Homology Spheres Research Articles

Bordered Floer homology and existence of incompressible tori in homology spheres

Let$Y$be a homology sphere which contains an incompressible torus. We show that$Y$cannot be an$L$-space, i.e. the rank of$\widehat{\text{HF}}(Y)$is greater than$1$. In fact, if the homology sphere$Y$is an irreducible$L$-space, then$Y$is$S^{3}$, the Poincaré sphere$\unicode[STIX]{x1D6F4}(2,3,5)$or hyperbolic.

Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

We construct an obstruction to the existence of embeddings of a homology 3-sphere into a homology S3×S1 under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to Fintushel and Stern. We make use of the Z-fold covering space of homology S3×S1 and the instantons on it.

Exceptional cosmetic surgeries on homology spheres

The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small Seifert Z/2Z-homology sphere or a toroidal irreducible non-Seifert surgery then there is at most one pair of exceptional truly cosmetic slope. We also prove that toroidal truly cosmetic surgeries on integer homology spheres must be integer homology spheres.

Orderability and Dehn filling

Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold $M$ with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of $M$ have left-orderable fundamental groups. Our technique uses certain representations from $\pi_1(M)$ into $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.

Cosmetic surgery and the $SL(2,\mathbb{C})$ Casson invariant for two-bridge knots

We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. We first verify by using previously known results that all the two-bridge knots of at most $9$ crossings admit no purely cosmetic surgery pairs except for the knot $9_{27}$. Then we show that any two-bridge knot corresponding to the continued fraction $[0, 2x, 2, -2x, 2x, 2, -2x]$ for a positive integer $x$ admits no cosmetic surgery pairs yielding homology 3-spheres, where $9_{27}$ appears when $x = 1$. Our advantage to prove this is using the $SL(2,\mathbb{C})$ Casson invariant.

On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere

We show that, if a rational homology 3-sphere Y bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by Y. To this end, we make use of constraints on definite forms bounded by Y induced from Donaldson's diagonalization theorem, and correction term invariants due to Frøyshov, and Ozsváth and Szabó. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.

Heegaard Floer correction terms, with a twist

We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped torsion spin ^c structures, generalising the correction terms (or d -invariants) defined by Ozsváth and Szabó for rational homology 3-spheres and, more generally, for 3-manifolds with standard HF ^\infty . Our twisted correction terms share many properties with their untwisted analogues. In particular, they provide restrictions on the topology of 4-manifolds bounding a given 3-manifold.

VANISHING THEOREMS FOR HYPERSURFACES IN THE UNIT SPHERE

AbstractLet Mn, n ≥ 3, be a complete hypersurface in $\mathbb{S}$n+1. When Mn is compact, we show that Mn is a homology sphere if the squared norm of its traceless second fundamental form is less than $\frac{2(n-1)}{n}$. When Mn is non-compact, we show that there are no non-trivial L2 harmonic p-forms, 1 ≤ p ≤ n − 1, on Mn under pointwise condition. We also show the non-existence of L2 harmonic 1-forms on Mn provided that Mn is minimal and $\frac{n-1}{n}$-stable. This implies that Mn has only one end. Finally, we prove that there exists an explicit positive constant C such that if the total curvature of Mn is less than C, then there are no non-trivial L2 harmonic p-forms on Mn for all 1 ≤ p ≤ n − 1.

Heegaard Floer homology and knots determined by their complements

In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology $\mathbb{R}P^3$'s are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in $L$-spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than $1$ in the Brieskorn sphere $\Sigma(2,3,7)$ are also determined by their complements.

On topological actions of finite groups on S3

We consider orientation-preserving actions of a finite group G on the 3-sphere S3 (and also on Euclidean space R3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal group SO(4) (or of SO(3) in the case of R3). On the other hand, there are topological actions with wildly embedded fixed point sets; such actions are not conjugate to smooth actions but one would still expect that the corresponding groups G are isomorphic to subgroups of the orthogonal groups SO(4) (or of SO(3), resp.). In the present paper, we obtain some results in this direction; we prove that the only finite, nonabelian simple group with a topological action on S3, or on any homology 3-sphere, is the alternating or dodecahedral group A5 (the only finite, nonabelian simple subgroup of SO(4)), and that every finite group with a topological, orientation-preserving action on Euclidean space R3 is in fact isomorphic to a subgroup of SO(3).

