For a closed 4-manifold X and closed 3-manifold M we investigate the smallest integer n (perhaps infinity) such that M embeds in the connected sum of n copies of X. It is proven that any lens space (or homology lens space) embeds topologically locally flatly in a connected sum of 8 copies of the complex projective plane.
It is known by the author that there exist 20 families of Dehn surgeries in the Poincaré homology sphere yielding lens spaces. In this paper, we give the concrete knot diagrams of the families and extend them to families of lens space surgeries in Brieskorn homology spheres. We illustrate families of lens space surgeries in $Σ(2,3,6n\pm1)$ and $Σ(2,2s+1,2(2s+1)n\pm1)$ and so on. As other examples, we give lens space surgeries in graph homology spheres, which are obtained by splicing two Brieskorn homology spheres.
Following the approach to grid homology of links in $S^3$, we prove combinatorially that the grid homology of links in lens spaces defined by Baker, Grigsby, and Hedden is a link invariant. Further, using the sign assignment defined by Celoria, we prove that the generalization of grid homology to integral coefficients is a link invariant.
It follows implicitly from recent work in Heegaard Floer theory that lens spaces are homology cobordant exactly when they are oriented homeomorphic. We provide a new combinatorial proof using the Heegaard Floer d-invariants, which themselves may be defined combinatorially for lens spaces.
A regular fiber of the Seifert fibering of the Poincaré homology sphere admits a Dehn surgery to $L(2,1)\#L(3,2)\#L(5,4)$. We prove that this is the only knot in the Poincaré homology sphere with a surgery to a connected sum of more than two lens spaces.
We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. In the proof of this corollary, we use Chebotarev density theorem.
In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let $M$ be a homology lens space with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot in $M$. We show that $K$ is determined by its complement if $M$ is non-hyperbolic, $K$ is hyperbolic, and $p$ is a prime more than 7, or, if $M$ is actually a lens space $L(p,q)$ and $K$ represents a generator of $H_1(L(p,q))$.
We present a combinatorial proof for the existence of the sign refined grid homology in lens spaces, and a self contained proof that $\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion in these groups.
We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincare homology sphere with a lens space surgery has either no symmetries or just a single strong involution.
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 element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of 2-bridge knots and any …
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 balls induced from Donaldson's diagonalization theorem and Heegaard Floer correction terms.
We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
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 $S^1\times D^3$'s. We give a simple procedure to construct rational homology cobordisms between plumbed 3-manifold. We introduce a family F of plumbed 3-manifolds with first Betti number equal to 1. By adapting an obstruction based on Donaldson's diagonalization theorem we characterize all manif…
We refine Matveev's result asserting that any two closed oriented 3-manifolds can be related by a sequence of borromean surgeries if and only if they have isomorphic first homology groups and linking pairings. Indeed, a borromean surgery induces a canonical isomorphism between the first homology groups of the involved 3-manifolds, which preserves the linking pairing. We prove that any such isomorphism is induced by a sequence of borromean surgeries. As an intermediate result, we prove that a given algebraic square finite presentation of the first homology group of a 3-manifold, which encodes t…
We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a corollary, we prove that if $Y$ is a linear combination of AR plumbed homology spheres with $μ(Y) = 1$, then $Y$ is not torsion in the homology cobordism group. A general computation of the involutive Heegaard Floer correction terms for these spaces is also included.
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 different lens space, then $L$ must be homeomorphic to $L(n,1)$, up to orientation-reversal. As an application, we classify ribbon $χ$-concordances between connected sums of $2$-bridge links. Our work builds on Lisca's work on embeddings of linear lattices.
We give some constraints on intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres in terms of the $\barμ$ invariant and compare them with those in terms of the $κ$ invariant. We also show that the difference between $κ$ and $-\barμ$ for a Seifert rational homology 3-sphere is at most 2, and under certain extra conditions the difference is 0.
This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under spin$^c$ conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed t…
We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to define an instanton Floer homology theory for sutured manifolds with tangles. As applications, we prove that the annular Khovanov homology (1) detects the unlink, (2) detects the closure of the trivial braid, and (3) distinguishes braid closures from other links; we also prove that …
Let $N$ be a prime 3-manifold that is not a closed graph manifold. Building on a result of Hongbin Sun and using a result of Asaf Hadari we show that for every $k\in\Bbb{N}$ there exists a finite cover $\tilde{N}$ of $N$ such that $|\operatorname{Tor} H_1(\tilde{N};\Bbb{Z})|>k$.
A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this paper, for a hyperbolic homology sphere $Y$ we derive explicit bounds on the relative grading between irreducible solutions to the Seiberg-Witten equations and the reducible one in terms of the spectral and Riemannian geometry of $Y$. Using this, we provide explicit bounds on some numerical invariants arising in monopole Floer homology (and its $\mathrm{Pin}(2)$-equivariant refinement). We apply this to st…
We study the homology of Riemannian manifolds of finite volume that are covered by an $r$-fold product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of hyperbolic planes. Using a variation of a method developed by Avramidi and Nguyen-Phan, we show that any such manifold $M$ possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic $r$-dimensional submanifolds whose fundamental classes are linearly independent in the homology group $H_r(M;\mathbb{Z})$.
We compute the Pin(2)-equivariant monopole Floer homology for the class of plumbed 3-manifolds with at most one "bad" vertex (in the sense of Ozsvath and Szabo). We show that for these manifolds, the Pin(2)-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Nemethi. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the Pin(2)-homology as an abelian group. As an application of this, we show that $β(-Y, s) = \barμ(Y, s)…
In the present paper, we construct a cusped hyperbolic $4$-manifold with all cusp sections homeomorphic to the Hantzsche-Wendt manifold, which is a rational homology sphere. By a result of Golénia and Moroianu, the Laplacian on $2$-forms on such a manifold has purely discrete spectrum. This shows that one of the main results of Mazzeo and Phillips from 1990 cannot hold without additional assumptions on the homology of the cusps. This also answers a question by Golénia and Moroianu from 2012. We also correct and refine the incomplete classification of compact orientable flat $3$-manifolds arisi…
