Dimensional Real Projective Space Research Articles

Index of Minimal Hypersurfaces in Real Projective Spaces

Abstract We prove that for an embedded unstable one-sided minimal hypersurface of the $(n+1)$-dimensional real projective space, the Morse index is at least $n+2$, and this bound is attained by the cubic isoparametric minimal hypersurfaces. We also show that there exist closed embedded two-sided minimal surfaces in the 3-dimensional real projective space of each odd index by computing the index of the Lawson surfaces.

Exchange Properties of Finite Set-Systems

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the $n$-dimensional real projective space with a subexponential number of vertices. They reduced the problem to finding a small downward closed set-system $\cal F$ covering an $n$-element ground set which satisfies the following condition: for any two disjoint members $A, B\in\cal F$, there exist $a\in A$ and $b\in B$ such that either $B\cup\{a\}\in\cal F$ and $A\cup\{b\}\setminus\{a\}\in\cal F$, or $A\cup\{b\}\in\cal F$ and $B\cup\{a\}\setminus\{b\}\in\cal F$. Denoting by $f(n)$ the smallest cardinality of such a family $\cal F$, they proved that $f(n)<2^{O(\sqrt{n}\log n)}$, and they asked for a nontrivial lower bound. It turns out that the construction of Adiprasito, Avvakumov, and Karasev is not far from optimal; we show that $2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log n}}$. We also study a variant of the above problem, where the condition is strengthened by also requiring that for any two disjoint members $A, B\in\cal F$ with $|A|>|B|$, there exists $a\in A$ such that $B\cup\{a\}\in\cal F$. In this case, we prove that the size of the smallest $\cal F$ satisfying this stronger condition lies between $2^{\Omega(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$.

Classification of Quantum Cellular Automata

There exists an index theory to classify strictly local quantum cellular automata in one dimension. We consider two classification questions. First, we study to what extent this index theory can be applied in higher dimensions via dimensional reduction, finding a classification by the first homology group of the manifold modulo torsion. Second, in two dimensions, we show that an extension of this index theory (including torsion) fully classifies quantum cellular automata, at least in the absence of fermionic degrees of freedom. This complete classification in one and two dimensions by index theory is not expected to extend to higher dimensions due to recent evidence of a nontrivial automaton in three dimensions. Finally, we discuss some group theoretical aspects of the classification of quantum cellular automata and consider these automata on higher dimensional real projective spaces.

Three ways to solve critical ϕ4 theory on 4 − 𝜖 dimensional real projective space: Perturbation, bootstrap, and Schwinger–Dyson equation

In this paper, we solve the two-point function of the lowest dimensional scalar operator in the critical [Formula: see text] theory on [Formula: see text] dimensional real projective space in three different methods. The first is to use the conventional perturbation theory, and the second is to impose the cross-cap bootstrap equation, and the third is to solve the Schwinger–Dyson equation under the assumption of conformal invariance. We find that the three methods lead to mutually consistent results but each has its own advantage.

STRUCTURE OF CHAMBERS CUT OUT BY VERONESE ARRANGEMENTS OF HYPERPLANES IN REAL PROJECTIVE SPACES

We study arrangements of $m$ hyperplanes in the $n$-dimensional real projective space, with a special focus on $m=n+3$ and $n=3$ or $n=4$.

On a Vertex-Minimal Triangulation of $\mathbb R \mathrm P ^4$

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}\mathrm{P}^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated $\mathbb{R}\mathrm{P}^4$ and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate $n$-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known $16$-vertex triangulation of $\mathbb{R}\mathrm{P}^4$. We also give a short, simple construction of the $22$-point Witt design, which is closely related to the complex we construct.

𝜖-expansion in critical ϕ3-theory on real projective space from conformal field theory

We use a compatibility between the conformal symmetry and the equations of motion to solve the one-point function in the critical [Formula: see text]-theory (a.k.a. the critical Lee–Yang model) on the [Formula: see text] dimensional real projective space to the first nontrivial order in the [Formula: see text]-expansion. It reproduces the conventional perturbation theory and agrees with the numerical conformal bootstrap result.

Bootstrapping Critical Ising Model on Three Dimensional Real Projective Space.

Given conformal data on a flat Euclidean space, we use crosscap conformal bootstrap equations to numerically solve the Lee-Yang model as well as the critical Ising model on a three dimensional real projective space. We check the rapid convergence of our bootstrap program in two dimensions from the exact solutions available. Based on the comparison, we estimate that our systematic error on the numerically solved one-point functions of the critical Ising model on a three dimensional real projective space is less than 1%. Our method opens up a novel way to solve conformal field theories on nontrivial geometries.

Minimal Dynamical Systems on Connected Odd Dimensional Spaces

AbstractLetbe a minimal homeomorphism (n ≥1). We show that the crossed producthas rational tracial rank at most one. Let Ω be a connected, compact, metric space with finite covering dimension and with. Suppose that,where Giis a finite abelian group, i = 0,1. Let β:Ω→Ωbe a minimal homeomorphism. We also show thathas rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on X✗Ω, where X is the Cantor set.

