arXiv research

Proof outlined for 4D smooth Poincaré conjecture.

This is the announcement of an alternative approach to the 3-dimensional Poincaré Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come with our purely 4-dimensional line of argument.

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

Study of torus surgeries on knot traces, finding exotic surfaces and traces.

Study topological 4-manifolds with specific fundamental groups.

We show that the Spivak normal fibration of an orientable 4-dimensional Poincaré complex has a vector bundle reduction.

Study of knots sharing 0-surgeries, classifying and computing their properties.

Study rules out exotic $S^4$ and $\# n \mathbb{CP}^2$ construction using zero surgery homeomorphisms.

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…

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set o…

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen's s-invariant, a slice obstruction within the gener…

We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.

Proves classification of 4D complete intersections up to diffeomorphism.

4-manifolds with specific groups have unique homotopy types.

Approaches 4D Schoenflies via pseudo-isotopy.

Study on 4D Einstein manifolds with Kähler conformal geometry.

In [Cheeger-Tian 2005], Cheeger-Tian proved an $ε$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional manifolds and higher dim…

This memoire consists of two main results. In the first one we describe Ricci flow theory and we give an educative way for proving Elliptization Conjecture and then we prove Poincare conjecture which is the second proof of Perelman for Poincare conjecture. In the second one which is the main propose of our memoire, we …

New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.

Minimal hypersurfaces are the only $H$-tensional in 4D space forms.

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

In this paper, by use of techniques associated to cobordism theory and Morse theory,we give a simple proof of Poincare conjecture, i.e. Every compact smooth simply connected 3-manifold is homeomorphic to 3-sphere.

In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate

We discuss some of the key ideas of Perelman's proof of Poincaré's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.

In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the cla…

Author discusses the Poincaré conjecture from 40 years ago.

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

In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré con…

Study eight categorifications of colored Jones polynomial, verifying physics conjectures.

The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial a…

CMS formulation solves Poincare conjecture for all dimensions.

A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds. From this classification we then prove the Poincaré conjecture.

Decomposition theory explores topological spaces and their quotient spaces.

We prove a version of the Arezzo-Pacard-Singer blow-up theorem in the setting of Poincaré type metrics. We apply this to give new examples of extremal Poincaré type metrics. A key feature is an additional obstruction which has no analogue in the compact case. This condition is conjecturally related to ensuring the metr…

This manuscript contains a detailed proof of the Poincare Conjecture. The arguments we present here are expanded versions of the ones given by Perelman in his three preprints posted in 2002 and 2003. This is a revised version taking in account the comments of the referees and others. It has been reformatted in the AMS …

Geometrization Theorem solves complex geometry problems.

We establish a number of foundational results on Poincaré spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair $(X,Y)$ satisfying relative Poincaré duality in dimension …

Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.

The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential in…

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

A vector field on a Kähler manifold is called c-projective if its flow preserves the J-planar curves. We give a complete local classification of Kähler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equiv…

This paper confirms volumes of geodesic balls can identify 4D space forms.

In this paper we define the bi-orthogonal sectional curvature and we present two modified Yamabe invariants for compact 4-dimensional manifolds. In particular we obtained a relationship between one of these invariants and a Hopf conjecture.

Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.

The paper conjectures a 4D characterization of tight contact structures and proves it for certain cases.

We prove an analogue of the result of Hsiang and Kleiner for 4-dimensional compact orbifolds with positive curvature and an isometric circle action. Additionally, we prove that when the underlying space is simply connected, then the orbifold fundamental group provides a bound on the failure of integer-valued Poincare D…

The A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture for free groups, is shown to admit a link-homotopy+ solution. The proof relies on geometric applications of the group-theoretic 2-Engel relation. Implications for the surgery conjecture are discussed.