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 $$({\widetilde{X}},E)\rightarrow (X,o)$$ ( X ~ , E ) → ( X , o ) is the resolution of a complex normal surface singularity and $$c_1:{\mathrm{Pic}}({\widetilde{X}})\rightarrow H^2({\widetilde{X}},{\mathbb {Z}})$$ c 1 : Pic ( X ~ ) → H 2 ( X ~ , Z ) is the Chern class map, then $${\mathrm{Pic}}^{l'}({\widetilde{X}}):= c_1^{-1}(l')$$ Pic l ′ ( X ~ ) : = c 1 - 1 ( l ′ ) has a (Brill–Noether type) stratification $$W_{l', k}:= \{{\mathcal {L}}\in {\mathrm{Pic}}^{l'}({\widetilde{X}})\,:\, h^1({\mathcal {L}})=k\}$$ W l ′ , k : = { L ∈ Pic l ′ ( X ~ ) : h 1 ( 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 $$W(l',k)$$ 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.

Euler classes of vector bundles over manifolds

Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

Vector Fields and Genus in Dimension 3

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.

AN INVARIANT OF LEGENDRIAN AND TRANSVERSE LINKS FROM OPEN BOOK DECOMPOSITIONS OF CONTACT 3-MANIFOLDS

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.

Entropy in uniformly quasiregular dynamics

Let $M$ be a closed, oriented, and connected Riemannian $n$ -manifold, for $n\geq 2$ , which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f:M\rightarrow M$ , the topological entropy $h(f)$ is $\log \deg f$ . This proves Shub’s entropy conjecture in this case.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.

APPLICATIONS OF INVOLUTIVE HEEGAARD FLOER HOMOLOGY

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms$\bar{d}$and$\text{}\underline{d}$for certain families of three-manifolds.

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].

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.

Invariants of open books of links of surface singularities

If M is the link of a complex normal surface singularity, then it carries a canonical contact structure ξcan, which can be identified from the topology of the 3-manifold M. We assume that M is a rational homology sphere. We compute the support genus, the binding number and the norm associated with the open books which support ζcan, provided that we restrict ourselves to the case of (analytic) Milnor open books. In order to do this, we determine monotonity properties of the genus and the Milnor number of all Milnor fibrations in terms of the Lipman cone.We generalize results of [3] valid for links of rational surface singularities, and we answer some questions of Etnyre and Ozbagci [7, section 8] regarding the above invariants.