arXiv research

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.

In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit…

New families of Brieskorn spheres bound rational homology balls.

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…

Constructs chiral rational homology spheres with hyperbolic groups.

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.

New Sasaki-Einstein 7-spheres found via Berglund-Hübsch transpose.

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

The paper examines conditions for contact surgeries on rational homology 3-spheres.

A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermor…

In this paper, we develop a method for constructing left-orders on the fundamental groups of rational homology 3-spheres. We begin by constructing the holonomy extension locus of a rational homology solid torus $M$, which encodes the information about peripherally hyperbolic $\widetilde{\text{PSL}_2\mathbb{R}}$ represe…

New findings restrict Heegaard Floer homology for certain rational homology spheres.

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

Study spectral gaps in hyperbolic rational homology spheres.

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behavi…

Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.

Fintushel and Stern showed that the Brieskorn sphere $Σ(2,3,7)$ bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…

Floer homology detects taut foliations in rational homology spheres.

The study of symmetries in manifolds derived from colored polytopes.

We give a new construction of monopole Floer homology for spin-c rational homology 3-spheres. As applications we define two invariants of certain smooth compact 4-manifolds with b_1=1 and b^+=0.

In this article, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules A equipped with their Blanchfield forms $φ$, and the modules A such that there is a unique isomorphism class of $(A,φ)$, and we prove that for the other modules A, there ar…

Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.

The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.

Local knots can't bound smaller surfaces in rational homology 3-spheres.

The study sets constraints on 4-manifold forms linked to specific invariants.

We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of …

We prove that the mod Z reduction of the torsion of a rational homology 3-sphere is completely determined by three data: a certain canonical spin^c structure, the linking form and a Q/Z-valued constant c. This constant is a new topological invariant of the rational homology sphere. Experimentations with lens spaces sug…

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

Using the Heegaard Floer homology of Ozsvath and Szabo we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.

We prove that the Seiberg-Witten invariants of a rational homology sphere are determined in a very explicit fashion by the Casson-Walker invariant and the Reidemeister torsion

Paper finds new 3D shapes that can be inside a 4D space.

New instanton invariants for rational homology spheres defined and shown to be functorial.

In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient …

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…

Knots generating infinite subgroup bound rational homology balls.

The paper constructs a hyperbolic 4-manifold with rational homology sphere cusp sections.

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel…

We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial …

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

Estimates Manolescu's κ-invariant using spin 4-orbifolds.

The study confirms the non-existence of rational homology ball symplectic fillings for certain Brieskorn spheres.

Study on negative Sasakian structures on specific 5-manifolds.

It was shown in my earlier article that the splice diagram of a rational homology sphere graph manifold determines the manifolds universal abelian cover. In this article we use the proof of this to give a condition on the splice diagram to determine when the universal abelian cover itself is a rational homology sphere.

We show that the perturbative ${\frak g}$ invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra ${\frak g}$, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …