Exotic Spheres Research Articles

Circle actions on homotopy spheres not bounding spin manifolds

Smooth circle actions are constructed on odd-dimensional homotopy spheres that do not bound spin manifolds. Examples are given in every dimension for which exotic spheres of the described type exist.

Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres

It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ∑ n is an exoticn-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ∑ n are tori of dimension ≤[(n+1)/2].

An Exotic Sphere With Nonnegative Sectional Curvature

An Exotic Sphere With Nonnegative Sectional Curvature

Branched coverings, open books and knot periodicity

Perhaps the most striking corollary of this analysis is a completely topological construction ofthe Brieskorn manifolds C(a, , . . . , a,) = V(f) n S’“+l (forf=zOso f . . * + z,““, zi complex variables, ai 2 1 positive integers, V(f) = zeros off in C”+r). These manifolds have been of extraordinary interest in recent years, producing examples of exotic spheres, lens spaces, new group actions and so on.

Torsion of a non-homogeneous solid of revolution

Solution has been obtained in the case of torsion of a solid of revolution of non-homogeneous material. The problem of a twisted sphere has been deduced from the general solution. The effect of non-homogeneity has been illustrated graphically.

Twisting of a non-homogeneous sphere

Effect of non-homogeneity of shear modulus on the stresses in a twisted sphere by surface couple has been exhibited in this paper.

Representing Handlebodies by Plumbing and Surgery

{/ 6 and n ^>2, JF is diffeomorphic to the sum of (TTI—- n,)-disk bundles over ^-spheres. This is obtained by constructing disjoint spheres S* each of which has D' = D X o as the one hemisphere and a disk D in Dm as the other hemisphere and by tying with bands the tubular neighbourhoods of the zi-sheres Sn{ = Dt}\UDjn. But, if 27i>77i, there arise non-empty intersections of the 7i-disks D. In this paper, we consider how handlebodies can be constructed from (m — ^)-disk bundles over n -spheres in the case when 2n'>m. This problem was motivated to analyze the structure of such m -manifolds with non-trivial homology groups only at dimensions 0, k, m — k, and m, (m^ 2k). The results are given in the section 6 as an application. In Hi 3], J. Milnor constructed the manifolds which are parallelizable and have exotic spheres as their boundaries. There, some 2A;-disk bundles over 2&-spheres are plumbed correspondingly with a suitable matrix, and in order to kill the fundamental group of the plumbing manifold a proper disk is attached to the boundary. In CllH, M. Kerevaire also constructed a manifold like that by plumbing 5-disk bundles over 5-spheres. In this paper, we generalize those constructions of J. Milnor and M. Kervaire. The generalization is done by plumbing (m — ra)-disk bundles over n,-spheres along (2n — m)-spheres or tori which are imbedded in the ^-spheres,

On Stable Diffeomorphism of Exotic Spheres in the Metastable Range

Let ⊖np+ 1denote the subgroup of the Kervaire-Milnor groupθnconsisting of thosen-spheres which imbed with trivial normal bundle in Euclidean(n + p+ 1)-space,n<2p.It is known that such imbeddings always exist [6], and that the normal bundle is independent of the imbedding[10].Following [2], we write Ωn,pfor the quotientθn/⊖np+ 1.The order of Ωn,pafter identifying each element with its inverse, is equal to the number of diffeomorphically distinct (orientation preserved) Σn×Sp[2; 5]. Indeed,Ωn,pis closely linked to the problem of determining the number of smooth structuresα(n, p)onSn×Sp.For instance, ifΩn,p= 0 thenα(n, p)equals the order ofθn+p[5]. Specific results are easily read off Table I and Theorem 2.1.

The nonexistence of free 𝑆¹ actions on some homotopy spheres

In this paper a necessary condition is given for the existence of a free differentiable action of the circle group S 1 {S^1} on a ( 4 k + 1 ) (4k + 1) -dimensional homotopy sphere. This includes a previously known criterion due to R. Lee and yields additional examples of exotic spheres for which no such actions exist.

Gromoll groups, ${\text{Diff}}\,S^n$ and bilinear constructions of exotic spheres

Gromoll groups, ${\text{Diff}}\,S^n$ and bilinear constructions of exotic spheres

On the Bound of the Dimensions of the Isometry Groups of All Possible Riemannian Metrics on an Exotic Sphere

On the Bound of the Dimensions of the Isometry Groups of All Possible Riemannian Metrics on an Exotic Sphere

The twisted sphere

The twisted sphere

Geometry of isoparametric hypersurfaces in Riemannian manifolds

In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of the Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with constant principal curvatures in general Riemannian manifolds. It turns out that in this case the focal submanifolds have the same properties as those in standard sphere, i.e., the shape operator with respect to any normal direction has common constant principal curvatures. Some necessary conditions involving Ricci curvature and scalar curvature are also derived.