Introduction to Vector Spaces, Vector Algebras, and Vector Geometries
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Commentary on Thurston's Work on Foliations
COMMENTARY ON FOLIATIONS* Quoting Thurston's definition of foliation [F11]. \Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry. ... A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension k foliation. In order that a manifold admit a codimension- k foliation, it must have a plane field of dimension (n − k)." Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb's foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [P], Bendixson [Be], Kaplan [Ka] and others. -
Group Theory
Group Theory Hartmut Laue Mathematisches Seminar der Universit¨at Kiel 2013 Preface These lecture notes present the contents of my course on Group Theory within the masters programme in Mathematics at the University of Kiel. The aim is to introduce into concepts and techniques of modern group theory which are the prerequisites for tackling current research problems. In an area which has been studied with extreme intensity for many decades, the decision of what to include or not under the time limits of a summer semester was certainly not trivial, and apart from the aspect of importance also that of personal taste had to play a role. Experts will soon discover that among the results proved in this course there are certain theorems which frequently are viewed as too difficult to reach, like Tate’s (4.10) or Roquette’s (5.13). The proofs given here need only a few lines thanks to an approach which seems to have been underestimated although certain rudiments of it have made it into newer textbooks. Instead of making heavy use of cohomological or topological considerations or character theory, we introduce a completely elementary but rather general concept of normalized group action (1.5.4) which serves as a base for not only the above-mentioned highlights but also for other important theorems (3.6, 3.9 (Gasch¨utz), 3.13 (Schur-Zassenhaus)) and for the transfer. Thus we hope to escape the cartesian reservation towards authors in general1, although other parts of the theory clearly follow well-known patterns when a major modification would not result in a gain of clarity or applicability. -
Ideal Theory of the Abelian Group-Algebra
IDEAL THEORY OF THE ABELIAN GROUP-ALGEBRA. ]~Y T. VENK&TARAYUDU, M.A. (University oŸ Madras.) Received September 30, 1937. (Communicated by Dr. R. Vaidyanathaswamy, ~t.A., D.SC.) 1. Introduction. la: is well known 1 that the group-algebra over a commutative corpus K defined by a finite group G is semi-simple, when the characteristic p of the corpus K is not a divisor of the order N of the group and that in this case, the group-algebra K [G] is the direct sum of simple algebras whose products in pairs ate all zero. Beyond these general properties, the structure of the group-algebra is not completely kuown even in the case wheu G is Abelian. For example, in the existing literature, the following questions have not been discussed :-- (1) In the decomposition of K [G] us the direct sum of simple algebras, the actual number of simple component algebras. (2) The radical of K [G] when p is a divisor of N. (3) The residue class ring of K [G] with respect to its radical when it exists. In the present paper, we consider the case when G is Abelian and we show that in this case K [G] is simply isomorphic with the residue class ring of the polynomial domain K [xi, xa, xr] with respect to the ideal (x~'--l, x~~-1, -x~, --1), where ni, na "n~ are the orders of a set of basis elements of the Abelian group G. From a study of this residue class ring, we deduce al1 the important structural properties of the Abelian group-aigebra K [G]. -
Math 6310, Fall 2017 Homework 8 1. Let I and J Be Comaximal 2-Sided
