Topology Of Manifolds Research Articles

When is the underlying space of an orbifold a manifold?

We classify orthogonal actions of finite groups on Euclidean vector spaces for which the corresponding quotient space is a topological, homological, or Lipschitz manifold, possibly with boundary. In particular, our results answer the question of when the underlying space of an orbifold is a manifold.

Infinite series in cohomology: attractors and Conley index

In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $$f :U \subseteq {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits a certain attractor–repeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if Z is a metric space and $$K \subseteq Z$$ is a compact, global attractor for a continuous map $$g :Z \rightarrow Z$$, we show how to interpret series of the form $$\sum _{j \ge 0} a_j (g^*)^j$$ as endomorphisms of the cohomology group of the pair (Z, K).

Morse–Smale Systems and Topological Structure of Carrier Manifolds

We review the results describing the connection between the global dynamics of Morse–Smale systems on closed manifolds and the topology of carrier manifolds. Also we consider the results related to topological classification of Morse–Smale systems.

The Kähler geometry of Bott manifolds

We study the Kähler geometry of stage n Bott manifolds, which can be viewed as n-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from [3,6], that any stage n Bott manifold Mn admits an extremal Kähler metric. We also give necessary conditions for Mn to admit a constant scalar curvature Kähler metric. We obtain more precise results for stage 3 Bott manifolds, including in particular some interesting relations with c-projective geometry and some explicit examples of almost Kähler structures.To place these results in context, we review and develop the topology, complex geometry and symplectic geometry of Bott manifolds. In particular, we study the Kähler cone, the automorphism group and the Fano condition. We also relate the number of conjugacy classes of maximal tori in the symplectomorphism group to the number of biholomorphism classes compatible with the symplectic structure.

Controlled Surgery and $$\mathbb {L}$$ L -Homology

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $\sigma^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds of dimension $n \geq 5$. Our proof uses essentially the geometrically defined $\mathbb{L}$-spectrum as described by Nicas (going back to Quinn) and some well known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map $H_n (B, \mathbb{L}) \rightarrow L_n (\pi_1 (B))$ in terms of forms in the case $n \equiv 0 (4)$. Finally, we explicitly determine the canonical map $H_n (B, \mathbb{L}) \rightarrow H_n (B, L_0)$.

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.

Dynamics of the Ericksen–Leslie Equations with General Leslie Stress II: The Compressible Isotropic Case

In this article, the non-isothermal compressible Ericksen–Leslie system for nematic liquid crystals subject to general Leslie stress is considered. It is shown that this system is locally well-posed within the $$L_q$$ -setting and that for initial data close to equilibria points (which are identical with the ones for the incompressible situation), the solution exists globally. Moreover, any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold. Note that no structural assumptions on the Leslie coefficients are imposed and, in particular, Parodi’s relation is not being assumed. The results can be viewed as an extension of the studies in Hieber and Pruss (Math Ann 369:977–996, 2017) for the incompressible case to the compressible situation.

On Topology of Manifolds Admitting a Gradient-Like Flow with a Prescribed Non-Wandering Set

We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension 4 and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly μ node and ν saddle equilibrium points of indices equal to either 1 or n — 1. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius $$g=\frac{\nu-\mu+2}{2}$$ . We also suggest an algorithm for constructing gradientlike flows on closed manifolds of dimension 3 and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices.

Space-time Singularities vs. Topologies in the Zeeman—Göbel Class

In this article we first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved spacetimes, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski spacetime. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class $\mathfrak{Z}-\mathfrak{G}$ of Zeeman-G\"obel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem fails to hold. The "problem" that L.C.T. does not hold can be resolved by "adding back" the light-cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the L.C.T. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a spacetime is vast and of no physical significance (Zeeman), more preferable from an appropriate topology in the class $\mathfrak{Z}-\mathfrak{G}$ under which a homeomorphism is an isometry (G\"obel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the spacetime manifold might play an important role in describing the transition from the quantum non-local theory to a classical local theory.

Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

Graded Artinian algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of a base ring with a fibre ring corresponds to a fibre bundle over a manifold. If the manifold is Kähler, then its cohomology ring satisfies the strong Lefschetz property, which means multiplication by a linear form has the largest possible Jordan type. In this paper, we study the behaviour of strong Lefschetz and Jordan type with respect to free extensions, using relative coinvariant rings of finite groups as prototypical models. We show that if V is a vector space and if the subgroup W of the general linear group Gl(V) is a non-modular finite reflection group and is a non-parabolic reflection subgroup, then the relative coinvariant ring cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings , some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples are related to open combinatorial questions proposed and partially solved by G. Almkvist.

Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber

A pendulum, which can oscillate in two directions, is subjected to a generalized external force. A non-smooth absorber is coupled to the pendulum with an arbitrary location and orientation. The equations of the system are derived and are treated with a multiple scale method. At fast time scale, the topology of the slow invariant manifold is described with its stable and unstable zones. The equilibrium and singular points of the system are detected at the first slow time scale. The responses of the main system, given as a function of the frequency of the external force, show reductions of the vibration levels. The analytic predictions are compared by direct numerical time integration of the equations of the system. They illustrate the operationality of the non-smooth absorber in several cases.

Residually finite rationally p groups

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

A note on fibrations of $$G_2$$ G 2 -manifolds

In this note, we first give some constructions of torsion-free $$G_2$$ -structures on some topological product manifolds. Then we provide a sufficient condition of 3-Calabi–Yau fibrations for $$G_2$$ -manifolds. Next we study the Gukov–Yau–Zaslow horizontal lifting for hyperKahler firbations of $$G_2$$ -manifolds, and discuss when the Gukov-Yau-Zaslow metric on this fibration is a $$G_2$$ -metric.

Constructing the virtual fundamental class of a Kuranishi atlas

Consider a space $X$, such as a compact space of $J$-holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of $X$ by representing $X$ via the zero set of a map $S_M: M\to E$, where $E$ is a finite dimensional vector space and the domain $M$ is an oriented, weighted branched topological manifold. Moreover, $S_M$ is equivariant under the action of the global isotropy group $\Gamma$ on $M$ and $E$. This tuple $(M,E, \Gamma, S_M)$ together with a homeomorphism $S_M^{-1}(0)/\Gamma \to X$ forms a single finite dimensional model (or chart) for $X$. The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However if $X$ is presented as the zero set of an sc-Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold $M$ that uses an sc-smooth partition of unity.

The Banach manifold Ck(M,N)

Let M be a compact manifold without boundary and let N be a connected manifold without boundary. For each k∈N the set of k times continuously differentiable maps between M and N has the structure of a smooth Banach manifold where the underlying manifold topology is the compact-open Ck topology. We provide a detailed and rigorous proof for this important statement which is already partially covered by existing literature.

Probing the Berry curvature and Fermi arcs of a Weyl circuit

The Weyl particle is the massless fermionic cousin of the photon. While no fundamental Weyl particles have been identified, they arise in condensed matter and meta-material systems, where their spinor nature imposes topological constraints on low-energy dispersion and surface properties. Here we demonstrate a topological circuit with Weyl dispersion at low-momentum, realizing a 3D lattice that behaves as a half-flux Hofstadter model in all principal planes. The circuit platform provides access to the complete complex-valued spin-texture of all bulk- and surface- states, thereby revealing not only the presence of Weyl points and the Fermi arcs that connect their surface-projections, but also, for the first time, the Berry curvature distribution through the Brillouin zone and the associated quantized Chiral charge of the Weyl points. This work opens a path to exploration of interacting Weyl physics in superconducting circuits, as well as studies of how manifold topology impacts band topology in three dimensions.

