The paper finds infinite families of exotic spheres with free actions.
problem Detecting smooth free and actions on exotic spheres.
method Topological modular forms.
result Infinite families of very exotic spheres with free actions.
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48 results for “exotic Kervaire spheres”
Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.
The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of…
This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy -sphere () carries an isoparametric function (with certain metric) with 2 points as the focal set,…
In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds, that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singula…
Study of Morse functions with constraints and their bordism groups.
problem Interpolating between Morse and generic functions' bordism groups.
method Elimination of cusps, Stein factorization, two-index theorem, handle extension theorem.
result Constrained bordism groups are related to connective bordism.
New metrics found on manifolds using Cheeger deformations.
problem Finding new metrics on manifolds.
method Cheeger deformations applied to -manifolds.
result New metrics of positive Ricci and almost non-negative curvature.
Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction speciali…
Completes classification of high-dimensional manifolds up to exotic sums.
problem Classifying high-dimensional manifolds up to exotic sums.
method Algebraic refinement of intersection forms and Witten genus.
result All but finitely many dimensions are resolved, completing the classification.
Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexesmath.GT
Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
Study detects Hopf elements and Kervaire classes in sphere spectra.
problem Detecting Hopf elements and Kervaire classes in stable homotopy groups.
method Using Hurewicz map and Real Johnson-Wilson theories.
result Certain families of elements are detected by Real Johnson-Wilson theories.
It is proved that there exists an integer such that a framed manifold of dimension , has the trivial Kervaire Invariant.
We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this resolves a lo…
Proves surface embedding theorem for 4-manifolds with good fundamental group.
problem Surface embedding in 4-manifolds with good fundamental group.
method Introduces dual spheres and Kervaire-Milnor invariant for computation.
result Combinatorial formula for Kervaire-Milnor invariant.
The paper explores smooth equivariant rigidity and finds infinitely many exotic smooth structures.
problem Smooth equivariant rigidity for certain group actions on manifolds.
method Analysis of -vector bundles and Atiyah-Singer index theorem.
result Infinitely many exotic smooth structures for certain group actions.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly t…
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
The study proves conditions for embedding manifolds into complex spaces.
problem Conditions for embedding manifolds into complex spaces.
method Analyzes CR-regular and totally real embeddings for n-manifolds.
result Establishes conditions for the existence of CR-regular embeddings and their implications.
A physicist explores exotic 7-spheres using fibre bundles and instantons.
problem Investigating the geometry and topology of exotic 7-spheres.
method Analytical and computational tools from theoretical physics, including fibre bundles and instantons.
result Derives an analytic expression for a family of Riemannian metrics on the Gromoll-Meyer sphere.
Study compares characteristic classes of bundles with exotic spheres.
problem Examines characteristic classes of bundles with exotic spheres.
method Compares characteristic classes of bundles with and without exotic spheres.
result Characteristic classes are isomorphic after inverting the order of the exotic sphere.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
New tools prove smooth actions on exotic spheres.
problem Existence of smooth actions on exotic spheres.
method Homotopy-theoretic tools, complex and quaternionic Mahowald invariants.
result Existence of smooth - and -actions on exotic spheres.
Proves existence of manifolds with Kervaire invariant one in specific dimensions.
problem Existence of smooth framed manifolds with Kervaire invariant one.
method Adams spectral sequence and combination of theorems.
result Smooth framed manifolds with Kervaire invariant one exist in dimensions 2, 6, 14, 30, 62, and 126.
We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.
Paper constructs exotic spacetimes with same physical properties.
problem Whether two topologically identical manifolds can have different geometries.
method Computational approach to produce physical models on exotic spheres.
result Lorentzian metrics on homeomorphic but not diffeomorphic manifolds with same physical properties.
Study finds infinite families of Sasaki-Einstein metrics on spheres.
problem Finding Sasaki-Einstein metrics on spheres and exotic spheres.
method Analyzing odd-dimensional spheres and exotic spheres that bound parallelizable manifolds.
result Infinitely many families of Sasaki-Einstein metrics on spheres and exotic spheres.
Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.
problem Constructing exotic embeddings of RP^2 and homotopy spheres.
method Using Montesinos knots and roll-spun knots to prove the existence of homotopy spheres.
result An infinite family of homotopy spheres and homotopy CP^2s are produced.
We prove the existence of exotic but homotopically trivial contact structures on spheres of dimension 8k-1. Together with previous results of Eliashberg and the second author this establishes the existence of such structures on all odd-dimensional spheres (of dimension at least 3).
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of…
We present Michel Kervaire work on knots in higher dimensions.
Einstein Metrics on Spheresmath.DG
We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularit…
Detecting exotic spheres involves analyzing framed configuration spaces.
problem Detecting exotic spheres through homotopy type of truncated Disc-presheaves.
method Using gluing results for Disc-presheaves, Atiyah duality, and computations of mapping class groups.
result Conditions for detecting exotic spheres through framed configuration spaces.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. In the second paper, we calculate the "smoothness structure" part of the path integral in quantum gravity for the exotic R^4 as non-compact manifold. We discuss the influence of th…
Study on stochastic flows on 7-dimensional spheres.
problem Stochastic processes on 7-dimensional spheres.
method Isometric stochastic flows of Stratonovich SDE on spheres.
result Properties of stochastic processes on Gromoll-Meyer exotic sphere.
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on where is a closed aspherical space form and is an exotic -sphere.
method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on when is 7-dimensional.
We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed…
The paper explores holographic structures on exotic spheres and their implications.
problem Understanding holographic structures on exotic spheres.
method Introducing and studying holographic structures on closed manifolds .
result Generalizing the Holography Theorem to fillable holographic structures on .
We construct Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard infranilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension on infranilma…
New method classifies spin 4-manifolds using Kervaire-Milnor invariant.
problem Classifying spin 4-manifolds up to stabilisation.
method Using Kervaire-Milnor invariant to compute Arf invariants and classify.
result New stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
Study on stochastic flows on exotic spheres, exploring their properties.
problem Investigating stochastic processes on exotic -dimensional spheres.
method Constructing exotic manifolds from disjoint unions and identifying points using maps.
result Explicit homeomorphisms and stochastic processes on exotic spheres.
Study on curvature properties of exotic 7-sphere using quaternionic geometry.
problem Examining curvature of an exotic 7-sphere.
method Using Kaluza-Klein Ansatz with quaternionic language, focusing on moduli spaces of instantons and metrics.
result Identified a center in the moduli space of instantons and computed the Ricci tensor for a metric of maximal isometry.
A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive secti…
We construct a co-dimension completely non-holonomic sub-bundle on the Gromoll-Meyer exotic sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7…
We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.