Math 6310, Fall 2017 Homework 8 1. Let I and J be comaximal 2-sided ideals of a ring R. Show that I \ J = IJ + JI. + + 2. The Euler function φ : Z ! Z is defined by ¯ φ(n) := jfi 2 Zn j gcd(i; n) = 1gj: × (a) Show that φ(n) = jZn j. (b) Show that if p is prime then φ(pa) = pa − pa−1. (c) Show that if gcd(ni; nj) = 1 for all i 6= j then φ(n1 ··· nk) = φ(n1) ··· φ(nk). a1 ak (d) Write n = p1 ··· pk where the pi are distinct primes. Show that 1 1 φ(n) = n(1 − ) ··· (1 − ): p1 pk + φ(n) 3. (a) Show that for any n 2 Z and a 2 Z with gcd(a; n) = 1, we have a ≡ 1 mod n. p (b) Show that for any prime p and a 2 Z, we have a ≡ a mod p (Fermat's little theorem). [0;1] 4. Let R := R be the ring of all functions [0; 1] ! R, and S := C[0; 1] the subring of continuous functions. (a) Show that neither R nor S are noetherian by constructing a strictly increasing chain of ideals in each. (b) Let I := ff 2 R j f(1=2) = 0g. Show that I is a principal ideal of R. (c) Let J := ff 2 S j f(1=2) = 0g. Show that the ideal J of S is not finitely generated and deduce again that S is not noetherian. [Suppose J = (f1; : : : ; fk), define g := maxfjf1j;:::; jfkjg. -
Morse Theory for Codimension-One Foliations
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 298. Number 1. November 1986 MORSE THEORY FOR CODIMENSION-ONE FOLIATIONS STEVEN C. FERRY AND ARTHUR G. WASSERMAN ABSTRACT. It is shown that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every smooth real- valued function on the manifold has a cusp singularity. Introduction. Morse theory is a method of obtaining information about a smooth manifold by analyzing the singularities of a generic real-valued function on the manifold. If the manifold has the additional structure of a foliation, one can then consider as well the singularities of the function restricted to each leaf. The resulting singular set becomes more complicated, higher dimensional, and less trivially placed. In exchange, one obtains information about the. foliation. In this paper we consider codimension-one smooth foliations and show that, in this case, the occurrence of one type of singularity, the cusp, is related to the existence of compact leaves. Specifically, we show that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every real-valued function on M has a cusp. The singularities of a function on a foliated manifold are very similar to the singularities of a map to the plane. See [F, H-W, L 1, T 1, Wa, W]. However, Levine [L 1] has shown that the only obstruction to finding a map of a compact manifold to the plane without a cusp singularity is the mod 2 Euler characteristic of the manifold. -
1 Symplectic Vector Spaces
Symplectic Manifolds Tam´asHausel 1 Symplectic Vector Spaces Let us first fix a field k. Definition 1.1 (Symplectic form, symplectic vector space). Given a finite dimensional k-vector space W , a symplectic form on W is a bilinear, alternating non degenerate form on W , i.e. an element ! 2 ^2W ∗ such that if !(v; w) = 0 for all w 2 W then v = 0. We call the pair (W; !) a symplectic vector space. Example 1.2 (Standard example). Consider a finite dimensional k-vector space V and define W = V ⊕ V ∗. We can introduce a symplectic form on W : ! : W × W ! k ((v1; α1); (v2; α2)) 7! α1(v2) − α2(v1) Definition 1.3 (Symplectomorphism). Given two symplectic vector spaces (W1;!1) and (W2;!2), we say that a linear map f : W1 ! W2 is a symplectomorphism if it is an isomorphism of vector ∗ spaces and, furthermore, f !2 = !1. We denote by Sp(W ) the group of symplectomorphism W ! W . Let us consider a symplectic vector space (W; !) with even dimension 2n (we will see soon that it is always the case). Then ! ^ · · · ^ ! 2 ^2nW ∗ is a volume form, i.e. a non-degenerate top form. For f 2 Sp(W ) we must have !n = f ∗!n = det f · !n so that det f = 1 and, in particular, Sp(W ) ≤ SL(W ). We also note that, for n = 1, we have Sp(W ) =∼ SL(W ). Definition 1.4. Let U be a linear subspace of a symplectic vector space W . Then we define U ? = fv 2 W j !(v; w) = 0 8u 2 Ug: We say that U is isotropic if U ⊆ U ?, coisotropic if U ? ⊆ U and Lagrangian if U ? = U. -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