On loose Legendrian knots in rational homology spheres

We prove that loose Legendrian knots in a rational homology contact 3-sphere, satisfying some additional hypothesis, are Legendrian isotopic if and only if they have the same classical invariants. The proof requires a result of Dymara on loose Legendrian knots and Eliashberg's classification of overtwisted contact structures on 3-manifolds.

A note on surgery obstructions and hyperbolic integer homology spheres

Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition.

On smooth manifolds with the homotopy type of a homology sphere

In this paper we study M(X), the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of X, via a map Ψ from M(X) into the quotient of K(X)=[X,BSO] by the action of the group of homotopy classes of simple self equivalences of X. The map Ψ describes which bundles over X can occur as normal bundles of manifolds in M(X). We determine the image of Ψ when X belongs to a certain class of homology spheres. In particular, we find conditions on elements of K(X) that guarantee they are pullbacks of normal bundles of manifolds in M(X).

Splice diagram singularities and the universal abelian cover of graph orbifolds

Given a rational homology 3-sphere M whose splice diagram $$\varGamma (M)$$ satisfies the semigroup condition, Neumann and Wahl define a complete intersection surface singularity called a splice diagram singularity. Under an additional hypothesis on M called the congruence condition they show that the link of this singularity is the universal abelian cover of M. They ask if this still holds if the congruence condition fails. In this article we generalize the congruence condition to orientable graph orbifolds. We show that under a small additional hypothesis this orbifold congruence condition implies that the link of the splice diagram singularity is the universal abelian cover. By showing that any two-node splice diagram satisfying the semigroup condition is the splice diagram of an orbifold satisfying the orbifold congruence condition, we answer the question of Neumann and Wahl affirmatively for two-node diagrams. However, examples show this approach to their question no longer works for three nodes.

A noncommutative generalization of Thurston’s gluing equations

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a noncommutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, Kashaev, [Formula: see text]-groupoids in knot theory, Geom. Dedicata 150(1) (2010) 105–130; Kashaev, Noncommutative teichmüller spaces and deformation varieties of knot completeness; Kashaev, Delta-groupoids and ideal triangulation in Chern–Simons gauge theory: 20 Years After, AMS/IP Studies Advanced Mathematics, Vol. 50 (American Mathematical Society, RI, 2011). In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyze some examples on other manifolds.

The surgery exact triangle in Pin(2)–monopole Floer homology

We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this triangle to describe the invariants associated to homology spheres obtained by (\pm1)-surgery on alternating knots.

Heegaard Floer homology of Matsumoto's manifolds

We consider a homology sphere Mn(K1,K2) presented by two knots K1, K2 with linking number 1 and framing (0,n). We call the manifold Matsumoto's manifold. We show that Mn(T2,3,K2) never bounds any contractible 4-manifold if n<2τ(K2) holds. We also give a formula of Ozsváth–Szabó's τ-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the δ-invariants of the twisted Whitehead doubles of torus knots and correction terms of the branched covers of the Whitehead doubles. By using Owens and Strle's obstruction we show that the 12-twisted Whitehead double of the (2,7)-torus knot and the 20-twisted Whitehead double of the (3,7)-torus knot are not slice but the double branched covers bound rational homology 4-balls. These are new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4-ball.

INVOLUTIVE HEEGAARD FLOER HOMOLOGY AND PLUMBED THREE-MANIFOLDS

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.

A distribution of rational homology 3-spheres captured by the CWL invariant Phase 1

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot [Formula: see text] with [Formula: see text].

Independence of satellites of torus knots in the smooth concordance group

The main goal of this article is to obtain a condition under which an infinite collection $\mathscr{F}$ of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining both the instanton moduli space over a 4-manifold with tubular ends and the corresponding Chern-Simons invariant of the adequate 3-dimensional portion of the 4-manifold. More specifically, the result is derived from Furuta's criterion for the independence of Seifert fibred homology spheres in the homology cobordism group of oriented homology 3-spheres. Indeed, we first associate to $\mathscr{F}$ the corresponding collection of 2-fold covers of the 3-sphere branched over the elements of $\mathscr{F}$ and then introduce definite cobordisms from the aforementioned covers of the satellites to a number of Seifert fibered homology spheres. This allows us to apply Furuta's criterion and thus obtain a condition that guarantees the independence of the family $\mathscr{F}$ in the smooth concordance group.