Homology Sphere Link Research Articles

Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.

The dimension of the image of the Abel map associated with normal surface singularities

Let (X, o) be a complex normal surface singularity with rational homology sphere link and let widetilde{X} be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes mathrm{Pic}(Z) of line bundles on Z. The set of possible values h^1(Z,mathcal {L}) for mathcal {L}in mathrm{Pic}(Z) can be understood in terms of the dimensions of the images of the Abel maps, as subspaces of mathrm{Pic}(Z). In this note we present two algorithms, which provide these dimensions. Usually, the dimension of mathrm{Pic}(Z) and of the dimension of the image of the Abel maps are not topological. However, we provide combinatorial formulae for them in terms of the resolution graph whenever the analytic structure on widetilde{X} is generic.

The relative Whitney trick and its applications

We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a 4-manifold with boundary.Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological 4-manifold. We further prove that any link in a homology sphere is order k Whitney tower concordant to a link in \(S^3\) for all k. Finally, we explore the minimum Gordian distance from a link in \(S^3\) to a homotopically trivial link. Extending this notion to links in homology spheres, we use the relative Whitney trick to make explicit computations for 3-component links and establish bounds in general.

The Abel map for surface singularities III: Elliptic germs

The present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the general theory, by the study of specific families we wish to show the power of this new method. Indeed, using the general theory of Abel maps applied for elliptic singularities in this note we are able to prove several key properties for elliptic singularities (e.g. the statements of the next paragraph), which by ‘old’ techniques were not reachable. If ({widetilde{X}},E)rightarrow (X,o) is the resolution of a complex normal surface singularity and c_1:{mathrm{Pic}}({widetilde{X}})rightarrow H^2({widetilde{X}},{mathbb {Z}}) is the Chern class map, then {mathrm{Pic}}^{l'}({widetilde{X}}):= c_1^{-1}(l') has a (Brill–Noether type) stratification W_{l', k}:= {{mathcal {L}}in {mathrm{Pic}}^{l'}({widetilde{X}}),:, h^1({mathcal {L}})=k}. In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. E.g., we show that the closure of any W(l',k) is an affine subspace. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.

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.

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.

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.

SOME SPLICE QUOTIENT DOUBLE POINTS

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links. In general, it can be difficult to determine whether or not a singularity is a splice quotient (an analytic condition). We consider splice quotient deformations of splice quotients of the form z2= xa+ yb, and show that in general not all equisingular deformations are splice quotients.

Nash problem for surface singularities is a topological problem

We address Nash problem for surface singularities using wedges. We give a refinement of the characterisation in A. Reguera-López (2006) [32] of the image of the Nash map in terms of wedges. Our improvement consists in a characterisation of the bijectivity of the Nash mapping using wedges defined over the base field, which are convergent if the base field is C, and whose generic arc has transverse lifting to the exceptional divisor. This improves the results of M. Lejeune-Jalabert and A. Reguera (2008) [16] for the surface case. In the way to do this we find a reformulation of Nash problem in terms of branched covers of normal surface singularities. As a corollary of this reformulation we prove that the image of the Nash mapping is characterised by the combinatorics of a resolution of the singularity, or, what is the same, by the topology of the abstract link of the singularity in the complex analytic case. Using these results we prove several reductions of the Nash problem, the most notable being that, if Nash problem is true for singularities having rational homology sphere links, then it is true in general.

On splice quotients of the form {zn = f (x,y )}

The splice quotients, defined by W. D. Neumann and J. Wahl, are an interesting class of normal surface singularities with rational homology sphere links. In general, it is difficult to determine whether or not a singularity is analytically isomorphic to a splice quotient, although there are certain necessary topological conditions. Let {zn = f(x, y)} define a surface Xf, n with an isolated singularity at the origin in . We show that for irreducible f, if (Xf, n, 0) satisfies the necessary topological conditions, then there exists a splice quotient of the form (Xg, n, 0), where the plane curve singularity defined by g = 0 has the same topological type as the one defined by f = 0. We also present an example of an (Xf, n, 0) that is not a splice quotient, but for which the universal abelian cover is a complete intersection of splice type together with a non-diagonal action of the discriminant group. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The end curve theorem for normal complex surface singularities

We prove the “End Curve Theorem,” which states that a normal surface singularity (X,o) with rational homology sphere link Σ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function (X,o) → (ℂ, 0) whose zero set intersects Σ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” (X,o) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in ℂ^t , where t is the number of leaves in the resolution graph for (X,o) , together with an explicit description of the covering transformation group. Among the immediate consequences of the End Curve Theorem are the previously known results: (X,o) is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).

Another proof of the end curve theorem for normal surface singularities

Neumann and Wahl introduced the notion of splice-quotient singularities, which is a broad generalization of quasihomogeneous singularities with rational homology sphere links, and proved the End Curve Theorem that characterizes splice-quotient singularities. The purpose of this paper is to give another proof of the End Curve Theorem. We use combinatorics of “monomial cycles” and some basic ring theory, whereas they applied their theory of numerical semigroups.

The geometric genus, the casson invariant conjecture and splice type singularities

AbstractThis is a survey of some results on splice-quotient singularities which are a natural and broad generalization of quasihomogeneous surface singularities with rational homology sphere links. From its topology (i.e., the link or the resolution graph), we can write down the “leading terms” of equations of a splice-quotient singularity, and compute the geometric genus. Applying the formula for the geometric genus, we can verify the Casson invariant conjecture for splice-quotient singularities.

Universal abelian covers of superisolated singularities

AbstractWe give explicit examples of Gorenstein surface singularities with integral homology sphere link, which are not complete intersections. Their existence was shown by Luengo–Velasco, Melle–Hernández and Némethi, thereby providing counterexamples to the universal abelian covering conjecture of Neumann and Wahl (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations''. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following.If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y 0, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y 0 and X is an equisingular deformation of the quotient Y 0/G.

ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

Signature invariants of covering links

We apply the theory of signature invariants of links in rational homology spheres to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, we derive an explicit formula to compute signature invariants of their covering links. Using the formula, we produce fused boundary links that are positive mutants of ribbon links but are not concordant to boundary links. We also show that for any finite collection of patterns, there are homology boundary links that are not concordant to any homology boundary links admitting a pattern in the collection.

Complex surface singularities with integral homology sphere links

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].

Links and analytic invariants of superisolated singularities

Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the “Seiberg-Witten invariant conjecture”(of Nicolaescu and the third author), the “Universal abelian cover conjecture” (of Neumann and Wahl) and the “Geometric genus conjecture” fail (at least at that generality in which they were formulated). Moreover, we also show that for Gorenstein singularities (even with integral homology sphere links) besides the geometric genus, the embedded dimension and the multiplicity (in particular, the Hilbert-Samuel function) also fail to be topological; and in general, the Artin cycle does not coincide with the maximal (ideal) cycle.

Complex surface singularities with integral homology sphere links

Complex surface singularities with integral homology sphere links