Branched coverings and nonzero degree maps between Seifert manifolds

This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with ℤ/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space

Let M n + 3 {M^{n + 3}} be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K 3 {K^3} with an effective circle action of hyperbolic type. Assume that the fiber N n {N^n} in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set S TOP ( N n × D k , ∂ ) {S_{\text {TOP}}}\,({N^n} \times {D^k},\partial ) of topological surgery vanishes for sufficiently large k k . Then the projective class and Whitehead groups of M M vanish and S TOP ( M n + 3 × D k , ∂ ) = 0 {S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0 if n + k ⩾ 3 n + k \geqslant 3 or if K 3 {K^3} is closed and n = 2 n = 2 . The UNil \text {UNil} groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNil j {\text {UNil}}_j vanishes in spite of the failure of Cappell’s sufficient condition.

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3 -manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C^{2} -smoothable.

On the monotonicity of amenable category

We study 3d mathcal{N}=2 supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q)b with rational squashing parameter b2 ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

Using the notion of rotation set for homeomorphisms of compact manifolds, we define the rotation homomorphism of a connected compact orientable Riemannian manifold and apply it to prove that the dimension of the isometry group of a connected compact orientable Riemannian 3-manifold without conjugate points is not greater than its first Betti number. In higher dimensions the same is true under the additional assumption that the fundamental cohomology class of the manifold is a cup product of integral 1-dimensional classes.

We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Seifert links in $S^3$. We also study the compatible contact structures of cablings along links in any 3-manifolds.

A $k$-submanifold $L$ of an open $n$-manifold $M$ is called {\it weakly integrable (WI)} [resp. {\it strongly integrable (SI)}] if there exists a submersion $\Phi:M\rightarrow {\Bbb{E}}^{n-k}$ such that $L\subset \Phi^{-1}(0)$ [resp. $L= \Phi^{-1}(0)$]. In this work we study the following problem, first stated in a particular case by Costa et al. ({\it Invent. Math.} 1988): which submanifolds $L$ of an open manifold $M$ are WI or SI? For general $M$, we explicitly solve the case $k=n-1$ and provide necessary and sufficient conditions for submanifolds to be WI and SI in higher codimension. As particular cases we recover the theorem of Bouma and Hector ({\it Indagationes Math.} 1983) asserting that any open orientable surface is SI in ${\Bbb{E}}^3$, and Watanabe's and Miyoshi's theorems ({\it Topology} 1993 and 1995) claiming that any link is WI in an open $3$-manifold. In the case $M={\Bbb{E}}^n$ we fully characterize WI and SI submanifolds, we provide examples of $3$- and $7$-manifolds which are not WI and we show that a theorem by Miyoshi ({\it Topology} 1995) which states that any link in ${\Bbb{E}}^3$ is SI does not hold in general. The right analogue to Miyoshi's theorem is also proved, implying in particular the surprising result that no knot in ${\Bbb{E}}^3$ is SI. Our results applied to the theory of foliations of Euclidean spaces give rise to some striking corollaries: using some topological invariants we classify all the submanifolds of ${\Bbb{E}}^n$ which can be realized as proper leaves of foliations; we prove that ${\Bbb{S}}^3$ can be realized as a leaf of a foliation of ${\Bbb{E}}^n$, $n \geq 7$, but not in ${\Bbb{E}}^5$ or ${\Bbb{E}}^6$, which partially answers a question by Vogt ({\it Math. Ann.} 1993); we construct open $3$-manifolds which cannot be leaves of a foliation of any compact $4$-manifold but are proper leaves in ${\Bbb{E}}^4$. The theory of WI and SI submanifolds is a framework where many classical tools of differential and algebraic topology play a prominent role: Phillips-Gromov h-principle, Hirsh-Smale theory, complete intersections, Seifert manifolds, the theory of immersions and embeddings, obstruction theory, etc.

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H 1( , ) with a free boundary condition of the type u(∂ ) ⊂ Γ, given a submanifold Γ⊂ . We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.

A condition ensuring metric subregularity (respectively, calmness) of general multifunctions between Banach spaces is derived. This condition is expressed solely in terms of the given data at the reference point and does not involve any information concerning the solution set of the corresponding inclusion given by the multifunction. In finite dimensions this condition can be expressed in terms of a derivative which appears to be a combination of the coderivative and the contingent derivative. It is further shown that this sufficient condition is in some sense the weakest possible first order condition sufficient for subregularity. We extend this condition under the additional assumption that one part of the multifunction is known to be subregular in advance. We also derive second order conditions for metric subregularity, both sufficient and necessary, for multifunctions associated with constraint systems as they occur in optimization. We show that the main difference between the necessary and sufficient conditions is the replacement of an inequality by a strict inequality, just as in the case of “no gap” second order optimality conditions in optimization.

Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let ℋ be a Hilbert space andA a von Neumann algebra. A dynamical semigroup Pt is a σ-weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup Pt possessing the property of preserving the identityI∈A is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA+. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra ofℒ∞(Rn) yields a series of new regularity conditions for both diffusion and jump processes.

CONTENTS Introduction Chapter I. Development of the Poincaré-Bendixson theory for flows on closed two-dimensional manifolds § 1. Possible types of trajectories. Singular and non-singular trajectories § 2. Classification of the connected components filled out by the non-singular trajectories Chapter II. Topological classification of flows S 3. The scheme of a flow on a sphere. Topological classification of Morse-Smale flows on two-dimensional manifolds § 4. The Poincaré rotation number of a flow on a torus and its properties. Transitive and singular flows on a torus § 5. The rotation homotopy class of half-trajectories of flows on closed orientable two-dimensional manifolds and its properties § 6. Necessary and sufficient conditions for the topological equivalence of transitive flows on closed orientable two-dimensional manifolds § 7. Necessary and sufficient conditions for the topological equivalence of non-trivial minimal sets of flows on closed orientable two-dimensional manifolds. Realization of non-trivial minimal sets by geodesic lines References

A recent result by Kardoš et al. (2023) related to the famous Berge–Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say O , in a bridgeless cubic graph, there exists a 1-factor intersecting all cycles in O in at least one edge. This remarkable result opens up natural generalizations in the case of an r -regular graph G and a t -factor F , with r and t being positive integers. In this paper, we start the study of this problem by proving necessary and sufficient conditions on G , t and r to assure the existence of a suitable F for any possible choice of the set O . First of all, we show that G needs to be 2-connected. Under this additional assumption, we highlight how the ratio t r seems to play a crucial role in assuring the existence of a t -factor F with the required properties by proving that t r ≥ 1 3 is a further necessary condition. We suspect that this condition is also sufficient, and we confirm it in the case t r = 1 3 , generalizing the case t = 1 and r = 3 proved by Kardoš, Máčajová, Zerafa, and in the case t r = 1 2 with t even. Finally, we provide further results for the case where even cycles are included.

A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming all the iterates \(f^n\) to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each \(p \ge 1\). It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and \(f^p\) is univalent in a neighbourhood of the boundary \( \partial D \) of D then f has no repelling p-periodic point on \( \partial D \). Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of \(f^n\) on W is either a parabolic periodic point or in the \(\omega \)-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the \(\omega \)-limit set of recurrent critical points.