Higher matrix-tree theorems and Bernardi polynomial

The classical matrix-tree theorem discovered by Kirchhoff in 1847 expresses the principal minor of the $$n \times n$$ Laplace matrix as a sum of monomials of matrix elements indexed by directed trees with n vertices. We prove, for any $$k \ge n$$, a three-parameter family of identities between degree k polynomials of matrix elements of the Laplace matrix. For $$k=n$$ and special values of the parameters, the identity turns to be the matrix-tree theorem. For the same values of parameters and arbitrary $$k \ge n$$, the left-hand side of the identity becomes a specific polynomial of the matrix elements called higher determinant of the matrix. We study properties of the higher determinants; in particular, they have an application (due to M. Polyak) in the topology of 3-manifolds.

Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems

The values of the Gibbs function of a system with <i>C</i> components create a 2‑dimensional topological manifold that is piecewise smooth and continuous. Each of the <i>C</i>+2 smooth elements of such a manifold represents the states of a phase within the system. The elements are glued together along the <i>C</i> types of phase transformation lines, which converge to a single point that represents the invariant state of the system (i.e. a state with zero degrees of freedom). Transformation lines, treated as edges, and the smooth elements of the manifold, i.e. faces, constitute a zero-vertex graph that represents the invariant state. This graph is referred to here as the <i>graph-map</i> of the invariant state. The distribution of each component in an invariant state depends on the configuration (distribution) of the phase transformation lines. Because the smoothness and continuity of the manifold makes certain configurations of the lines forbidden, some forms of invariant states are also forbidden, even though they satisfy the Gibbs phase rule. Some academic handbooks do not take this fact into account, and provide forbidden configurations as examples of invariant states. States that only differ in terms of the permutation of two or more of their components will belong to the same class. This study shows that all real graph-maps can be represented by <i>C</i>-vertex graphs with <i>C</i>+2 edges that have an even value of the vertex valence. The number of such graphs, i.e. the number of classes of invariant states, <i>h</i>_o(<i>C</i>), is shown to meet the recurrence relation h_o(2<i>k</i>+1) = 2*h_o(2<i>k</i>) - h_o(2<i>k</i>-1), where <i>k</i> = 1, 2, 3, 4. Knowing the number <i>h</i>_o(<i>C</i>) for several small values of <i>C</i> allows us to determine the number of invariant states in a thermodynamic system using the above equation, regardless of the complexity of the system.

Toric Topology of the Complex Grassmann Manifolds

The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}/T^n$ by developing the methods of toric geometry and toric topology. We use a subdivision of $G_{n,k}$ into the strata $W_{\sigma}$ and determine all regular and singular points of the moment map $\mu$, introduce the notion of the admissible polytopes $P_\sigma$ such that $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}}$ and the notion of the spaces of parameters $F_{\sigma}$, which together describe $W_{\sigma}/T^{n}$ as the product $\stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$. To find the appropriate topology for the set $\cup _{\sigma} \stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$ we introduce the notions of the universal space of parameters $\tilde{\mathcal{F}}$ and the virtual spaces of parameters $\tilde{F}_{\sigma}\subset \tilde{\mathcal{F}}$ such that there exist the projections $\tilde{F}_{\sigma}\to F_{\sigma}$. Hence, we propose a method for the description of the orbit space $G_{n,k}/T^n$. Earlier we proved that the orbit space $G_{4,2}/T^4$, defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\partial \Delta _{4,2}\ast \mathbb{C} P^1$. We prove here that the orbit space $G_{5,2}/T^5$, defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space obtained by attaching the disc $D^8$ to the space $\Sigma ^{4}\mathbb{R} P^2$ by the generator of the group $\pi _{7}(\Sigma ^{4}\mathbb{R} P^2)=\mathbb{Z} _{4}$. In particular, $(G_{5,2}/G_{4,2})/T^5$ is homotopy equivalent to $\partial \Delta _{5,2}\ast \mathbb{C} P^2$. The methods and the results of this paper are fundaments for our theory of $(2l,q)$-manifolds.

The classification of 2‐connected 7‐manifolds

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form.