Homotopy Spheres Research Articles

Realizing congruence subgroups inside the diffeomorphism group of a product of homotopy spheres

Let M be a smooth manifold which is homeomorphic to the n-fold product of Sk, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of Hk(M;Z). We prove that the image of this homomorphism contains a congruence subgroup of SLn(Z) whenever n is at least 3.

Brieskorn manifolds, positive Sasakian geometry, and contact topology

Abstract Using S 1 ${S^{1}}$ -equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn–Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of S 2 × S 3 ${S^{2}\times S^{3}}$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki–Einstein metrics on certain homotopy spheres. Finally, a new family of Sasaki–Einstein metrics of real dimension 20 on S 5 ${S^{5}}$ is exhibited.

Group actions on spheres with rank one isotropy

Let G G be a rank two finite group, and let H \mathcal {H} denote the family of all rank one p p -subgroups of G G for which rank p ⁡ ( G ) = 2 \operatorname {rank}_p(G)=2 . We show that a rank two finite group G G which satisfies certain explicit group-theoretic conditions admits a finite G G -CW-complex X ≃ S n X\simeq S^n with isotropy in H \mathcal {H} , whose fixed sets are homotopy spheres. Our construction provides an infinite family of new non-linear G G -CW-complex examples.

ON ALGEBRAIC INVARIANTS FOR FREE ACTIONS ON HOMOTOPY SPHERES

We investigate conjectures and questions regarding topological phenomena related to free actions on homotopy spheres and present some affirmative answers.

Isoparametric functions on exotic spheres

This paper extends widely the work in [11]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells–Kuiper projective planes are established. In particular, every homotopy n-sphere (n>4) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [26] (only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres). As an application, we improve a beautiful result of Bérard-Bergery [2] (see also pp. 234–235 of [3]).

Groups of Homotopy Spheres: I [reprinted from Annals of Mathematics, Vol. 77, No. 3, May 1963

Groups of Homotopy Spheres: I [reprinted from Annals of Mathematics, Vol. 77, No. 3, May 1963

On Three-Dimensional Alexandrov Spaces

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of the real projective plane; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincare Conjecture in dimension three, in the context of Alexandrov spaces, in the two forms it is usually formulated for manifolds. We first show that the only three-dimensional Alexandrov space that is also a homotopy sphere is the 3-sphere; then we give examples of closed, geometric, simply connected three-dimensional Alexandrov spaces for five of the eight Thurston geometries, proving along the way the impossibility of getting such examples for the Nil, $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ and Sol geometries. We conclude the paper by proving the analogue of the geometrization conjecture for closed three-dimensional Alexandrov spaces.

Construction of semi-free S^3 actions on homotopy sphere with untwisted fixed point set

In his paper "Surgery and the theory of differentiable transformation groups", William Browder developed surgery techniques to study semi-free actions of S1 on homotopy spheres, under the additional assumption that the fixed point set is a homotopy sphere. He used this surgery to show how to construct such actions. In this paper, I discuss a similar theory about semi-free actions of S3 on homotopy spheres. An open problem is raised at the end of the paper.

Farrell–Jones spheres and inertia groups of complex projective spaces

Abstract We introduce and study a new class of homotopy spheres called Farrell–Jones spheres. Using Farrell–Jones sphere we construct examples of closed negatively curved manifolds M 2 n ${M^{2n}}$ , where n = 7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C. S. Aravinda and F. T. Farrell. We show that every exotic sphere not bounding a spin manifold (Hitchin sphere) is a Farrell–Jones sphere. We also discuss the relationship between inertia groups of ℂ ℙ n ${\mathbb {C}\mathbb {P}^n}$ and Farrell–Jones spheres.

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engstr\"om to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.

Homotopy Sphere Representations for Matroids

For any rank r oriented matroid M, a construction is given of a “topological representation” of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to S r–1 . The construction is completely explicit and depends only on a choice of maximal flag in M. If M is orientable, then all Folkman-Lawrence representations of all orientations of M embed in this representation in a homotopically nice way.

On the moduli space of positive Ricci curvature metrics on homotopy spheres

We show that the moduli space of Ricci positive metrics on a certain family of homotopy spheres has infinitely many components.

ON GROUPS OF TYPE $\mathcal{L}^{-1}$

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property [Formula: see text] which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

Cappell-Shaneson homotopy spheres are standard

We show here that an infinite sequence of homotopy 4-spheres constructed by Cappell-Shaneson are all diffeomorphic to S 4 . This generalizes previous results of Akbulut-Kirby and Gompf.

A SURVEY ON POLYGONAL PORTRAITS OF MANIFOLDS

Planar portraits are geometric representations of smooth manifolds defined by their generic maps into the plane. A simple subclass called the polygonal portraits is introduced, their realisations, and relations of their shapes to the topology of source manifolds are discussed. Generalisations and analogies of the results to other planar portraits are also mentioned. A list of manifolds which possibly admit polygonal portraits is given, up to diffeomorphism and up to homotopy spheres. This article is intended to give a summary on our research on the topic, and hence precise proofs will be given in other papers.

A survey on polygonal portraits of manifolds

AbstractPlanar portraits are geometric representations of smooth manifolds defined by their generic maps into the plane. A simple subclass called the polygonal portraits is introduced, their realisations, and relations of their shapes to the topology of source manifolds are discussed. Generalisations and analogies of the results to other planar portraits are also mentioned. A list of manifolds which possibly admit polygonal portraits is given, up to diffeomorphism and up to homotopy spheres. This article is intended to give a summary on our research on the topic, and hence precise proofs will be given in other papers.

On the torus cobordant cohomology spheres

Let G be a compact Lie group. In 1960, P A Smith asked the following question: “Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?” A result due to Atiyah and Bott proves that the answer is ‘yes’ for ℝ p and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on S n which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in ℝ-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith’s question.

Indefinite Extrinsic Spheres

We study the lightlike geometry of the second fundamental form of the intersection between an algebraic hypersurface in $\mathbb{C}_{s}^{n}$ and a pseudosphere $S_{2s}^{2n-1}(2/\sqrt{k})$ including a class of extrinsic spheres which are not homotopy spheres.

Subgroups of Group of homotopy spheres

Let &#952n denote the group of h-cobordism classes of homotopy n-sphere under the connected sum operation. H(p, q) is the subgroup of &#952p consisting of those homotopy p-spheres &#8721 p such that &#8721 p x Sw is diffeomorphic to Sp x Sq. Also bPp+1 is the subgroup of homotopy p-sphere which bounds parallelizable manifolds. In this paper, we will prove that H(p,q)bPp+1 is isomorphic to the Cokernel of Hopf-Whitehead homorphism J: &#928pS0(q+1) &#8594 &#928p + Q + 1 (Sq+1>) JONAMP Vol. 11 2007: pp. 47-50

Formula omitted]-free rank two finite groups act freely on a homotopy product of two spheres

A classic result of Swan states that a finite group G acts freely on a finite homotopy sphere if and only if every abelian subgroup of G is cyclic. Following this result, Benson and Carlson conjectured that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n . Recently, Adem and Smith have shown that every rank two finite p -group acts freely on a finite complex with the homotopy type of two spheres. In this paper we will make further progress, showing that rank two groups that are Qd ( p ) -free act freely on a finite homotopy product of two spheres.