DOCSLIB.ORG

Solutions to Math 147 Midterm Exam — April 30Th, 2014

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solutions to Math 147 Midterm Exam — April 30Th, 2014 Solutions to Math 147 Midterm Exam | April 30th, 2014 1. (30 points) Suppose that f : M ! N is a smooth map between smooth manifolds of equal dimension with the property that dfx is nonsingular for every x 2 M. (a) Show that f is an open mapping, i.e., any open set U ⊂ M is mapped to an open set f(U) ⊂ N. Let y 2 N be an arbitrary point in the image of f with f(x) = y. Since M and N have the same dimension and dfx is nonsingular,dfx is an isomorphism. Choosing parametrizations about x and y (and possibly shrinking their domain), by the Inverse Function Theorem there is a neighborhood U of x on which f is a diffeomorphism onto its image f(U). Since y was arbitrary, for any U 0 ⊂ M, f(U 0) is open in N by the definition of an open subset. (b) Is f a diffeomorphism? Give a proof or counterexample. In general, f can be fail to be injective or surjective, so need not be a diffeomorphism. For the latter case, consider the inclusion of any submanifold into a manifold with the same dimension is a counterexample, e.g., take the inclusion of a single point into the disjoint union of 2 points, or the inclusion of an interval (a; b) into the reals, R; the derivative is nonsingular everywhere, 1 2 ∼ but the map is not surjective. For the former case, consider the map R ! S ⊂ R = C given by x 7! eix; the derivative of this map is nonsingular everywhere, but the map is not injective since eix = ei(x+2π). 2. (30 points) Consider the function 1 det: Mn(R) ! R that sends an n × n matrix to its determinant. n2 (a) Use this function to show that the special linear group, denoted SLn(R) ⊂ Mn(R) = R , consisting of n × n matrices of determinant 1 is a smooth manifold. It might help to use that Det(tA) = tnDet(A). n2 Let A 2 Mn(R). First we explain the source and target of the derivative. Since Mn(R) = R , n2 n2 and T Rx = R , we get that T (Mn(R))A = Mn(R). So we get d(det)A : Mn(R) ! R: Now we compute the derivative at A in the A direction, det(A + tA) − det(A) (1 + t)ndet(A) − det(A) d(det)A(A) = lim = lim t!0 t t!0 t and we may factor out the det(A) to get (1 + t)n − 1 d(det)A(A) = det(A) lim = det(A) · n: t!0 t Since this is nonzero for A nonsingular, d(det)A is surjective for A nonsingular. In particular, 1 −1 2 is a regular value and so det (1) = SLn(R) is a manifold of dimension n − 1. (b) Show that the tangent space of SLn(R) at the identity matrix consists of all matrices with zero trace. It might help to use the first few terms of the Taylor expansion, Det(I +tA) = 1+tTr(A)+ O(t2). The tangent space to SLn(R) at I consists of elements in the kernel of ddetI . To compute this kernel, we use that det(I + tA) = 1 + tTr(A) + O(t2); so that det(I + tA) − det(I) 1 + tTr(A) + O(t2) − 1 ddetI (A) = lim = lim = Tr(A): t!0 t t!0 t Math 147, Spring 2014 Solutions to Midterm Exam | April 30th, 2014 Page 2 of 2 So the kernel of ddetI , and hence the tangent space T SLn(R)I , consists of matrices with zero trace. 3. (30 points) Let X and Y be manifolds with boundary and f : X ! Y a diffeomorphism. Prove that the restriction of f to the boundary of X is again a diffeomorphism, @f : @X ! @Y from the boundary of X to the boundary of Y . Note that the restriction of f to any submanifold Z ⊂ X is a diffeomorphism to f(Z). In the homework, we saw that @X is a submanifold of dimension one less, so if we restrict f to @X, we obtain a diffeomorphism fj@X : @X ! f(@X). Hence the only thing that is left to show is f(@X) = @Y . Let x 2 @X, we want to show f(x) 2 @Y . In order to show f(x) 2 @Y , we have to find a chart around f(x), say ( (V ); f(x); ) where is a diffeomorphism : V ! (V ), V is an open subset −1 of H and (f(x)) 2 @H. Since x 2 @X, we have a chart around x, say (φ(U); x; φ) where φ is a −1 diffeomorphism φ : U ! φ(U), U is an open subset of H and φ (x) 2 @H. Now we can choose V to −1 −1 be U and to be f ◦ φ. Obviously (f ◦ φ) (f(x)) = φ (x) 2 @H. Thus f(@X) ⊂ @Y ; if we repeat the same argument for f −1, we deduce f −1(@Y ) ⊂ @X, hence f(@X) = @Y . 