Odd-dimensional Spheres Research Articles

The Whitehead conjecture, the tower ofS1conjecture, and Hecke algebras of type A

In the early 1980s, the author proved G. W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990s, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd-dimensional spheres. In this paper, we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.

Complex-projective and lens product spaces

Let \(t\) be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal \(S^1\)-action, and of the \(t\)-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the \(t\)th roots of unity. For a commutative complex-oriented cohomology theory \(h^*\), we determine the \(h^*\)-cohomology ring of these spaces (in terms of the \(t\)-series for \(h^*\), in the case of \(t\)-torsion lens product spaces). When \(h^*\) is singular cohomology with mod 2 coefficients, we also determine the action of the Steenrod algebra. We show that these spaces break apart after a suspension as a wedge of desuspensions of usual stunted complex-projective (\(t\)-torsion lens) spaces. We estimate the category and topological complexity of complex-projective and lens product spaces, showing in particular that these invariants are usually much lower than predicted by the usual dimensional bounds. We extend Davis’ analysis of manifold properties such as immersion dimension, (stable) span, and (stable) parallelizability of real-projective product spaces to the complex-projective and lens product cases.

Complex tangencies to embeddings of Heisenberg groups and odd-dimensional spheres

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real n-dimensional manifolds into ℂn. The generic topological structure of the set complex tangents to such embeddings Mn ↪ ℂn takes the form of a (stratified) (n-2)-dimensional submanifold of Mn. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres S2n-1 ↪ ℂ2n-1 by first considering the situation for the higher-dimensional analogues of the Heisenberg group.

Inverse Semigroups and Sheu's Groupoid for Odd Dimensional Quantum Spheres

Abstract.In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid C*-algebras. We show that the C*-algebra is generated by an inverse semigroup T of partial isometries. We show that the groupoid 𝓖tight associated with the inverse semigroup T by Exel is exactly the same as the groupoid considered by Sheu.

Kropina metrics and Zermelo navigation on Riemannian manifolds

The present paper studies globally defined Kropina metrics as solutions of the Zermelo’s navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.

CR immersions and Lorentzian geometry

Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersionscoveredbyconnection-preservingequivariantimmersionsamongthetotal spacesofthecanonicalcirclebundles)thusrelatingthegeometryofapseudohermitian immersion from a strictly pseudoconvex CR manifold M into an odd dimensional sphere, to the spectrum of the sublaplacian on M.

CR immersions and Lorentzian geometry

Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersions covered by connection-preserving equivariant immersions among the total spaces of the canonical circle bundles) thus relating the geometry of a pseudohermitian immersion from a strictly pseudoconvex CR manifold $$M$$ into an odd dimensional sphere, to the spectrum of the sublaplacian on $$M$$ .

Cellular decomposition and free resolution for split metacyclic spherical space forms

Given a free isometric action of a split metacyclic group on odd dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere equivariant with respect to the group action. A cell decomposition of the factor space and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group follow. In particular, the construction provides a simple explicit 4-periodic free resolution for the split metacyclic groups.

A note on sub-Riemannian structures associated with complex Hopf fibrations

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorems for a general corank 1 sub-Riemannian structure with symmetries done by C. Li and I. Zelenko (2011) in [2].

Trivializable sub-Riemannian structures on spheres

We classify the trivializable sub-Riemannian structures on odd-dimensional spheres SN that are induced by a Clifford module structure of RN+1. The underlying bracket generating distribution is of step two and spanned by a set of global linear vector fields X1,…,Xm. As a result we show that such structures only exist in the cases where N=3,7,15. The corresponding hypo-elliptic sub-Laplacians Δsub are defined as the (negative) sum of squares of the vector fields Xj. In the case of a trivializable rank four distribution on S7 and a trivializable rank eight distribution on S15 we obtain a part of the spectrum of Δsub. We also remark that in both cases there is a relation between the eigenfunctions and Jacobi polynomials.

Sub-Riemannian geodesics and heat operator on odd dimensional spheres

In this article we study the sub-Riemannian geometry of the spheres S2n+1 and S4n+3, arising from the principal S1-bundle structure defined by the Hopf map and the principal S3-bundle structure given by the quaternionic Hopf map, respectively. The S1 action leads to the classical contact geometry of S2n+1, while the S3 action gives another type of sub-Riemannian structure, with a distribution of corank 3. In both cases the metric is given as the restriction of the usual Riemannian metric on the respective horizontal distributions. For the contact S7 case, we give an explicit form of the intrinsic sub-Laplacian and obtain a commutation relation between the sub-Riemannian heat operator and the heat operator in the vertical direction.

Characterizations of Einstein manifolds and odd-dimensional spheres

In this paper, we use a Killing vector field on a Riemannian manifold to characterize odd-dimensional spheres and Einstein manifolds.

Construction of Hamiltonian-Minimal Lagrangian submanifolds in Complex Euclidean Space

We describe several families of Lagrangian submanifolds in complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.

The Borsuk–Ulam theorem for homotopy spherical space forms

In this work, we show for which odd-dimensional homotopy spherical space forms the Borsuk–Ulam theorem holds. These spaces are the quotient of a homotopy odd-dimensional sphere by a free action of a finite group. Also, the types of these spaces which admit a free involution are characterized. The case of even-dimensional homotopy spherical space forms is basically known.

Hopf fibration: Geodesics and distances

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of three-dimensional sphere and some partial results are obtained in the general case. The Carnot–Carathéodory distance is calculated. We also present some motivations related to quantum mechanics.

Quantum double suspension and spectral triples

In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. The notion of quantum double suspension of a C ⁎ -algebra was introduced by Hong and Szymanski. Here we introduce the quantum double suspension of a spectral triple and show that the WHKAE is stable under quantum double suspension. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd-dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.

Sequences of minimal surfaces in S 2n+1

For a minimal surface immersed into an odd-dimensional unit sphere S2n+1 with the first (n−2) higher-order ellipses of curvature being a circle, we construct a sequence of such surfaces and investigate if some two minimal surfaces in such a sequence can be congruent by an orientation-reversing isometry.

Fredholm theory and transversality for the parametrized and for the $S_1$-invariant symplectic action

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the _L_2-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic _S_1-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define _S_1-equivariant Floer homology. As an intermediate result of independent interest, we generalize Aronszajn’s unique continuation theorem to a class of elliptic integro-differential inequalities of order two.

Regularity and dimension spectrum of the equivariant spectral triple for the odd-dimensional quantum spheres

The odd-dimensional quantum sphere S_q^{2ℓ + 1} is a homogeneous space for the quantum group \mathrm{SU}_q(ℓ + 1) . A generic equivariant spectral triple for S_q^{2ℓ + 1} on its L_2 -space was constructed by Chakraborty and Pal in [4]. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give a detailed construction of its smooth function algebra and some related algebras that help proving regularity and in the computation of the dimension spectrum. Following the idea of Connes for \mathrm{SU}_q(2) , we first study another spectral triple for S_q^{2ℓ + 1} equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the \mathrm{SU}_q(ℓ + 1) -equivariant triple in the case q = 0 from those for the torus equivariant triple. For the case q ≠ 0 , we deduce regularity and dimension spectrum from the case q = 0 .

Erratum to: Ricci solitons and odd-dimensional spheres

Erratum to: Ricci solitons and odd-dimensional spheres