Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

A spectral sequence is established whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology in Khovanov's classification.

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n − 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; {\mathbb Z} )$. Our main result is a readily calculable classification of embeddings $N \to {\mathbb R}^7$ up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9.

The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$, with an indeterminacy.

Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008.

For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$.

Following Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds.

We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve.

As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specializing to half-dimensional torus actions.

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$-type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic $4$-manifolds have sets $\pi _0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi _1$. The result about $\pi _0$ is well-known and elementary, but the result about $\pi _1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$. We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

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.

Building upon our earlier work with M. C. Hughes, we construct many new smooth structures on closed simply connected nonspin $4$-manifolds with positive signature. We also provide numerical and asymptotic upper bounds on the function $\lambda (\sigma )$ that was defined in our earlier work.

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.

We use the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the Khovanov complex to give an alternative way to compute Rasmussen s-invariants of knots. By disregarding generators away from homological degree 0, we can considerably improve the efficiency of the algorithm. With a slight modification, we can also apply it to a refinement of Lipshitz–Sarkar.

We observe an inductive structure in a large class of Artin groups of finite real, complex and affine types and exploit this information to deduce the Farrell–Jones isomorphism conjecture for these groups.

We obtain a new theorem for the non-properness set $S_f$ of a non-singular polynomial mapping $f:\mathbb C^n \to \mathbb C^n$. In particular, our result shows that if f is a counterexample to the Jacobian conjecture, then $S_f\cap Z \neq \emptyset $, for every hypersurface Z dominated by $\mathbb C^{n-1}$ on which some non-singular polynomial $h: \mathbb C^{n}\to \mathbb C$ is constant. Also, we present topological approaches to the Jacobian conjecture in $\mathbb C^n$. As applications, we extend bidimensional results of Rabier, Lê and Weber to higher dimensions.

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.