4. (30 points) Use Sard's Theorem to show that for n ≥ 2, any map S1 ! Sn is homotopic to a constant 1 n n map. [HINT: First show that S ! S is not surjective, and then use stereographic projection to R to produce the required homotopy.] 1 n 1 Following the hint, we first show that any map f : S ! S fails to be surjective. Since dim(TSx) < n 1 n dim(TSy ) for any x 2 S and y 2 S , all values of f are critical. By Sard's Theorem critical values have measure zero (so in particular, cannot have full measure) f cannot be surjective. Now, let z 2 Sn be a point not in the image of f. By choosing an isotopy of Sn through rotations, without loss of generality we may assume that z is the north pole. Since the image of f lies in the 1 n domain of stereographic projection, we obtain a map f~: S ! R , where f~ is f postcomposed with stereographic projection. Now take the homotopy F~(t; x) = tf~(x) for t 2 [0; 1]. At t = 0, we have the map f~ and at t = 0, the map is constant. We obtain a homotopy of the original map by defining F (t; x) := h ◦ F~ ◦ h−1(t; x) where h is stereographic projection. By construction, F (1; x) = f(x) and F (0; x) is the constant map. 5. (30 points) A connected manifold M is contractible if the identity map on M is smoothly homotopy to a constant map. n (a) Show that R is contractible for every n. n n n Consider the map F : [0; 1] × R ! R given by F (t; v) = tv for t 2 [0; 1] and v 2 R . For t = 1, we have F (1; v) = v is the identity map, and at t = 0, we have F (0; v) = 0 is the constant map n to zero. Hence, R is contractible. (b) Show that no compact connected manifold without boundary is contractible, other than the one point space. The mod 2 degree of the constant map is 0 (as calculated by taking the inverse image of a point that is not in the image, and using that we don't have a 1 point space). On the other hand, the mod 2 degree of the identity map is 1. Since these degrees are different, the mod 2 degree theorem concludes that the corresponding maps cannot be smoothly homotopic. Hence, the given manifold is not contractible..
Load more

Recommended publications
  • Low-Dimensional Representations of Matrix Groups and Group Actions on CAT (0) Spaces and Manifolds
    Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifolds Shengkui Ye National University of Singapore January 8, 2018 Abstract We study low-dimensional representations of matrix groups over gen- eral rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds. 1 Introduction Low-dimensional representations are studied by many authors, such as Gural- nick and Tiep [24] (for matrix groups over fields), Potapchik and Rapinchuk [30] (for automorphism group of free group), Dokovi´cand Platonov [18] (for Aut(F2)), Weinberger [35] (for SLn(Z)) and so on. In this article, we study low-dimensional representations of matrix groups over general rings. Let R be an associative ring with identity and En(R) (n ≥ 3) the group generated by ele- mentary matrices (cf. Section 3.1). As motivation, we can consider the following problem. Problem 1. For n ≥ 3, is there any nontrivial group homomorphism En(R) → En−1(R)? arXiv:1207.6747v1 [math.GT] 29 Jul 2012 Although this is a purely algebraic problem, in general it seems hard to give an answer in an algebraic way. In this article, we try to answer Prob- lem 1 negatively from the point of view of geometric group theory. The idea is to find a good geometric object on which En−1(R) acts naturally and non- trivially while En(R) can only act in a special way. We study matrix group actions on CAT(0) spaces, spheres and acyclic manifolds. We prove that for low-dimensional CAT(0) spaces, a matrix group action always has a global fixed point (cf.
