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 .

Slopes of links and signature formulas

We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima–Yamasaki η \eta -function (in the univariate case) and Cochran invariants, on the other hand.

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.

CYCLIC BRANCHED COVERINGS OF BRIESKORN SPHERES BOUNDING ACYCLIC 4-MANIFOLDS

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.

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.

Concordance, Crossing Changes, and Knots in Homology Spheres

Any knot in $S^{3}$ can 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 that 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.

Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

We compute the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

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.