arXiv research

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

New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.

Classifies two families of simply connected 7-manifolds with minimal homological complexity.

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in H^4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-s…

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

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.

Classifies torus bundles bounding 4-manifolds with rational homology.

Study $\mathrm{G}_2$-instantons on 7-sphere from 4-sphere instantons.

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

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…

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

New 3-manifolds bound rational 4-balls through specific operations.

Homological stability fails for Cremona groups, rational varieties, and function fields.

Classifies surgeries on torus knots and cables that bound rational homology balls.

Researchers found the second homology of Torelli groups for large g.

Study shows no smooth embeddings of rational homology balls into complex projective plane.

New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.

The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other …

Study shows Seifert fibered spaces don't bound rational homology balls.

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…

In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk $D^4$ (called a ration…

Study on rational projective planes with small index singularities.

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,…

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…

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

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 …

New families of Brieskorn spheres bound rational homology balls.

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology $S^1\times S^2$'s bound rational homology $S^1\times D^3$'…

We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.

Constructs chiral rational homology spheres with hyperbolic groups.

Knots generating infinite subgroup bound rational homology balls.

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

New knots bound rational homology balls, using Alexander polynomials.

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

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries…

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

Paper proves triple linking form vanishes under specific conditions.

Floer homology detects taut foliations in rational homology spheres.

The Farey tree helps embed rational balls and lens spaces into complex projective space.

Study Markov staircases in symplectic embeddings of rational homology ellipsoids.

Generalizes Lefschetz fibrations with rational homology disk smoothings.

We prove that there are rational homology balls $B_p$ smoothly embedded in the $2$-handlebodies associated to certain knots. Furthermore we show that, if we rationally blow up the $2$-handlebody along the embedded rational homology ball $B_p$, then the resulting $4$-manifold cannot be obtained just by a sequence of ord…

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