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
    LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1.
    [Show full text]
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
    [Show full text]
  • GEOMETRY and GROUPS These Notes Are to Remind You of The
    GEOMETRY AND GROUPS These notes are to remind you of the results from earlier courses that we will need at some point in this course. The exercises are entirely optional, although they will all be useful later in the course. Asterisks indicate that they are harder. 0.1 Metric Spaces (Metric and Topological Spaces) A metric on a set X is a map d : X × X → [0, ∞) that satisfies: (a) d(x, y) > 0 with equality if and only if x = y; (b) Symmetry: d(x, y) = d(y, x) for all x, y ∈ X; (c) Triangle Rule: d(x, y) + d(y, z) > d(x, z) for all x, y, z ∈ X. A set X with a metric d is called a metric space. For example, the Euclidean metric on RN is given by d(x, y) = ||x − y|| where v u N ! u X 2 ||a|| = t |an| n=1 is the norm of a vector a. This metric makes RN into a metric space and any subset of it is also a metric space. A sequence in X is a map N → X; n 7→ xn. We often denote this sequence by (xn). This sequence converges to a limit ` ∈ X when d(xn, `) → 0 as n → ∞ . A subsequence of the sequence (xn) is given by taking only some of the terms in the sequence. So, a subsequence of the sequence (xn) is given by n 7→ xk(n) where k : N → N is a strictly increasing function. A metric space X is (sequentially) compact if every sequence from X has a subsequence that converges to a point of X.
    [Show full text]
  • 10 Group Theory
    10 Group theory 10.1 What is a group? A group G is a set of elements f, g, h, ... and an operation called multipli- cation such that for all elements f,g, and h in the group G: 1. The product fg is in the group G (closure); 2. f(gh)=(fg)h (associativity); 3. there is an identity element e in the group G such that ge = eg = g; 1 1 1 4. every g in G has an inverse g− in G such that gg− = g− g = e. Physical transformations naturally form groups. The elements of a group might be all physical transformations on a given set of objects that leave invariant a chosen property of the set of objects. For instance, the objects might be the points (x, y) in a plane. The chosen property could be their distances x2 + y2 from the origin. The physical transformations that leave unchanged these distances are the rotations about the origin p x cos ✓ sin ✓ x 0 = . (10.1) y sin ✓ cos ✓ y ✓ 0◆ ✓− ◆✓ ◆ These rotations form the special orthogonal group in 2 dimensions, SO(2). More generally, suppose the transformations T,T0,T00,... change a set of objects in ways that leave invariant a chosen property property of the objects. Suppose the product T 0 T of the transformations T and T 0 represents the action of T followed by the action of T 0 on the objects. Since both T and T 0 leave the chosen property unchanged, so will their product T 0 T . Thus the closure condition is satisfied.
    [Show full text]
  • Mixed States from Diffeomorphism Anomalies Arxiv:1109.5290V1
    SU-4252-919 IMSc/2011/9/10 Quantum Gravity: Mixed States from Diffeomorphism Anomalies A. P. Balachandran∗ Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA and International Institute of Physics (IIP-UFRN) Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil Amilcar R. de Queirozy Instituto de Fisica, Universidade de Brasilia, Caixa Postal 04455, 70919-970, Brasilia, DF, Brazil October 30, 2018 Abstract In a previous paper, we discussed simple examples like particle on a circle and molecules to argue that mixed states can arise from anoma- lous symmetries. This idea was applied to the breakdown (anomaly) of color SU(3) in the presence of non-abelian monopoles. Such mixed states create entropy as well. arXiv:1109.5290v1 [hep-th] 24 Sep 2011 In this article, we extend these ideas to the topological geons of Friedman and Sorkin in quantum gravity. The \large diffeos” or map- ping class groups can become anomalous in their quantum theory as we show. One way to eliminate these anomalies is to use mixed states, thereby creating entropy. These ideas may have something to do with black hole entropy as we speculate. ∗[email protected] [email protected] 1 1 Introduction Diffeomorphisms of space-time play the role of gauge transformations in grav- itational theories. Just as gauge invariance is basic in gauge theories, so too is diffeomorphism (diffeo) invariance in gravity theories. Diffeos can become anomalous on quantization of gravity models. If that happens, these models cannot serve as descriptions of quantum gravitating systems. There have been several investigations of diffeo anomalies in models of quantum gravity with matter in the past.
