Homology Spheres Research Articles

Dual submanifolds in rational homology spheres

Let $\Sigma$ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds $M_+, M_-$ in $\Sigma$ are called dual to each other if the complement $\Sigma - M_+$ strongly homotopy retracts onto $M_-$ or vice-versa. In this paper we will give a complete answer of which integral triples $(n; m_+, m_-)$ can appear, where $n=dim \Sigma -1$, $m_+={codim}M_+ -1$ and $m_-={codim}M_- -1$.

Four‐dimensional analogues of Dehn's lemma

AbstractWe investigate certain 4‐dimensional analogues of the classical 3‐dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential 2‐sphere in the boundary of a simply connected 4‐manifold such that is null‐homotopic in need not extend to an embedding of a ball in . However, if has abelian fundamental group with boundary a homology sphere, then bounds a topologically embedded ball in . Additionally, we give examples where such an does not bound any smoothly embedded ball in . In a similar vein, we construct incompressible tori where is a contractible 4‐manifold such that extends to a map of a solid torus in , but not to any embedding of a solid torus in . Moreover, we construct an incompressible torus in the boundary of a contractible 4‐manifold such that extends to a topological embedding of a solid torus in but no smooth embedding. As an application of our results about tori, we address a question posed by Gompf about extending certain families of diffeomorphisms of 3‐manifolds; he has recently used such families to construct infinite order corks.

Knot Concordance and Homology Sphere Groups

We study two homomorphisms to the rational homology sphere group. If $\psi$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $\psi$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $\psi$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $\beta$ denotes the homomorphism from the knot concordance group defined by taking double branched covers of knots, we prove that the kernel of $\beta$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls.

Embedding 3‐manifolds in spin 4‐manifolds

An invariant of orientable 3-manifolds is defined by taking the minimum n such that a given 3-manifold embeds in the connected sum of n copies of S 2 × S 2 , and we call this n the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.

Foliations, orders, representations, L-spaces and graph manifolds

We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard–Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions—using representations, using left-orders, using foliations, and using Heegaard–Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology 3-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that Heegaard–Floer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology 3-sphere graph manifolds as well as other interesting problems in Heegaard–Floer theory.

A class of knots with simple SU(2)-representations

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number $\geq 3$ which are $SU(2)$-simple. One expects the instanton knot Floer homology $I^\natural(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\det(K)$. In fact, the complex underlying $I^\natural(K)$ is of rank equal to $\det(K)$, provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram. With the methods we use, we show that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group.

Hyperbolic rational homology spheres not admitting fillable contact structures

In this short note, we exhibit an infinite family of hyperbolic rational homology $3$--spheres which do not admit any fillable contact structures. We also note that most of these manifolds do admit tight contact structures.

Bounds for Entries of $\gamma$-Vectors of Flag Homology Spheres

We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $\Delta$, we show that its $\gamma$-vector $\gamma^\Delta{\,=\,}(1,\gamma_1,\gamma_2,\ldots)$ satisfies: $\gamma_j{\,=\,}0$ for all $j>\gamma_1$, $\gamma_2\leq\binom{\gamma_1}{2}$, $\gamma_{\gamma_1}\in\{0,1\}$, and $\gamma_{\gamma_1-1}\in\{0,1,2,\gamma_1\}$, supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $\Delta$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $\gamma$-vectors of flag homology spheres (of any dimension); these are the first examples of this kind. In addition, we prove a flag analog of Perles' 1970 theorem on $k$-skeleta of polytopes with “few” vertices, specifically, the number of combinatorial types of $k$-skeleta of flag homology spheres with $\gamma_1\leq b$ of any given dimension, is bounded independently of the dimension.

Analytic singularities supported by a specific integral homology sphere link

The main question we target is the following: If one fixes a topological type of a complex normal surface singularity then what are the possible analytic types supported by it, and/or, what are the possible values of the geometric genus? We answer the question for a specific (in some sense pathological) topological type, which supports rather different analytic structures. These structures are listed together with some of their key analytic invariants.

A surgery formula for the asymptotics of the higher dimensional Reidemeister torsion and Seifert fibered spaces

We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits of the leading coefficients. There are finitely many possibilities as the limits of the leading coefficients for a Seifert fibered space. We also show that the maximum is given by the product of (-log 2) and the Euler characteristic of the base orbifold for a Seifert fibered space. These limits of the leading coefficients give a locally constant function on a character variety. This function takes the maximum only on the top-dimensional components of the SU(2)-character varieties for Seifert fibered homology spheres.

