arXiv research

The paper constructs homotopy 4-spheres using pochette surgery.

Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.

New homotopy 4-spheres and real projective 4-spaces created.

Calegari's 4-spheres from fibered knots are proven standard.

We prove that the infinite family of homotopy 4-spheres constructed by Daniel Nash are all diffeomorphic to 4-sphere.

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…

Standard $S^4$ proved to be diffeomorphic to a curious homotopy sphere.

Study pochette surgery on 4-manifolds, focusing on 4-spheres.

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 $S^4$ and hence homeomorphic to $S^4$ but possibly not diffeomorphic to $S^4$). It is still unknown if these new examples are in fact exoti…

In relation to the 4-dimensional smooth Poincaré conjecture we construct a tentative invariant of homotopy 4-spheres using embedded contact homology (ECH) and Seiberg-Witten theory (SWF). But for good reason it is a constant value independent of the sphere, so this null-result demonstrates that one should not try to us…

New theory proves infinite homology 3-spheres in homology 4-spheres.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Gluck twisting certain knots results in standard 4-spheres.

The Price twist creates three 4-manifolds from a 4-sphere.

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.

Standardizes Dunfield-Gong's 4-sphere, solves knot sliceness problem.

The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $\#^n(S^1\times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diff…

New research finds 145 infinite families of CS spheres are standard.

D.Nash defined a family of homotopy 4-spheres in [11]. Proving that his manifolds ${\mathcal S}_{m,n,m',n'}$ are all real $S^4$, we find that they have handle decomposition with no 1-handles, two 2-handles and two 3-handles. The handle structures give new potential counterexamples of Property 2R conjecture.

We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.

The paper proves properties of branched covers of specific knots and tori.

A key open problem in M-theory is the mechanism of "gauge enhancement", which supposedly makes M-branes exhibit the nonabelian gauge degrees of freedom that are seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan-Paton gauge fields on D-…

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 mani…

Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akb…

We show the homotopy spheres $Σ_{n} = -W\smile_{f^{n}}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f: \partial W \to \partial W$ is $S^4$. To do this we first show that $Σ_{n}$ are obtained by Gluck twistings of $S^4$; then from this we show how to cancel $3$-han…

We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the $n$-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.

New bounds and examples for sphere unknotting numbers.

Perelman's proof confirmed, new method uses 4D topology.

New non-orientable 4-manifolds created via knotting operations.

Paper proves every stable 4-sphere has a unique diffeomorphism class.

A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.

Study on Ricci flow on 4-spheres, proving standard sphere convergence.

Turing complete flow on 4-sphere preserves volume.

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-hand…

Researchers found non-smoothable surfaces in a 4-sphere, solving K3 problems.

A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show…

New spanning 3-disks found for unlink in 4-sphere.

Here we study two interesting smooth contractible manifolds, whose boundaries have non-trivial mapping class groups. The first one is a non-Stein contractible manifold, such that every self diffeomorphism of its boundary extends inside; implying that this manifold can not be a loose cork. The second example is a Stein …

Smoothly knotted 5RP^2 found in 4-sphere.

Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some contractible 4-manifold in the 4-sphere.

New methods detect nontrivial loops of spheres in 4-manifolds.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

New findings about twists in 4-sphere diffeomorphisms.

3-balls in 4-sphere become isotopic in 5-ball.

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…

Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.

In this paper, Problem 4.17 on R. Kirby's problem list is solved by constructing infinitely many aspherical 4-manifolds that are homology 4-spheres

Researchers create infinite Brunnian links of 3-balls in 4-sphere.