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We give an elementary cut-and-paste proof of results of Bott and Shchepin: the third finite subset space of the circle is homeomorphic to the 3-sphere and the inclusion of the first finite subset s...

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3$-sphere branched over links. A feature of surface bundles coming from our construction is that the monodromies can be read off the braids obtained from the links as the branched set., 10 pages, 8 figures; minor changes from previous version

We introduce non-Abelian Kuramoto model on \begin{document}$ S^3 $\end{document} in the most general form. Following an analogy with the classical Kuramoto model (on the circle \begin{document}$ S^1 $\end{document} ), we study some interesting variations of the model on \begin{document}$ S^3 $\end{document} that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on \begin{document}$ S^3 $\end{document} . We briefly address two particular models: Kuramoto models on \begin{document}$ S^3 $\end{document} with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population. Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.

Associated with an immersion φ : S3 → ℂP3, we can define a canonical bundle endomorphism F : TS3 → TS3 by the pull back of the Kahler form of ℂP3. In this article, related to F we study equivariant minimal immersions from S3 into ℂP3 under the additional condition (∇XF)X = 0 for all X ∈ ker (F). As main result, we give a complete classification of such kinds of immersions. Moreover, we also construct a typical example of equivariant non-minimal immersion φ: S3 → ℂP3 satisfying (∇XF)X = 0 for all X ∈ ker (F), which is neither Lagrangian nor of CR type.

On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n >1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Amp��re equation up to the boundary is obstructed by a local curvature invariant of the boundary, the CR obstruction density $\mathcal{O}$. While local examples of obstruction flat CR manifolds are plentiful, the only known compact examples are the spherical CR manifolds. We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable. While in the nonembeddable case we may no longer interpret the obstruction density $\mathcal{O}$ in terms of the boundary regularity of Fefferman's equation (or the logarithmic singularity of the Bergman kernel) the equation $\mathcal{O}\equiv 0$ is still of great interest, e.g., since it corresponds to the Bach flat equation of conformal gravity for the Fefferman space of the CR structure (a conformal Lorentzian 4-manifold). Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable., 25 pages

In this paper, we consider decompositions of 3-manifolds with three handlebodies. We classify such decompositions of the 3-sphere and lens spaces with small genera. These decompositions admit operations called stabilizations. We also determine whether these decompositions are stabilized., 39 pages, 14 figures

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition. Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition. In this paper, we firstly relate 2-metrics of the class to surfaces in the 3-sphere: for a 2-metric of the class, a 5-dimensional set of (non-isometric) analytic surfaces in the 3-sphere is determined such that any surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surface and the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean 4-space. Secondly, we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces., 39 pages

In this paper we give an explicit parametrisation of the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere. As Hitchin proved, a harmonic map of a 2-torus is described by its spectral data, which consists of a hyperelliptic curve together with a pair of differentials and a line bundle. The space of spectral data is naturally a fibre bundle over the space of spectral curves. For homogeneous tori the space of spectral curves is a disc and the bundle is trivial. For tori with a one-dimensional invariance group, we enumerate the path connected components of the space of spectral curves and show that they are either `helicoids' or annuli, and that they densely foliate the parameter space. The bundle structure of the moduli space of spectral data over the annuli components is nontrivial. In the two cases, the spectral data require only elementary and elliptic functions respectively and we give explicit formulae at every stage. Homogeneous tori and the Gauss maps of Delaunay cylinders are used as illustrative examples., 37 pages, 7 figures

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere $S^3$ in $\mathbb{R}^4$, closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in $\mathbb{R}^3$ are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands., 29 pages, 1 figure

We show that the extrinsic diameter of immersed flat tori in the 3-sphere is $\pi$ under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration., Comment: 21 pages, 18 figures

In the present work, we study the decompositions of codimension-one transitions that alter the singular set the of stable maps of $S^3$ into $\mathbb{R}^3,$ the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets., 17 pages, 13 figures, 4 tables

We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case for when the sphere is 3-dimensional and where we take the group of symmetries to be $SO(4)$. As the 3-sphere is a group, both left and right multiplication on itself are commuting symmetries which together generate the full symmetry group. This gives rise to a notion of left and right momenta for the problem, and allows for a reduction in stages, first by the left and then the right, or vice versa. The intermediate reduced spaces obtained by left or right reduction are shown to be coadjoint orbits of the special Euclidean group $SE(4)$. The full reduced spaces are generically 4-dimensional and we describe these spaces and their singular strata. The dynamics of the 2-body problem descend through a double cover to give a dynamical system on $SO(4)$, which after reduction is the same as that of a 4-dimensional spinning top with symmetry. This connection allows us to `hit two birds with one stone' and derive results about both the spinning top and the 2-body problem simultaneously. We provide the equations of motion on the reduced spaces and fully classify the relative equilibria and discuss their stability., Comment: 21 pages

