arXiv research

Study shows mapping class groups differ for $h$-cobordant manifolds.

The study explores conditions for $h$-cobordisms between smooth 4-manifolds.

We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.

We study the homeomorphism types of manifolds h-cobordant to a fixed one. Our investigation is partly motivated by the notion of special manifolds introduced by Milnor in his study of lens spaces. In particular we revisit and clarify some of the claims concerning h-cobordisms of these manifolds.

Study $h$-cobordisms of complexity 2 in 5D, finding obstructions and examples.

Constructs a functor for equivariant smooth h-cobordisms.

We consider ribbon $R^4$'s, that is, smooth open 4-manifolds, homeomorphic to $R^4$ and associated to $h$-cobordisms between closed 4-manifolds. We show that any generalized ribbon $R^4$ associated to a sequence of $h$-cobordisms between non-diffeomorphic 4-manifolds is exotic. Notion of a positive ribbon $R^4$ is defi…

In this note we classify the diffeomorphism classes rel. boundary of smooth h-cobordisms between two fixed 1-connected 4-manifolds in terms of isometries between the intersection forms.

We prove that any two smooth h-cobordant simply-connected 4-manifolds can be obtained by taking two manifolds with boundary, one of which is contractible, and gluing them along the boundary via two different attaching maps.

Alexander trick applied to homology spheres for manifold homeomorphisms.

We give a sufficient and necessary condition of the fundamental group homomorphism of a map between manifolds to induce homology equivalences. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.

This is primarily an exposition, combining work of several authors (Curtis, Hsiang, Freedman, Stong, Matveyev, and Bizaca), of the proof that a smooth 5-dimensional h-cobordism between simply connected 4-manifolds is a product off of a contractible piece which itself is diffeomorphic to the 5-ball.

We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from X to X. As an application, we give results restricting the minimal number of do…

The abstract explains counterexamples in 4-manifold topology.

In this paper, we develop a geometric procedure for producing a reverse to Quillen's plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudo-collars, which are stackings of …

Computes invariant for smooth h-cobordisms families, proving duality and vanishing theorems.

The paper constructs infinitely many $G$-smoothings of a $G$-manifold.

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nas…

We study the Whitehead torsions of inertial h-cobordisms, and identify various types representing a nested sequence of subsets of the Whitehead group. A number of examples are given to show that these subsets are all different in general.

We construct a map from the suspension $G$-spectrum $Σ_G^\infty M$ of a smooth compact $G$-manifold to the equivariant $A$-theory spectrum $A_G(M)$, and we show that its fiber is, on fixed points, a wedge of stable $h$-cobordism spectra. This map is constructed as a map of spectral Mackey functors, which is compatible …

New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.

Let (W,M,M'), dim W > 5, be a non-trivial h-cobordism (i.e., the Whitehead torsion of (W,V) is non-zero). We prove that every smooth function f: W --> [0,1], f(M)=0, f(M')=1 has at least 2 critical points. This estimate is sharp: W possesses a function as above with precisely two critical points.

New examples of manifolds that are homotopy but not simple homotopy equivalent.

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conje…

Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Ca…

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-…

New complexity measure for shake-slice knots established.

The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A rather striking amount of work on pseudoisotopy theory [Hatcher, …

The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds. Related conjectures on…

New examples of manifolds with similar homotopy but different simple homotopy types.

We interpret heterotic M-theory in terms of h-cobordism, that is the eleven-manifold is a product of the ten-manifold times an interval is translated into a statement that the former is a cobordism of the latter which is a homtopy equivalence. In the non-simply connected case, which is important for model building, the…

Homology handles with trivial Alexander polynomial bound a 3D sphere.

The study bounds invariants of PL manifolds and counts complexity of lens spaces.

Let $Λ$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $Λ$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $Λ$, and such that the generating family homology is canonically isomorphic to the endomorphism algebra of this she…

Kawauchi defined a group structure on the set of homology $S^1$\times$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injecti…

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…

Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. We classify the set of equivariant homeomorphism classes of free $C_\ell$-actions on the product $S^1 \times S^n$ of spheres, up to indeterminacy bounded in $\ell$. The description is expressed in terms of number theory. The techniques are various appl…

Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.

Extends Tian theorem to Vaisman manifolds for approximations.

In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…

Study on a new type of manifolds that generalize almost C-manifolds.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

New class of complex manifolds defined, properties studied.

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…

Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Condition for intersection of real flag manifolds in complex flag manifold.

The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.