Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

FUNCTORIAL PROPERTIES OF THE HOPF INVARIANT I

Abstract

Homotopy operations Θ: [ΣY, U] → [ΣY, V] which are natural in Y are considered. In particular a technique used in the definition of the Hopf invariant (as treated by Berstein-Hilton) shows that any fibration p: E → B with fiber V, when provided with a homotopy section of Ωp, determines such a homotopy operation [ΣY, E] → [ΣY, V]. More generally, starting from a track class of homotopies α º f ? β º g we adapt this fibration technique to construct a homotopy operation [ΣY, M(f,g)] → [ΣY, F _α * F _β] called a Hopf invariant. The intervening fibration in the definition of this Hopf invariant arises via the fiberwise join construction.     

Simultaneous reduction of a lattice basis and its reciprocal basis

Given a latticeL we are looking for a basisB=[b ₁, ...b _n ] ofL with the property that bothB and the associated basisB ^*=[b ₁ ^* , ...,b _n ^* ] of the reciprocal latticeL ^* consist of short vectors. For any such basisB with reciprocal basisB ^* let . Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)exp(c ₁·n ^1/3) for a constantc ₁ independent ofn. We improve this upper bound toS(B)exp(c ₂·(lnn)²) withc ₂ independent ofn.We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity . In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R~L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R~L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd_RX = GpdS(S?_R~L X). Similar result is obtained for Ding projective dimension of the S-complex S?_R~L X.     

On Two Families of Hopf Algebras of Dimension 2 m

Abstract

In this paper we construct two families of semisimple Hopf algebras of dimension 2 ⁿ⁺¹, n ≥ 3. They are all constructed as Radford's biproducts. For these examples and their duals we compute their grouplike elements, centers, character algebras and Grothendieck rings. Comparing these facts we are able to show that depending on the dimension, representatives of one of the families are selfdual. We also prove that Hopf algebras from these families are neither triangular nor cotriangular and that their cocycle deformations are trivial.     

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S ¹⁵ → S ⁸, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S ¹ subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.     

THE FUNDAMENTAL GROUP IN FIBRATIONS

A generalized Mayer-Vietoris sequence involving crossed homomorphisms is established and the construction is applied to the homotopy sequence of the CW-pair (X.X¹) to relate the homotopy sequences of (X.X¹) and the fibre bundle F → E → X in low dimensions. If there is a partial cross-section of E → X over X², the classical form, π₁ E ～ π₁ [xtilde] π₁ F as a semidirect product, results. In case there is no extension over X² of any cross-section of the restricted bundle χ:π₂ (x², x¹) → X¹ the corresponding obstruction map X_E:π₂(x²,x¹) → π₁F is non-trivial and in case F → E → X is an SO(n)-bundle (n ≥ 3), χ_E maps into a subgroup of the centre, Z(π₁ F), of order at most 2.     

A theory of PWD-structures

A general study is undertaken of product-wedge-diagonal (=PWD) structures on a space. In part this concept may be viewed as arising from G.W. Whitehead's fat-wedge characterization of Lusternik-Schnirelmann category. From another viewpoint PWD-structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD-structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map X→Y. A principal result of the paper uses Hopf invariants to formulate a Berstein-Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join U*V that has PWD-structure defined canonically via the join structure in terms of diagonal maps on U and V.     

THOUGHTS ON THE CANTOR-BERNSTEIN THEOREM

The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation f U g^?1. Considering Tarski's Fixpoint Theorem as the implicit basic ingredient of such proofs. We examine several classical proofs/starting with Dedekind (1887), and illuminate their common feature by means of the categorical notion of a natural fixpoint. We consider a categorical form (CBT) of the theorem (with h ? f Ug^?1) in a variety of contexts, obtaining some examples of categories where CBT holds and others where it fails. Among other results we prove for a topos E, (1) CBT holds if E is Boolean, and conversely if E has a natural number object; (2) The Axiom of Choice in E implies a dual version of CBTI and conversely if E has splitting supports and a natural number object.     

Techniques for Classifying Hopf Algebras and Applications to Dimension p 3

Classifying Hopf algebras of a given finite dimension n over ? is a challenging problem. If n is p, p², 2p, or 2p² with p prime, the classification is complete. If n = p³, the semisimple and the pointed Hopf algebras are classified, and much progress on the remaining cases was made by the second author but the general classification is still open. Here we outline some results and techniques which have been useful in approaching this problem and add a few new ones. We give some further results on Hopf algebras of dimension p³ and finish the classification for dimension 27.     

A family of quadratic forms associated to quadratic mappings of spheres

The general form of a real quadratic mapping of spheres can be determined by studying the diagonalization of each form in an associated family of quadratic forms. In particular, the eigenvalues provide a means for detecting maps which are of the Hopf type. When the eigenvalues are nonzero for every form in the family, the forms associated to $\text{?:}\text{S}{}^{\mathrm{n}}\text{→}\text{S}{}^{\mathrm{m}}$ give rise to a quadratic form on the tangent bundle of the unit sphere Sⁿ. If ? is of the Hopf type, nondegeneracy of each form occurs only when n=1,3,7,15.     

On a property of functions on the sphere and its application

Given a continuous function f:S^m+n−2→R^m, and n points u₁,u₂,…,u_n∈S^m+n−2; does there exist a rotation r∈SO(m+n−1) such that f(ru₁)=f(ru₂)=?=f(ru_n)? In this paper, we study the property of a continuous map from a sphere to a Euclidean space by using the theory of Smith periodic transformation and Brouwer degree of map theorem. The conjecture is proved under the case of n=2 and m being even. Furthermore, this conjecture is proved for the case when u_j⋅u_j+1=λ and the dimension of the sphere is not less than m+n−2. This paper generalizes the Borsuk-Ulam theorem and then presents its application.     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

Self-maps of the product of two spheres fixing the diagonal

We compute the monoid of essential self-maps of Sn×Sn fixing the diagonal. More generally, we consider products S×S, where S is a suspension. Essential self-maps of S×S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.     

On the Structure of Weak Hopf Algebras

We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo, a trivial automorphism. We find a sufficient condition in terms of Tr(S²) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple.     

Compatible associative bialgebras

We introduce a non-symmetric operad 𝒩, whose dimension in degree n is given by the Catalan number c_n?1. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a 𝒩-algebra. The data (As,As²,𝒩) is a good triple of operads, in J.-L. Loday’s sense. Our construction induces another triple of operads (As,As₂,As), where As₂ is the operad of matching dialgebras. Motivated by A. Goncharov’s Hopf algebra of paths P(S), we introduce the notion of bi-matching dialgebras and show that the Hopf algebra P(S) is a bi-matching dialgebras.     

Blocks of minimal dimension

Any block with defect group P of a finite group G with Sylow-p-subgroup S has dimension at least |S|²/|P|; we show that a block which attains this bound is nilpotent, answering a question of G. R. Robinson. Received: 20 November 2006     

On Semisimple Hopf Algebras of Dimension pq2, II

We obtain further classification results for semisimple Hopf algebras of dimension pq ² over an algebraically closed field k of characteristic zero. We complete the classification of semisimple Hopf algebras of dimension 28.