We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e. regular neighborhoods of embedded graphs, where a knot in a 3-subspace of $S^3$ is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in $S^3$. To show this, we discuss relation between certain group-theoretic and homotopic properties of knots in a compact 3-manifold, which can be of independent interest. Further, using the notion of transient knot, we define an integer-valued invariant of knots in $S^3$ that we call the transient number. We then show that the union of the sets of knots of unknotting number one and tunnel number one is a proper subset of the set of knots of transient number one., Comment: 22 pages, 14 figures; minor changes

We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equations does not have an integral solution, we show that some $2$-dimensional CW complexes cannot be embedded into the 3-sphere., 10 page, 11 figures

Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is the unknot., version to appear in Israel J. Math

We begin by describing the relationship between minimal immersions without umbilical points into the 3-sphere and solutions of the sinh-Gordon equation explicitly, especially to obtain the \(\mathfrak {sl}(2,\mathbb {C})\)-valued connection form αλ corresponding to the zero-curvature representation of the sinh-Gordon equation; from the integration of αλ we will obtain spectral data for periodic solutions of the sinh-Gordon equation.

An automorphism $f$ of a closed orientable surface $\Sigma$ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma$ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to an embedding $\Sigma \hookrightarrow S^3$ such that $\Sigma$ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa and the second author plays a key role., Comment: 23 pages, 16 figures; typos and errors corrected

In this paper we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,\varepsilon,\theta)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations., Comment: 40 pages, 6 figures, many details

For a surface in 3-sphere, by identifying the conformal round 3-sphere as the projectivized positive light cone in Minkowski 5-spacetime, we use the conformal Gauss map and the conformal transform to construct the associate homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local fundamental theorem for a surface in conformal round 3-sphere from that of the associate 4-surface in Minkowski 5-spacetime. More importantly, following the idea of Fefferman and Graham, we construct local scalar invariants for a surface in conformal round 3-sphere. One distinct feature of our construction is to link the classic work of Blaschke to the works of Bryan and Fefferman-Graham., 37 pages

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Fr\o yshov, and Ozsv\'ath and Szab\'o. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property., Comment: 17 pages, 5 figures; Typos fixed. We added more results including properties on spherical 3-manifolds. The version to appear in Topology and its Applications

We prove that amongst the equivariant constant mean curvature tori in the 3-sphere, the Clifford torus is the only local minimum of the Willmore energy. All other equivariant minimal tori in the 3-sphere are local maxima of the Willmore energy.

We study compatible contact structures of fibered, positively twisted graph multilinks in S 3 and prove that the contact structure of such a multilink is tight if and only if the orientations of its link components are all consistent with or all opposite to the orientation of the fibers of the Seifert fibrations of that graph multilink. As a corollary, we show that the compatible contact structures of the Milnor fibrations of real analytic germs of the form $(f\bar{g},O)$ are always overtwisted.

Perturbing the classical metric on the round 3-sphere \(S^3\) by the Killing vector fields tangent to Hopf fibers, one gets a class of Finsler metrics of Randers type with constant flag curvature \(\mathbf{K}=1\), depending on one parameter \(k>1\), called Bao–Shen’s (J Lond Math Soc 66:453–467, 2002) metrics. The corresponding spheres will be called Bao–Shen’s spheres, which are proper candidates of positively curved Finsler space forms. In this paper, we study the minimal surfaces in Bao–Shen’s spheres. We first study submanifolds isometrically immersed in a Randers manifold by the method of Zermelo’s navigation. Then we give a clear formula of the mean curvature of the surface in a Bao–Shen’s sphere by introducing the volume ratio function to show its relation with the mean curvature of the surface in round 3-sphere. As an application, we find an interesting family of minimal surfaces with respect to Busemann–Hausdorff volume form in Bao–Shen’s sphere called helicoids. This family contains the compact minimal surfaces \(\tau _{m,n}\) in round 3-sphere constructed by Lawson (Ann Math 92(3):335–374, 1970), including great 2-spheres, Clifford torus, Klein bottles, etc. Moreover, two rigidity results are given.

Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $S^3\backslash L$. The main result is obtained by translating a growth condition on $\pi_1(M)$ into the existence of a supersolution to the $4$-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces., Comment: 44 pages

By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twistings. In spite of this phenomenon, we show that there exists a compact 3-submanifold of the 3-sphere which cannot be unknotted by twistings. This shows that the Fox's re-embedding cannot always be replaced with twistings., 8 pages, 5 figures. In v2, more details on the Proof of Theorem 2.6 are given, and concluding remarks are added. In v3, Definitions 1.1 and 1.2 are clarified, and Corollary 2.4 is deleted by the ambiguous statement

Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hormander's classes of operators Psi(m)(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L-2-boundedness and Sobolev L-2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere S-3 and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K backslash G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Psi(m)(K backslash G) can be lifted to a pseudo-differential operator in Psi(m)(G), extending the known results on invariant partial differential operators.

