkeit in German) first appeared in Riemann's doctoral thesis in 1851. At the early times, manifolds are defined extrinsi- cally: they are the set of all possible values of some variables with certain constraints. Translated into modern language,\smooth manifolds" are objects that are (locally) de- fined by smooth equations and, according to last lecture, are embedded submanifolds in Euclidean spaces. In 1912 Weyl gave an intrinsic definition for smooth manifolds. A natural question is: what is the difference between the extrinsic definition and the intrinsic definition? Is there any \abstract" manifold that cannot be embedded into any Euclidian space? In 1930s, Whitney and others settled this foundational problem: the two ways of defining smooth manifolds are in fact the same. In fact, Whitney's result is much more stronger than this. He showed that not only one can embed any smooth manifold into some Euclidian space, but that the dimension of the Euclidian space can be chosen to be (as low as) twice the dimension of the manifold itself! Theorem 0.1 (The Whitney embedding theorem). Any smooth manifold M of di- mension m can be embedded into R2m+1. Remark. In 1944, by using completely different techniques (now known as the \Whitney trick"), Whitney was able to prove Theorem 0.2 (The Strong Whitney Embedding Theorem). Any smooth man- ifold M of dimension m ≥ 2 can be embedded into R2m (and can be immersed into R2m−1). We will not prove this stronger version in this course, but just mention that the Whitney trick was further developed in h-cobordism theory by Smale, using which he proved the Poincare conjecture in dimension ≥ 5 in 1961! Remark. -
1. Whitney Embedding Theorem 1.0.1. Let M N Be a Smooth Manifold
1. Whitney embedding Theorem 1.0.1. Let M n be a smooth manifold. Then M n admits a smooth 2n+1 proper embedding into R . Proof. We will only give a proof when M is compact. It's enough to show 2n+1 that there exists a smooth 1−1 immersion f : M ! R . Such an f will be automatically a proper embedding due to compactness of M . Lemma 1.0.2. Let M n be a compact n-manifold. There exists a a smooth k 1 − 1 immersion f : M ! R for some large k depending on M . 0 Proof of Lemma 1.0.2. For any point p choose a coordinate ball Bp con- 0 0 0 taining p and let xp : Bp ! Vp be a local coordinate chart where Vp is n 0 an open ball in R . Let Bp ⊂ Bp be a smaller coordinate ball containing 0 p and such that xp(Bp) = Vp is a concentric ball to Vp of smaller radius. ¯ 0 Then Bp is compact and is contained in Bp . We have that [p2M = M is an open cover of M . By compactness we can choose a finite subcover by 0 0 coordinate balls B1;:::;Bm such that xi : Bi ! Vi ; i = 1; : : : m are local coordinate charts. Let 'i : M ! R be a smooth function such that 'i = 1 ¯ 0 0 ¯ on Bi , 'i = 0 on MnBi and 0 < 'i < 1 on BinBi . m(n+1) Let f : M ! R be given by f(p) = ('1(p)x1(p);:::;'m(p)xm(p);'1(p);:::;'m(p)) Claim 1: f is 1 − 1. -
Sphere Immersions Into 3–Space
Algebraic & Geometric Topology 6 (2006) 493–512 493 Regular homotopy and total curvature II: sphere immersions into 3–space TOBIAS EKHOLM We consider properties of the total curvature functional on the space of 2–sphere immersions into 3–space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8 and we establish a non-injectivity result for local minima. 53C42; 53A04, 57R42 1 Introduction An immersion of manifolds is a map with everywhere injective differential. Two immersions are regularly homotopic if there exists a continuous 1–parameter family of immersions connecting one to the other. The Smale–Hirsch h–principle[10;4] says that the space of immersions M N , dim.M / < dim.N / is homotopy equivalent to ! the space of injective bundle maps TM TN . In contrast to differential topological ! properties, differential geometric properties of immersions do not in general satisfy h–principles, see[3, (A) on page 62]. In this paper and the predecessor[2], we study some aspects of the differential geometry of immersions and regular homotopies in the most basic cases of codimension one immersions. We investigate whether or not it is possible to perform topological constructions while keeping control of certain geometric quantities. Consider immersions S 2 R3 . In this case the h–principle implies a famous theorem ! of Smale[9]: all immersions S 2 R3 are regularly homotopic. In particular, there ! exists sphere eversions (ie, regular homotopies connecting the unit 2–sphere in R3 to the same immersion with the opposite (co)orientation). -
Lecture 5: Submersions, Immersions and Embeddings
LECTURE 5: SUBMERSIONS, IMMERSIONS AND EMBEDDINGS 1. Properties of the Differentials Recall that the tangent space of a smooth manifold M at p is the space of all 1 derivatives at p, i.e. all linear maps Xp : C (M) ! R so that the Leibnitz rule holds: Xp(fg) = g(p)Xp(f) + f(p)Xp(g): The differential (also known as the tangent map) of a smooth map f : M ! N at p 2 M is defined to be the linear map dfp : TpM ! Tf(p)N such that dfp(Xp)(g) = Xp(g ◦ f) 1 for all Xp 2 TpM and g 2 C (N). Remark. Two interesting special cases: • If γ :(−"; ") ! M is a curve such that γ(0) = p, then dγ0 maps the unit d d tangent vector dt at 0 2 R to the tangent vectorγ _ (0) = dγ0( dt ) of γ at p 2 M. • If f : M ! R is a smooth function, we can identify Tf(p)R with R by identifying d a dt with a (which is merely the \derivative $ vector" correspondence). Then for any Xp 2 TpM, dfp(Xp) 2 R. Note that the map dfp : TpM ! R is linear. ∗ In other words, dfp 2 Tp M, the dual space of TpM. We will call dfp a cotangent vector or a 1-form at p. Note that by taking g = Id 2 C1(R), we get Xp(f) = dfp(Xp): For the differential, we still have the chain rule for differentials: Theorem 1.1 (Chain rule). Suppose f : M ! N and g : N ! P are smooth maps, then d(g ◦ f)p = dgf(p) ◦ dfp. -
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.