Related papers: The Farey tree and embeddings of lens spaces and r…

We introduce a family of 3-variable "Farey polynomials" that are closely connected with the geometry and topology of $3$-manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates…

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…

Motivated by prominent problems like the Hilali conjecture Yamaguchi--Yokura recently proposed certain estimates on the relations of the dimensions of rational homotopy and rational cohomology groups of fibre, base and total spaces in a…

In this note we study the Seifert rational homology spheres with two complementary legs, i.e. with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with 3 exceptional fibers and…

When does the double cover of the three-sphere branched along an alternating link bound a rational homology ball? Heegaard Floer homology generates a necessary condition for it to bound: the link's chessboard lattice must be cubiquitous,…

Motivated by Akbulut-Larson's construction of Brieskorn spheres bounding rational homology 4-balls, we explore plumbed 3-manifolds that bound rational homology circles and use them to construct infinite families of rational homology…

We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice…

The rational homology balls $B_n$ appeared in Fintushel and Stern's rational blow-down construction [FS2]. Later, Symington [Sy1], defined this operation in the symplectic category. In [Kh2], the author defined the inverse procedure, the…

We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$…

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…

We study rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements…

We use elementary arguments to prove that none of the Stein rational homology 4-balls shown by the authors and Brendan Owens to embed smoothly but not symplectically in the complex projective plane admit such almost complex embeddings. In…

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls into the standard complex projective plane $\mathbb{C}\mathrm{P}^2$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb{C}\mathrm{P}^2$,…

We characterize when some small Seifert fibered spaces can be the convex boundaries of symplectic rational homology balls and give strong restrictions for others to bound such manifolds. In particular, we show that the only spherical…

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and…

We determine when there exists a ribbon rational homology cobordism between two connected sums of lens spaces, i.e. one without $3$-handles. In particular, we show that if a lens space $L$ admits a ribbon rational homology cobordism to a…

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into $\mathbb{R}^n$. We view the space of embeddings as the value of a certain functor at…

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner's lambda lengths and $SL_2$-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian…

We examine the structure of Farey maps, which are a class of maps (graph embeddings on surfaces) that have received significant attention recently. We describe how they are related to each other through regular coverings and parallel…

We present a handlebody construction of small symplectic caps, and hence of small closed symplectic 4-manifolds. We use this to construct handlebody descriptions of symplectic embeddings of rational homology balls in…