Homology Spheres Research Articles

The Signature of a Splice

We study the behavior of the signature of colored links [Flo05, CF08] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it.

On the Iwasawa $\mu$-invariants of branched $\mathbf{Z}_{p}$-covers

Following the analogies between knots and primes, we establish relative genus theory for a branched cover of rational homology 3-spheres. Then we formulate analogues of Iwasawa’s theorems on $\mu$-invariants for branched $\mathbf{Z}_{p}$-covers of rational homology 3-spheres, by using relative genus theory.

Cyclic group actions on contractible 4–manifolds

There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.

Satellite operators as group actions on knot concordance

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.

Some arithmetic properties of the CWL invariant

To derive information about rational homology 3-spheres via the Casson–Walker–Lescop (CWL) invariant λ, this paper focuses on the fractional part of the invariant, completes the determination of the fractional parts of λ for homology lens spaces, shows that the fractional part is an invariant of the linking forms mod Z for homology lens spaces and gives a complete set of invariants derived from λ to classify the linking forms mod Z.

The E8-boundings of homology spheres and negative sphere classes in E(1)

We define topological invariants of homology 3-sphere, ds and ds_, which are the maximal and minimal second Betti number divided by 8 among definite spin boundings of the homology sphere. We also define similar invariants g8 and g8_ by the maximal (or minimal) product sum of the quadratic form E8 of bounding 4-manifolds. The aim of these invariants is to measure the size of bounding definite spin 4-manifold. We give several ways to construct definite spin boundings. In particular, we construct uncommon E8-boundings for Σ(2,3,12n+5) by using handle decomposition. As a by-product of this construction, we show that some negative 2nd homology classes k[f]−[s] in E(1) are represented by a sphere, where f and s are a fiber and sectional class of E(1).

The geometric genus of hypersurface singularities

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.

Pretzel knots with L-space surgeries

A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.

Non-left-orderable surgeries on twisted torus knots

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in the 3-sphere, it is natural to ask what conditions on the knot group are sufficient to imply that the quotient associated to Dehn surgery is not left-orderable. Clay and Watson develop a criterion for determining the left-orderability of this quotient group and use it to verify the conjecture for surgeries on certain L-space twisted torus knots. We generalize a recent theorem of Ichihara and Temma to provide another such criterion. We then use this new criterion to generalize the results of Clay and Watson and to verify the conjecture for a much broader class of L-space twisted torus knots.

Stein fillings of homology 3-spheres and mapping class groups

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology 3-sphere supported by an open book with page a 4-holed sphere admits a unique Stein filling up to symplectic deformation. Furthermore, according to a property of deforming symplectic fillings of rational homology 3-spheres into strong symplectic fillings, we also show that a symplectically fillable integral homology 3-sphere supported by an open book with page a 4-holed sphere admits a unique symplectic filling up to symplectic deformation and blow-up.

Extending smooth cyclic group actions on the Poincaré homology sphere

Let $X_0$ denote a compact, simply-connected smooth $4$-manifold with boundary the Poincar\'e homology $3$-sphere $\Sigma(2,3,5)$ and with even negative definite intersection form $Q_{X_0}=E_8$. We show that free $\mathbb{Z}/p$ actions on $\Sigma(2,3,5)$ do not extend to smooth actions on $X_0$ with isolated fixed points for any prime $p>7$. The approach is to study the equivariant version of the Yang-Mills instanton-one moduli space for $4$-manifolds with cylindrical ends. As an application we show that for $p>7$ a smooth $\mathbb{Z}/p$ action on $\#^8 S^2 \times S^2$ with isolated fixed points does not split along a free action on $\Sigma(2,3,5)$. The results hold for $p=7$ if the action is homologically trivial.

The SL(2, C) Casson Invariant for Knots and the Â-polynomial

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

Knot lattice homology in L-spaces

We show that the knot lattice homology of a knot in an [Formula: see text]-space is chain homotopy equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring [Formula: see text]). Suppose that [Formula: see text] is a negative definite plumbing tree which contains a vertex [Formula: see text] such that [Formula: see text] is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology [Formula: see text] is isomorphic to the Heegaard Floer homology [Formula: see text] of the corresponding rational homology sphere [Formula: see text].

Linking forms revisited

We show that the Q/Z-valued linking forms on rational homology spheres are (anti-) symmetric and we compute the linking form of a 3-dimensional rational homology sphere in terms of a Heegaard splitting. Both results have been known to a larger or lesser degree, but it is difficult to find rigorous down-to-earth proofs in the literature.

NONSURJECTIVE SATELLITE OPERATORS AND PIECEWISE-LINEAR CONCORDANCE

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.

Random Methods in 3-Manifold Theory

We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface.

Random methods in 3-manifold theory

We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface.

Floer homology and splicing knot complements

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $\widehat{\mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $\widehat{\mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$\big|(h_\infty^1-h_1^1)(h_\infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)\big|.$$ We also show that if splicing the complement of a knot $K\subset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.

On left-orderability and double branched covers of Kanenobu’s knots

We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.

Brieskorn manifolds, positive Sasakian geometry, and contact topology

Abstract Using S 1 ${S^{1}}$ -equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn–Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of S 2 × S 3 ${S^{2}\times S^{3}}$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki–Einstein metrics on certain homotopy spheres. Finally, a new family of Sasaki–Einstein metrics of real dimension 20 on S 5 ${S^{5}}$ is exhibited.