    [Show full text]
  • Area-Preserving Diffeomorphisms of the Torus Whose Rotation Sets Have
    Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior Salvador Addas-Zanata Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Rua do Mat˜ao 1010, Cidade Universit´aria, 05508-090 S˜ao Paulo, SP, Brazil Abstract ǫ In this paper we consider C1+ area-preserving diffeomorphisms of the torus f, either homotopic to the identity or to Dehn twists. We sup- e pose that f has a lift f to the plane such that its rotation set has in- terior and prove, among other things that if zero is an interior point of e e 2 the rotation set, then there exists a hyperbolic f-periodic point Q∈ IR such that W u(Qe) intersects W s(Qe +(a,b)) for all integers (a,b), which u e implies that W (Q) is invariant under integer translations. Moreover, u e s e e u e W (Q) = W (Q) and f restricted to W (Q) is invariant and topologi- u e cally mixing. Each connected component of the complement of W (Q) is a disk with diameter uniformly bounded from above. If f is transitive, u e 2 e then W (Q) =IR and f is topologically mixing in the whole plane. Key words: pseudo-Anosov maps, Pesin theory, periodic disks e-mail: [email protected] 2010 Mathematics Subject Classification: 37E30, 37E45, 37C25, 37C29, 37D25 The author is partially supported by CNPq, grant: 304803/06-5 1 Introduction and main results One of the most well understood chapters of dynamics of surface homeomor- phisms is the case of the torus.
    [Show full text]
  • Right-Angled Artin Groups in the C Diffeomorphism Group of the Real Line
    Right-angled Artin groups in the C∞ diffeomorphism group of the real line HYUNGRYUL BAIK, SANG-HYUN KIM, AND THOMAS KOBERDA Abstract. We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG ∞ group, embeds into the C diffeomorphism group of the real line. 1. Introduction The right-angled Artin group on a finite simplicial graph Γ is the following group presentation: A(Γ) = hV (Γ) | [vi, vj] = 1 if and only if {vi, vj}∈ E(Γ)i. Here, V (Γ) and E(Γ) denote the vertex set and the edge set of Γ, respec- tively. ∞ For a smooth oriented manifold X, we let Diff+ (X) denote the group of orientation preserving C∞ diffeomorphisms on X. A group G is said to embed into another group H if there is an injective group homormophism G → H. Our main result is the following. ∞ R Theorem 1. Every right-angled Artin group embeds into Diff+ ( ). Recall that a finitely generated group G is a limit group (or a fully resid- ually free group) if for each finite set F ⊂ G, there exists a homomorphism φF from G to a nonabelian free group such that φF is an injection when restricted to F . The class of limit groups fits into a larger class of groups, which we call residually RAAG groups. A group G is in this class if for each arXiv:1404.5559v4 [math.GT] 16 Feb 2015 1 6= g ∈ G, there exists a graph Γg and a homomorphism φg : G → A(Γg) such that φg(g) 6= 1.