This paper deals with the low-dimensional dynamics in the general non-Abelian Kuramoto model of mutually interacting generalized oscillators on the 3-sphere. If all oscillators have identical intrinsic generalized frequencies and the coupling is global, the dynamics is fully determined by several global variables. We state that these generalized oscillators evolve by the action of the group G H of (quaternionic) Mobius transformations that preserve S 3. The global variables satisfy a certain system of quaternion-valued ordinary differential equations, that is an extension of the Watanabe-Strogatz system. If the initial distribution of oscillators is uniform on S 3, additional symmetries arise and the dynamics can be restricted further to invariant submanifolds of (real) dimension four.

We show that a free period three action on a lens space is standard, i.e. the quotient is homeomorphic to a lens space. This is an extension of the result for period three actions on the three-sphere, arXiv:math.GT/0204077, by the author and J. Hyam Rubinstein., 67 pages, 54 pictures

In order to investigate to what extent the calculus of classical (pseudo-)Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civita's theorem, stating that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.

Let f,g: (C2,0) → (C,0) be two holomorphic germs with isolated singularities and no common branches and let Lf, [Formula: see text] be their links. We prove that the real analytic germ [Formula: see text] has an isolated singularity at 0 if and only if the link Lf ∪ -Lg is fibred. This was conjectured by Rudolph in [14]. If this condition holds, then the underlying Milnor fibration is an open-book fibration of the link Lf ∪ -Lg which coincides with [Formula: see text] in a tubular neighbourhood of this link. This enables one to realize a large family of fibrations of plumbing links in S3 as the Milnor fibrations of some real analytic germs [Formula: see text].

We show that a free period three action on the three-sphere is standard, i.e. the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts., Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper11.abs.html

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to [Formula: see text] in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in [Formula: see text] and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point.

Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere $S^3$ and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all St\"ackel systems in these varieties. This allows us to recover the known list of separation coordinates on $S^3$ in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron $K_4$.

We consider mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness., Comment: arXiv admin note: substantial text overlap with arXiv:1309.4278

We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of immersions by using Ekholm-Takase's formula in terms of singular Seifert surfaces. In order to construct singular Seifert surfaces for the immersions, we use the Kirby calculus., 19 pages, 17 figures

We derive colour spaces of the hue-colourfulness-luminance type, on the basis of a four-dimensional hypercube. We derive a chromatic 2D hue that goes together with chromatic saturation and luminance, a toroidal hue that goes together with a toroidal saturation and colourfulness, and also, a 3D tint that goes together with colourfulness.

We give a straightforward method that helps recognize when a noncompact hyperbolic 3-manifold is a link complement in the 3-sphere and automatically produces the link diagram. The method is based on converting a side-pairing to a handle decomposition.

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface \Sigma, such that f can be reconstructed in terms of algebraic data on \Sigma. Particularly interesting examples of constrained Willmore tori are the tori with constant mean curvature (CMC) in a 3-dimensional space form. It is shown in [14] and in [16] that the spectral curves of these tori are hyperelliptic. In this paper we show under mild conditions that a constrained Willmore torus f in the 3-sphere is a CMC torus in a 3-dimensional space form if its spectral curve has the structure of a CMC spectral curve., Comment: 13 pages

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of M\"obius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb C$ and classify equivariant constrained Willmore tori by the genus g of their spectral curve. In this case the spectral genus satisfies $g \leq 3.$, Comment: 24 pages

An infinite family of pseudo-Anosov diffeomorphisms over Heegaard surface of the 3-sphere is constructed, when genus is at least 3.

We define the thin fundamental Gray 3-groupoid $S_3(M)$ of a smooth manifold $M$ and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps $S_3(M) \to C(H)$, where $H$ is a 2-crossed module of Lie groups and $C(H)$ is the Gray 3-groupoid naturally constructed from $H$. As an application, we define Wilson 3-sphere observables., Definition of a 2-crossed module corrected. Other minor corrections. 48 Pages

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of reduction of group representations. A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. The deck transformations form a subgroup, and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold. A particular class of spherical 3-manifolds arises from the Platonic polyhedra. The harmonic analysis on the Poincare dodecahedral 3-manifold was analyzed along these lines. For comparison we construct here the harmonic analysis on simplicial spherical manifolds of dimension n=1,2,3. Harmonic analysis applied to the cosmic microwave background by selection rules can provide evidence for multiply connected cosmic topologies., 28 pages, 2 figures, minor corrections