1. Overview Over the Basic Notions in Symplectic Geometry Definition 1.1
AN INTRODUCTION TO SYMPLECTIC GEOMETRY MAIK PICKL 1. Overview over the basic notions in symplectic geometry Definition 1.1. Let V be a m-dimensional R vector space and let Ω : V × V ! R be a bilinear map. We say Ω is skew-symmetric if Ω(u; v) = −Ω(v; u), for all u; v 2 V . Theorem 1.1. Let Ω be a skew-symmetric bilinear map on V . Then there is a basis u1; : : : ; uk; e1 : : : ; en; f1; : : : ; fn of V s.t. Ω(ui; v) = 0; 8i and 8v 2 V Ω(ei; ej) = 0 = Ω(fi; fj); 8i; j Ω(ei; fj) = δij; 8i; j: In matrix notation with respect to this basis we have 00 0 0 1 T Ω(u; v) = u @0 0 idA v: 0 −id 0 In particular we have dim(V ) = m = k + 2n. Proof. The proof is a skew-symmetric version of the Gram-Schmidt process. First let U := fu 2 V jΩ(u; v) = 0 for all v 2 V g; and choose a basis u1; : : : ; uk for U. Now choose a complementary space W s.t. V = U ⊕ W . Let e1 2 W , then there is a f1 2 W s.t. Ω(e1; f1) 6= 0 and we can assume that Ω(e1; f1) = 1. Let W1 = span(e1; f1) Ω W1 = fw 2 W jΩ(w; v) = 0 for all v 2 W1g: Ω Ω We claim that W = W1 ⊕ W1 : Suppose that v = ae1 + bf1 2 W1 \ W1 . We get that 0 = Ω(v; e1) = −b 0 = Ω(v; f1) = a and therefore v = 0. -
The Asymptotic Behavior of the Codimension Sequence of Affine G-Graded Algebras
THE ASYMPTOTIC BEHAVIOR OF THE CODIMENSION SEQUENCE OF AFFINE G-GRADED ALGEBRAS YUVAL SHPIGELMAN Abstract. Let W be an affine PI algebra over a field of characteristic zero graded 1 by a finite group G: We show that there exist α1; α2 2 R, β 2 2 Z, and l 2 N such β n G β n that α1n l ≤ cn (W ) ≤ α2n l . Furthermore, if W has a unit then the asymptotic G β n 1 behavior of cn (W ) is αn l where α 2 R, β 2 2 Z, l 2 N. Introduction Throughout this article F is an algebraically closed field of characteristic zero and W is an affine associative F - algebra graded by a finite group G. We also assume that W is a PI-algebra; i.e., it satisfies an ordinary polynomial identity. In this article we study G-graded polynomial identities of W , and in particular the corresponding G-graded codimension sequence. Let us briefly recall the basic setup. Let XG be a countable G G set of variables fxi;g : g 2 G; i 2 Ng and let F X be the free algebra on the set X . Given a polynomial in F XG we say that it is a G-graded identity of W if it vanishes upon any admissible evaluation on W . That is, a variable xi;g assumes values only from the corresponding homogeneous component Wg. The set of all G-graded identities is an ideal in the free algebra F XG which we denote by IdG(W ). -
Resolving Wild Embeddings of Codimension-One Manifolds in Manifolds of Dimensions Greater Than 3*
Topology and its Applications 24 (1986) 13-40 13 North-Holland RESOLVING WILD EMBEDDINGS OF CODIMENSION-ONE MANIFOLDS IN MANIFOLDS OF DIMENSIONS GREATER THAN 3* Fredric D. ANCEL Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA Received 25 June 1985 Revised 10 January 1986 For ns4, every embedding of an (n-1)-manifold in an n-manifold has a S-resolution for each S > 0. Consequently, for n z 4, every embedding of an (n - I)-manifold in an n-manifold can be approximated by tame embeddings. AMS (MOS) Subj. Class.: Primary 57N45; generalized manifold with boundary 1. Introduction The Codimension-One Tame Approximation Theorem in dimension n states that every embedding of an (n - 1)-manifold in an n-manifold can be approximated by tame embeddings. Bing proved this theorem in dimension 3 in [4], and went on to exploit it to great effect in his study of 3-manifolds. The theorem was established in dimensions 25 by Ancel and Cannon in [3]. It is proved in the remaining dimension, 4, in the present paper. The Codimension-One Tame Approximation Theorem is contained in a more comprehensive proposition which is the principal result of this paper: a Resolution Theorem for Wild Codimension-One Embeddings. The latter theorem is founded on the notion of a b-resolution of a wild embedding. For 6 > 0, a &resolution of a wild embedding e : M + N of a manifold M in a manifold N is, roughly speaking, a cell-like map G : N + N which moves no point of N farther than S and to which is associated a tame embedding f: M + N such that G of= e. -