An invariant of rational homology 3–spheres via vector fields

We define an invariant of rational homology 3-spheres via vector fields. The construction of our invariant is a generalization of both that of the Kontsevich-Kuperberg-Thurston invariant and that of Watanabe's Morse homotopy invariant, which implies the equivalence of these two invariants.

Cosmetic surgery in L-spaces and nugatory crossings

The cosmetic crossing conjecture (also known as the “nugatory crossing conjecture”) asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.

Unified quantum invariants for integral homology spheres associated with simple Lie algebras

For each finite dimensional, simple, complex Lie algebra $\mathfrak g$ and each root of unity $\xi$ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $\tau_M^{\mathfrak g}(\xi)\in \mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant $J_M$ of integral homology spheres $M$ with values in the cyclotomic completion $\widehat {\mathbb Z [q]}$ of the polynomial ring $\mathbb Z [q]$, such that the evaluation of $J_M$ at each root of unity gives the WRT quantum invariant of $M$ at that root of unity. This result generalizes the case ${\mathfrak g}=sl_2$ proved by the first author. It follows that $J_M$ unifies all the quantum invariants of $M$ associated with $\mathfrak g$, and represents the quantum invariants as a kind of "analytic function" defined on the set of roots of unity. For example, $\tau_M(\xi)$ for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. It follows that WRT quantum invariants $\tau_M(\xi)$ for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at $q=1$, and hence by the Le-Murakami-Ohtsuki invariant. Another consequence is that the WRT quantum invariants $\tau_M^{ \mathfrak g}(\xi)$ are algebraic integers. The construction of the invariant $J_M$ is done on the level of quantum group, and does not involve any finite dimensional representation, unlike the definition of the WRT quantum invariant. Thus, our construction gives a unified, "representation-free" definition of the quantum invariants of integral homology spheres.

Surgery on tori in the 4–sphere

AbstractWe investigate the operation of torus surgery on tori embedded inS4. Key questions include which 4–manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3–manifolds into 4–manifolds by viewing Dehn surgery as a cross section of a surgery on a surface. In particular, we give new embeddings of homology spheres intoS4.

Surgery obstructions and Heegaard Floer homology

Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

We study in detail the large N expansion of {mathrm {SU}}(N) and {mathrm {SO}}(N)/{mathrm {Sp}}(2N) Chern–Simons partition function Z_N(M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a certain scalar, linear, non-local Riemann-Hilbert problem (RHP). We develop tools necessary to address a class of such RHPs involving finite subgroups of mathrm{PSL}_{2}({mathbb {C}}). We associate with such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. These techniques are applied to the RHP relevant for Chern–Simons theory on Seifert spaces. When pi _1(M) is finite—i.e., for manifolds M that are quotients of {mathbb {S}}_{3} by a finite isometry group of type ADE—we find that the Weyl group associated with the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large N expansion of Z_N(M) is computed by the topological recursion. This has consequences for the analyticity properties of {mathrm {SU}}/{mathrm {SO}}/{mathrm {Sp}} perturbative invariants of knots along fibers in M.

The Turaev and Thurston norms

In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a 3-manifold. Inspired by this definition, Turaev introduced in 2002 a analogous norm on the first cohomology group of a finite 2-complex. We show that if N is the exterior of a link in a rational homology sphere, then the Thurston norm agrees with a suitable variation of Turaev's norm defined on any 2-skeleton of N.

Towards Noncommutative Topological Quantum Field Theory: New invariants for 3-manifolds

We present some ideas for a possible Noncommutative Topological Quantum Field Theory (NCTQFT for short) and Noncommutative Floer Homology (NCFH for short). Our motivation is two-fold and it comes both from physics and mathematics: On the one hand we argue that NCTQFT is the correct mathematical framework for a quantum field theory of all known interactions in nature (including gravity). On the other hand we hope that a possible NCFH will apply to practically every 3-manifold (and not only to homology 3-spheres as ordinary Floer Homology currently does). The two motivations are closely related since, at least in the commutative case, Floer Homology Groups constitute the space of quantum observables of (3+1)-dim Topological Quantum Field Theory. Towards this goal we define some new invariants for 3-manifolds using the space of taut codim-1 foliations modulo coarse isotopy along with various techniques from noncommutative geometry.

L-space surgery and twisting operation

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient condition for { K_n } to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family { K_n } contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres $\Sigma (p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$.