Non-orientable Surfaces Research Articles

On the topological complexity of manifolds with abelian fundamental group

Abstract We find conditions which ensure that the topological complexity of a closed manifold M with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik–Schnirelmann category of the cofibre of the diagonal map Δ : M → M × M {\Delta:M\to M\times M} for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds.

Existence of metrics maximizing the first eigenvalue on non-orientable surfaces

We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in [21].

On the structure of the planar subgraphs of obstruction graphs of a nonorientable surface with a given genus

The problem of studying the structure of planar graphs with sets of points, which should be critical concerning the distance between cells on the boundaries of which the elements of a given set are located in operations of removing vertices or edges of a graph, is considered. Knowing the structure of these planar graphs, it is possible to construct a finite set of planar graphs with given characteristics required for the construction of obstruction graphs of a given nonorientable genus. The main result is to use the constructed list of plane graphs critical concerning distance 2 to construct obstruction graphs of a given nonorientable genus.

Surface codes and color codes associated with non-orientable surfaces

Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.

The projective 3-annihilating-ideal hypergraphs

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph [Formula: see text] of a commutative ring [Formula: see text] and the topological embedding of [Formula: see text] to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings R (up to isomorphism) such that [Formula: see text] is a projective graph.

Lefschetz fibrations on nonorientable 4-manifolds

Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk, whose regular fiber is a nonorientable surface with nonempty boundary. This is an analogue of a result of Harer obtained in the orientable case. As a corollary, we obtain a $4$-dimensional proof of the fact that every nonorientable closed $3$-manifold admits an open book decomposition, which was first proved by Berstein and Edmonds using branched coverings. Moreover, the monodromy of the open book we obtain for a given $3$-manifold belongs to the twist subgroup of the mapping class group of the page. In particular, we construct an explicit minimal open book for the connected sum of arbitrarily many copies of the product of the circle with the real projective plane. We also obtain a relative trisection diagram for $W$, based on the nonorientable Lefschetz fibration we construct, similar to the orientable case first studied by Castro. As a corollary, we get trisection diagrams for some closed $4$-manifolds, e.g. the product of the $2$-sphere with the real projective plane, by doubling $W$. Moreover, if $X$ is a closed nonorientable $4$-manifold which admits a Lefschetz fibration over the $2$-sphere, equipped with a section of square $\pm 1$, then we construct a trisection diagram of $X$, which is determined by the vanishing cycles of the Lefschetz fibration. Finally, we include some simple observations about low-genus Lefschetz fibrations on closed nonorientable $4$-manifolds.

P-Groups of automorphisms of compact non-orientable Riemann surfaces

We study p-groups of automorphisms of compact non-orientable Riemann surfaces of topological genus gge 3. We obtain upper bounds of the order of such groups in terms of p, g and the minimal number of generators of the group. We also determine those values of g for which these bounds are sharp. Furthermore, the same kind of results are obtained when the p-group acts as the full automorphism group of the surface.

Enumeration of unsensed r-regular maps on the projective plane and the Klein bottle

The paper is devoted to the problem of enumerating r-regular maps on two simplest non-orientable surfaces, the projective plane and the Klein bottle, up to all symmetries (so-called unsensed maps). We obtain general formulas that reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve the problem of explicitly describing all cyclic orbifolds for such surfaces. We also derive recurrence relations for rooted quotient maps on orbifolds that can be orientable or non-orientable surfaces with branch points and/or boundary components. These results enable us to obtain explicit formulas for the numbers of unsensed maps on the projective plane and the Klein bottle by the number of edges.

On a class of orientable and nonorientable strip surfaces

In this paper a class of regular surfaces, generated by rotation and twist of a rigid line segment, is considered. Firstly a complete description of those surfaces by means of local charts is obtained and then its orientability is characterized in terms of the number of half-twists, taking into account their direction (clockwise or counterclockwise) since changes in the sense of twist are allowed. Indeed a procedure to construct nonorientable surfaces is provided. Illustrative examples and computer-generated plots using the free software Maxima are included.

Large N behaviour of the two-dimensional Yang–Mills partition function

AbstractWe compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$ . Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B303(1) 95–98).

Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, .... A natural toy model for further investigations is the Möbius strip, a non-orientable surface with Euler characteristic 0, and particularly the ‘‘square’’ Möbius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns.

On non‐orientable surfaces embedded in 4‐manifolds

We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz "one is enough" theorem to the case of non-orientable surfaces.

Enumeration of rooted 4-regular maps without planar loops

With the help of composition formula we obtain a very simple expression that enables us to find the generating function for rooted 4-regular maps without planar loops on an arbitrary orientable or non-orientable surface of a given genus g, assuming that the generating function for all rooted 4-regular maps on this surface is known. As an application of this result, we derive explicit expressions for the corresponding generating functions for maps on the sphere, the projective plane, the torus, and the Klein bottle. We also provide an explicit formula for the number of unlabelled 4-regular maps without planar loops on the sphere, counted up to orientation-preserving homeomorphisms.

A Magnus extension for locally indicable groups

A group G possesses the Magnus property if for every two elements u, v∈G with the same normal closure, u is conjugate to v or v−1. O. Bogopolski and J. Howie proved independently that the fundamental groups of all closed orientable surfaces possess the Magnus property. The analogous result for closed non-orientable surfaces was proved by O. Bogopolski and K. Sviridov except for one case that was later covered by the author. In this article, we generalize those results, which can be viewed as Magnus extensions for free groups, to a Magnus extension for locally indicable groups and consider the influence of adding a group as a direct factor. For this purpose, we also prove versions of the Freiheitssatz for locally indicable groups and of a result by M. Edjvet adding a group as a direct factor.

Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, M\"obius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.

Lower central series, surface braid groups, surjections and permutations

AbstractGeneralising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

Degenerating sequences of conformal classes and the conformal Steklov spectrum

AbstractLet $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

Effective topological complexity of orientable-surface groups

We use rewriting systems to spell out cup-products in the (twisted) cohomology groups of a product of surface groups. This allows us to detect a non-trivial obstruction bounding from below the effective topological complexity of an orientable surface with respect to its antipodal involution. Our estimates are at most one unit from being optimal, and are closely related to the (regular) topological complexity of non-orientable surfaces.

Exhausting curve complexes by finite superrigid sets on nonorientable surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\mathcal {C}(N)$ be the curve complex of $N$. We prove that if $(g, n) \neq (1,2)$ and $g + n \neq 4$, then there is an exhaustion of $\mathcal {C}(N)$

Liouville Foliations of Topological Billiards with Slipping

In the paper, a new class of integrable billiards, namely, billiards with slipping, is studied. At the reflection from the boundary, a billiard particle of such a system may not only change its velocity, but also move some distance along the border. Some laws of slipping preserve the integrability of flat confocal and circular billiards and billiard books glued from them, i.e., billiards on cell complexes. In the paper, the topology of the Liouville foliations for several integrable billiards with slipping, both flat and locally flat, is studied. Two such systems are Liouville equivalent to integrable geodesics flows of small degrees on nonorientable two-dimensional surfaces, namely, the projective plane and the Klein bottle. This shows that the nonorientability of a two-dimensional surface, in itself, is not an obstacle to its implementability by an appropriate integrable billiard.