Locally Flat Imbeddings of Topological Manifolds in Codimension Three
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 157, June 1971 LOCALLY FLAT IMBEDDINGS OF TOPOLOGICAL MANIFOLDS IN CODIMENSION THREE BY GLENN P. WELLERO) Abstract. This paper presents an imbedding theorem for one topological mani- fold M" in another topological manifold Q", provided that the codimension (q —n) is at least three. The result holds even if the manifolds are of the recently discovered non-piecewise-linear type. Denote the boundaries of M and Q by M and Q respec- tively. Suppose that M is 2n—q connected and Q is 2n —q + \ connected. It is then proved that any map /: (M, M) —>(ß, Q) such that f\M is a locally flat imbedding is homotopic relative to M to a proper locally flat imbedding g: M —>Q. It is also shown that if Mis closed and 2n—q + \ connected and Q is 2«—q + 2 connected, then any two homotopic locally flat imbeddings are locally flatly concordant. Introduction. The category of topological manifolds has recently been shown to include manifolds possessing no piecewise linear (P.L.) structure [8]. These non-P.L. manifolds must be investigated in their own right. One approach to the study of topological manifolds is to obtain results analogous to those already known for differentiable and P.L. manifolds. In the study of imbeddings of manifolds, one of the principal results in the P.L. category is the following: Irwin's Imbedding Theorem [7]. If n¿¡q-3, Mn is a compact, 2n-q con- nected P.L. manifold, Qf is a 2n—q+\ connected P.L. -
Arxiv:1907.11121V2 [Math.AG]
CRITERIA FOR COMPLETE INTERSECTIONS MIHAI HALIC Abstract. We obtain criteria for detecting complete intersections in projective varieties. Motivated by a conjecture of Hartshorne concerning subvarieties of projective spaces, we investigate situations when two-codimensional smooth subvarieties of rational homogeneous varieties are complete intersections. Introduction Hartshorne [20] conjectured that smooth subvarieties of projective spaces of sufficiently low codimension are necessarily complete intersections. He also stated that this is indeed the case as soon as the degree of the subvariety is small compared to the dimension of the ambient projective space. The goal of this article is to investigate this problem in a more general setting. We ask whether a similar statement holds for smooth subvarieties X of other ambient varieties V as well. From the beginning, one distinguishes two cases: One assumes that the normal bundle of X decomposes into a direct sum of line bundles. ´ This condition is clearly satisfied for complete intersections. or There are no a priori hypotheses. ´ In the first case, the property of being a complete intersection is reduced to verifying the splitting of the normal bundle of the subvariety; this feature is, at least theoretically, much easier to verify. Faltings [13] showed that a smooth subvariety of the projective space Pn, of codimension at most n 2, is a complete intersection as soon as its normal bundle splits. By analysing his proof, one{ observes that at its base there is a Hauptlemma—main lemma—which is valid in great generality. However, its ‘implementation’ is strongly adapted to subvarieties of projective spaces. This may explain why, to our knowledge, the arguments have not been extended to a wider framework.