    [Show full text]
  • Differential Topology: Exercise Sheet 3
    Prof. Dr. M. Wolf WS 2018/19 M. Heinze Sheet 3 Differential Topology: Exercise Sheet 3 Exercises (for Nov. 21th and 22th) 3.1 Smooth maps Prove the following: (a) A map f : M ! N between smooth manifolds (M; A); (N; B) is smooth if and only if for all x 2 M there are pairs (U; φ) 2 A; (V; ) 2 B such that x 2 U, f(U) ⊂ V and ◦ f ◦ φ−1 : φ(U) ! (V ) is smooth. (b) Compositions of smooth maps between subsets of smooth manifolds are smooth. Solution: (a) As the charts of a smooth structure cover the manifold the above property is certainly necessary for smoothness of f : M ! N. To show that it is also sufficient, consider two arbitrary charts U;~ φ~ 2 A and V;~ ~ 2 B. We need to prove that the map ~ ◦ f ◦ φ~−1 : φ~(U~) ! ~(V~ ) is smooth. Therefore consider an arbitrary y 2 φ~(U~) such that there is some x 2 M such that x = φ~−1(y). By assumption there are charts (U; φ) 2 A; (V; ) 2 B such that x 2 U, −1 f(U) ⊂ V and ◦ f ◦ φ : φ(U) ! (V ) is smooth. Now we choose U0;V0 such ~ ~ that x 2 U0 ⊂ U \ U and f(x) 2 V0 ⊂ V \ V and therefore we can write ~ ~−1 ~ −1 −1 ~−1 ◦ f ◦ φ j = ◦ ◦ ◦ f ◦ φ ◦ φ ◦ φ : (1) φ~(U0) As a composition of smooth maps, we showed that ~ ◦ f ◦ φ~−1j is smooth. As φ~(U0) x 2 M was chosen arbitrarily we proved that ~ ◦ f ◦ φ~−1 is a smooth map.
    [Show full text]
  • Matrix Lie Groups and Their Lie Algebras
    Matrix Lie groups and their Lie algebras Alen Alexanderian∗ Abstract We discuss matrix Lie groups and their corresponding Lie algebras. Some common examples are provided for purpose of illustration. 1 Introduction The goal of these brief note is to provide a quick introduction to matrix Lie groups which are a special class of abstract Lie groups. Study of matrix Lie groups is a fruitful endeavor which allows one an entry to theory of Lie groups without requiring knowl- edge of differential topology. After all, most interesting Lie groups turn out to be matrix groups anyway. An abstract Lie group is defined to be a group which is also a smooth manifold, where the group operations of multiplication and inversion are also smooth. We provide a much simple definition for a matrix Lie group in Section 4. Showing that a matrix Lie group is in fact a Lie group is discussed in standard texts such as [2]. We also discuss Lie algebras [1], and the computation of the Lie algebra of a Lie group in Section 5. We will compute the Lie algebras of several well known Lie groups in that section for the purpose of illustration. 2 Notation Let V be a vector space. We denote by gl(V) the space of all linear transformations on V. If V is a finite-dimensional vector space we may put an arbitrary basis on V and identify elements of gl(V) with their matrix representation. The following define various classes of matrices on Rn: ∗The University of Texas at Austin, USA. E-mail: [email protected] Last revised: July 12, 2013 Matrix Lie groups gl(n) : the space of n
    [Show full text]
  • Classification of Partially Hyperbolic Diffeomorphisms Under Some Rigid
    Classification of partially hyperbolic diffeomorphisms under some rigid conditions Pablo D. Carrasco ∗1, Enrique Pujals y2, and Federico Rodriguez-Hertz z3 1ICEx-UFMG, Avda. Presidente Antonio Carlos 6627, Belo Horizonte-MG, BR31270-90 2CUNY Graduate Center, 365 Fifth Avenue, Room 4208, New York, NY10016 3Penn State, 227 McAllister Building, University Park, State College, PA16802 June 30, 2020 Abstract Consider a three dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) either an Anosov diffeomorphism, a (gener- alized) skew-product or the time-one map of an Anosov flow, thus recovering a well known classification conjecture of the second author to this restricted setting. 1 Introduction and Main Results Let M be a manifold. One of the central tasks in global analysis is to understand the structure of Diffr(M), the group of diffeomorphisms of M. This is of course a very com- plicated matter, so to be able to make progress it is necessary to impose some reduc- tions. Typically, as we do in this article, the reduction consists of studying meaningful subsets in Diffr(M), and try to classify or characterize elements on them. We will consider partially hyperbolic diffeomorphisms acting on three manifolds. We choose to do so due to their flexibility (linking naturally algebraic, geometric and dynamical aspects), and because of the large amount of activity that this particular research topic has nowadays. Let us spell the precise definition that we adopt here, and refer the reader to [CRHRHU17, HP16] for recent surveys.
    [Show full text]