Standard S4 proved to be diffeomorphic to a curious homotopy sphere.
problem Determining the diffeomorphism of a curious homotopy sphere to the standard S4. method Proof based on properties of homotopy spheres and loose corks.
result The curious homotopy sphere is diffeomorphic to the standard S4. Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
The paper constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
New homotopy 4-spheres and real projective 4-spaces created.
problem Creating new homotopy 4-spheres and real projective 4-spaces.
method Extending Cappell-Shaneson's construction to produce new sets of smooth 4-manifolds.
result Produces new collections of homotopy 4-spheres and real projective 4-spaces.
Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. 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.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
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 U(1)- and Sp(1)-actions on exotic spheres. We prove that the infinite family of homotopy 4-spheres constructed by Daniel Nash are all diffeomorphic to 4-sphere.
Paper studies spaces of flattenings of simplicial spheres and their homotopy type.
problem Existence and uniqueness of differentiable structures on simplicial spheres.
method Analyzes spaces of flattenings of simplicial spheres and their homotopy type.
result Spaces of flattenings have the homotopy type of the orthogonal group.
Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.
problem Characterizing and comparing simple spines of homotopy 2-spheres.
method Proving ambient isotopy of simple spines representing the same homology class.
result Simple spines of homotopy 2-spheres are unique.
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. Complete obstruction found for 2-spheres in 5-manifolds to be isotopic.
problem Obstructing homotopic embeddings of 2-spheres into 5-manifolds from being isotopic.
method Using a level preserving Whitney move in codimension 3 to eliminate double points, and classical methods.
result New results for simply-connected 5-manifolds and 2-spheres with algebraic dual 3-spheres.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
In the paper of Montgomery, D. and Yang, C.T. [5], they discuss the de-suspension of smooth free actions of S1 on (2n+1)-dimensional homotopy spheres. In this paper we discuss the de-suspension of smooth free actions of S3 on (4n + 3)-dimensional homotopy spheres.
A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers p for which a given homotopy sphere admits a special generic map into Rp. By means of t…
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …
Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.
problem Understanding the topology of almost complex structures on the 6-sphere.
method Analyzes the fundamental and rational homotopy groups, computes homotopy fiber and groups, and generalizes to 6-manifolds.
result Induces isomorphism on fundamental and rational homotopy groups of the 6-sphere.
Invariant detects triple points in sphere immersions.
problem Detecting regular homotopy classes of triple-point-free sphere immersions.
method Defining an invariant using a directed tree and integer-valued function.
result Space of triple-point-free spheres has infinitely many regular homotopy classes.
Study homotopy groups of spaces of long links and knots, finding new generators.
problem Understanding homotopy groups of spaces of long links and knots.
method Graphing map increases dimensions, split injections from homotopy groups of spheres, and analyzing knotting effects.
result Generators for homotopy groups in a new degree for spaces of equidimensional long links.
Abstract: Study gyrations of sphere products and connected sums, generalizing Fico's Lemmata.
problem Understanding the homotopy type of gyrations of sphere products and connected sums.
method Recasting Fico's Lemmata into modern homotopy theoretic setting.
result Generalization of Fico's Lemmata to sphere products and connected sums.
Calegari's 4-spheres from fibered knots are proven standard.
problem Proving Calegari's 4-spheres from fibered knots are standard.
method 5-dimensional handlebody techniques and mapping class groups of 3-dimensional handlebodies.
result All Calegari's homotopy 4-spheres from fibered knots are diffeomorphic to the standard 4-sphere.
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.
We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\script…
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.
Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.
problem Understanding the homotopy properties of 6-manifolds over 4-manifolds.
method Analyzes the homotopy equivalence and rational homotopy of the total space of a sphere bundle over a 4-manifold.
result Looping a 6-manifold over a 4-manifold is homotopy equivalent to a product of loops on spheres.
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric com…
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.
We show 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.
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
Derivative map for disk diffeomorphisms induces nontrivial homotopy groups.
problem Proving nontriviality of homomorphisms induced by derivative maps.
method Combining recent results on homotopy spheres, plumbing approach, and explicit constructions.
result Non-zero homomorphism between specific homotopy groups.
Study shows moduli space of fibrations has specific homotopy types.
problem Understanding the homotopy types of moduli spaces of fibrations.
method Analyzes diffeomorphism group quotient and uses vector field models.
result Moduli space homotopy types are two-spheres or real projective planes.
With any (open or closed) cover of a space T we associate certain homotopy classes of maps T into n-spheres. These homotopy invariants can be considered as obstructions for extensions of covers of a subspace A to a space X. We using these obstructions for generalizations of the classic KKM (Knaster-Kuratowski-Mazurkiew…
Study of Willmore energy on sphere sublevel sets and flow singularities.
problem Understanding the Willmore energy landscape and singularities of the Willmore flow.
method Gluing different instances of the Willmore flow and using an invariant for triple-point-free spheres.
result Classification of initial surfaces with energy at most 12π leading to unavoidable singularities.
The space of metrics with positive Ricci curvature on spheres has special algebraic structures.
problem Understanding the space of metrics with positive Ricci curvature on spheres.
method Using H-space and loop space structures, and operad theory.
result The space of metrics with positive Ricci curvature on spheres is homotopy equivalent to an n-fold loop space.
We use surgery along 2-tori embedded in a union of two copies of a product of punctured 2-tori to produce a new collection of homotopy 4-spheres (4-manifolds homotopy equivalent to S4 and hence homeomorphic to S4 but possibly not diffeomorphic to S4). It is still unknown if these new examples are in fact exoti…
Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between…
We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.
The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.
problem Understanding the homotopy types of diffeomorphism groups and sphere embeddings.
method Cerf's upgraded proof, scanning maps, canceling handles, Embedding Calculus.
result The monoid of Schoenflies spheres forms a group under connect-sum.
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except 15,31,63,127 is obtained. It is proved that for n>127 in the stable homotopy group o…
Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.
problem Understanding the automorphisms of sphere complexes associated with graphs.
method Analyzing the group of proper homotopy equivalences and constructing an exhaustion of the sphere complex.
result The automorphism group of the sphere complex is isomorphic to the mapping group of the graph.
For dimensions n greater than or equal to 3, we show that the space of metrics of positive scalar curvature on the n-sphere is homotopy equivalent to a subspace which takes the form of a H-space with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum con…
Innovates a three-component link homotopy invariant.
problem Classifying three-component link maps up to homotopy.
method Developed tools and invariants for distinguishing three-component link maps.
result Found three-component link maps that are not homotopic.
The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras as…
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvatu…