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Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$., Comment: 10 pages

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 Gol\'enia 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 Gol\'enia and Moroianu from 2012. We also correct and refine the incomplete classification of compact orientable flat $3$-manifolds arising from cube colourings provided earlier by the last two authors., Comment: 15 pages, 1 figure, 1 table; SageMath worksheets available at https://github.com/sashakolpakov/24-cell-colouring

Given a vector field on a 3D rational homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. This provides a formula for the genus of a transverse surface, and in particular, of a Birkhoff section. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to helicity.

We introduce a generalization of the Lisca–Ozsváth–Stipsicz–Szabó Legendrian invariant ${\mathfrak L}$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link L in a contact 3-manifold ${(M,\xi)}$ with a diagram D, given by an open book decomposition of ${(M,\xi)}$ adapted to L, and we construct a chain complex ${cCFL^-(D)}$ with a special cycle in it denoted by ${\mathfrak L(D)}$ . Then, given two diagrams ${D_1}$ and ${D_2}$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes that induces an isomorphism in homology and sends ${\mathfrak L(D_1)}$ into ${\mathfrak L(D_2)}$ . Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of ${\xi}$ on their complement is tight.

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions., 15 pages. Final version - to appear in the Glasgow Mathematical Journal

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., 15 pages, 7 figures. Comments welcome. Aside from minor edits, Version 3, rewrite of appendix and minor revisions in the body suggested by the referee, to appear in Pacific J. Math., Version 2 includes an appendix by Neil R. Hoffman which answers Question 10(2) of Version 1

Kricker defined an invariant of knots in homology 3-spheres which is a rational lift of the Kontsevich integral, and proved with Garoufalidis that this invariant satisfies splitting formulas with respect to a surgery move called null-move. We define a functorial extension of the Kricker invariant and prove splitting formulas for this functorial invariant with respect to null Lagrangian-preserving surgery, a generalization of the null-move. We apply these splitting formulas to the Kricker invariant., Minor corrections. References updated. To appear in Algebraic & Geometric Topology

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere., Comment: 10 pages, 1 figure. Changes from Version 1: Theorem 1.6 from version 1 was previously proven by the same technique by Austin-Rolfsen. The result of Theorem 1.9 frmo version 1 appears in a remark of Daemi

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact--embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields., Comment: The presentation has been improved thanks to a referee's report; version coinciding with the published one. (Comments v2: The preprint has been reworked. Most important changes: Section 4 has been improved, a mistake in the old statements of Theorem G and Corollary H has been rectified, and the form has been ameliorated throughout all the manuscript. 50 pages. Comments v1: 73 pages)

We prove the Lipman-Zariski conjecture for complex surface singularities of genus one, and also for those of genus two whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal compact complex surfaces whose smooth locus has trivial tangent bundle. We also deduce that all complex-projective surfaces with locally free and generically nef tangent sheaf are smooth, and we classify them., Comment: Published version is slightly shortened

Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M({{\mathcal {T}}})$$\end{document}M(T) is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document}T. We consider the combinatorial multivariable Poincaré series associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document}T and its counting functions, which encode rich topological information. Using the ‘periodic constant’ of the series (with reduced variables associated with an arbitrary subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}$$\end{document}I of the set of vertices) we prove surgery formulae for the normalized Seiberg–Witten invariants: the periodic constant associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}$$\end{document}I appears as the difference of the Seiberg–Witten invariants of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M({{\mathcal {T}}})$$\end{document}M(T) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M({{\mathcal {T}}}{\setminus }{{\mathcal {I}}})$$\end{document}M(T\I) for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}$$\end{document}I.

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study a family of singularities called weighted Le–Yomdin, which provide infinitely many examples of surface singularities with a rational homology sphere link. To end this paper we also study a family of surface singularities generalizing Brieskorn–Pham singularities in a different direction. This family contains infinitely many examples of integral homology sphere links, answering a question by Nemethi.

We compare three naturally occurring multi-indexed filtrations of ideals on the local ring of a Newton nondegenerate hypersurface surface singularity with rational homology sphere, which in many cases are all distinct. These are the divisorial, the order, and the image filtrations. These filtrations are indexed by the lattice associated with a toric partial resolution of the singularity, or equivalently, the free Abelian group generated by the compact facets of the Newton polyhedron.

A three-dimensional orbifold $(\Sigma, \gamma_i, n_i)$, where $\Sigma$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where $C$ is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients., Comment: 22 pages;referee remarks incorporated; definitions of "orbifold" and "singular pair" contrasted with alternative usage of terms; algebro-geometric proof of existence of UALC (Theorem 3.1); Section 7 example corrected

For each integral homology sphere $Y$, a function $\Gamma_Y$ on the set of integers is constructed. It is established that $\Gamma_Y$ depends only on the homology cobordism of $Y$ and it recovers the Fr{\o}yshov invariant. A relation between $\Gamma_Y$ and Fintushel-Stern's $R$-invariant is stated. It is shown that the value of $\Gamma_Y$ at each integer is related to the critical values of the Chern-Simons functional. Some topological applications of $\Gamma_Y$ are given. In particular, it is shown that if $\Gamma_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself., Comment: 44 pages. Comments are welcome! v2: Remark 1.6 (due to Levin and Lidman) added to the paper, references updated

We introduce techniques to study the topology of Stein fillings of a given contact three-manifold (Y,ξ) which are not negative definite. For example, given a spinc rational homology sphere (Y,s) with s self-conjugate such that the reduced monopole Floer homology group HM∙(Y,s) has dimension one, we show that any Stein filling which is not negative definite has b+2=1 or 2, and b−2 is determined in terms of the Froyshov invariant. The proof of this uses Pin(2)-monopole Floer homology. More generally, we prove that analogous statements hold under certain assumptions on the contact invariant of ξ and its interaction with Pin(2)-symmetry. We also discuss consequences for finiteness questions about Stein fillings.

Let G be S_{5} or SL(2, 5) and leet Sigma be a homology sphere with smooth G-action such that the G-fixed point set consists of oddnumber points. Then the dimension of Sigma could be restrictive. In this article, we report results on the dimension of Sigma and on the tangential G-representation of a G-fixed point in Sigma.

In the paper we investigate continuity of Orlicz–Sobolev mappings $$W^{1,P}(M,N)$$ of finite distortion between smooth Riemannian n-manifolds, $$n\ge 2$$ , under the assumption that the Young function P satisfies the so-called divergence condition $$\int _1^\infty P(t)/t^{n+1}\, \hbox {d}t=\infty $$ . We prove that if the manifolds are oriented, N is compact, and the universal cover of N is not a rational homology sphere, then such mappings are continuous. That includes mappings with $$Df\in L^n$$ and, more generally, mappings with $$Df\in L^n\log ^{-1}L$$ . On the other hand, if the space $$W^{1,P}$$ is larger than $$W^{1,n}$$ (for example if $$Df\in L^n\log ^{-1}L$$ ), and the universal cover of N is homeomorphic to $$\mathbb {S}^n$$ , $$n\ne 4$$ , or is diffeomorphic to $$\mathbb {S}^n$$ , $$n=4$$ , then we construct an example of a mapping in $$W^{1,P}(M,N)$$ that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: Both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold N.