Nielsen coincidence numbers, Hopf invariants and spherical space forms

Given two maps between smooth manifolds, the obstruction to removing their coincidences (via homotopies) is measured by minimum numbers. In order to determine them we introduce and study an infinite hierarchy of Nielsen numbers N_i, i = 0, 1, ..., \infty. They approximate the minimum numbers from below with decreasing accuracy, but they are (in principle) more easily computable as i grows. If the domain and the target manifold have the same dimension (e.g. in the fixed point setting) all these Nielsen numbers agree with the classical definition. However, in general they can be quite distinct. While our approach is very geometric the computations use the techniques of homotopy theory and, in particular, all versions of Hopf invariants (a la Ganea, Hilton, James..). As an illustration we determine all Nielsen numbers and minimum numbers for pairs of maps from spheres to spherical space forms. Maps into even dimensional real projective spaces turn out to produce particularly interesting coincidence phenomena.

L–S categories of vector bundles over projective spaces

I.M. James calculated vecat(kγR), the L–S category of kγR, for k=2t and γR, the canonical line bundle over the (2t−1)-dimensional real projective space. In this paper, we calculate vecat(kγR) for integers k and γR, the canonical line bundle over projective spaces. We also calculate vecatFξ, the L–S category of ξ as an F-vector bundle, where F=R or C, for ξ, multiple Whitney sums of canonical line bundles over complex and quaternion projective spaces.

Estimation of the minimal number of periodic points for smooth self-maps of odd dimensional real projective spaces

Let f be a smooth self-map of a closed connected manifold of dimension m⩾3. The authors introduced in [G. Graff, J. Jezierski, Minimizing the number of periodic points for smooth maps. Non-simply connected case, Topology Appl. 158 (3) (2011) 276–290] the topological invariant NJDr[f], where r is a fixed natural number, which is equal to the minimal number of r-periodic points in the smooth homotopy class of f. In this paper smooth self-maps of real projective space RPm, where m>3 is odd, are considered and the estimations from below and above for NJDr[f] are given.

Maps into projective spaces: Liquid crystal and conformal energies

Variational problems for the liquid crystal energy of mappings from three-dimensional domains into the real projective plane are studied. More generally, we study the dipole problem, the relaxed energy, and density properties concerning the conformal $p$-energy of mappings from $n$-dimensional domains that are constrained to take values into the $p$-dimensional real projective space, for any positive integer $p$. Furthermore, a notion of optimally connecting measure for the singular set of such class of maps is given.

Preservers of generalized orthogonality on finite dimensional real vector and projective spaces

We describe the general form of bijective orthogonality preserving maps on ℝ n equipped with a pair of generalized indefinite inner products. The relations between the projective space and vector space versions of this result are examined and an example is given showing that the hypotheses of our main theorem are essential.

Surfaces in $S^{n}$ with Prescribed Gauss Map

Let $G$ be a $C^{\infty }$-mapping from a connected Riemann surface $M$ into the complex quadric $Q_{n-1}$ in the $n$-dimensional complex projective space. We give a condition for the existence of a surface in the $n$-dimensional Euclidean unit sphere $S^{n}$ such that the Gauss map is $G$. Under this condition, if $M$ is a torus, there exists a surface in $S^{n}$ such that the Gauss map is $G$. We also show that for a connected Riemann surface $M$ there exists an immersion $X:M\rightarrow RP^{n}$ such that a neighborhood of each point of $X(M)$ is covered by a surface in $S^{n}$ with prescribed Gauss map $G$ where $RP^{n}$ is the $n$-dimensional real projective space.

Global structure of the mod two symmetric algebra,H∗(BO; 𝔽2), over the Steenrod algebra

The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^m-1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstable A-algebras coalesce to produce the Dickson algebras of general linear group invariants, and we speculate about possible related topological realizability. Our methods also produce a related simple minimal A-module presentation of the cohomology of infinite dimensional real projective space, with filtered quotients the unstable modules F(2^p-1)/A bar{A}_{p-2}, as described in an independent appendix.

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.

A conjecture of Barratt-Jones-Mahowald concerning framed manifolds having Kervaire invariant one

To SETTLE the question of the existence or non-existence of a framed manifold having a nontrivial Kervaire invariant (or Arf invariant) is one of the main long-standing problems in algebraic topology. The Kervaire invariant is a Z/Zvalued invariant which may be formulated in many contexts. Originally it occurred as an invariant in framed surgery theory and for this approach the reader may consult Cl23 for example. It is reformulated in [7] in terms of the Adams spectral sequence for the stable homotopy of spheres. In particular the only open cases were reduced to determining whether ht E Exts ‘*+‘(Z/2,2/2) is an infinite cycle, producing a non-trivial element 8, in the 2’[+’ 2 stem of the stable homotopy of spheres. More recently the Kahn-Priddy theorem [8] and the algebraic Kahn-Priddy theorem [lo] have been used to convert it to a problem in the stable homotopy of infinite dimensional real projective space [WP”. The Kahn-Priddy theorem gives a stable map T: [WF’aSO which is a split surjection of stable homotopy groups (localized at the prime 2)

On Pu's theorem for odd dimensional real projective spaces

On Pu's theorem for odd dimensional real projective spaces

Symmetric spaces which are real cohomology spheres

This is a survey in which we collate some known results using semi-standard techniques, dropping the condition of simple connectivity in Kostant's work [2] and proving Theorem 1. Let M be a compact connected riemannian symmetric space. Then M is a real cohomology (dim M)-sphere if and only if (1) M is an odd dimensional sphere or real projective space; or (2) M = M/Γ where (a) M = S 2 r i X •. X STM, rt > 0, product of m > 1 even dimensional spheres, and (b) Γ consists of all γ = γλX X γm, where γ{ is the identity map or the antipodal map of S 2 r , and the number of yt which are antipodal maps, is even or (3) M = SU(3)/SO(3) orM = {SU(3)/Z3}/SO(3) or (4) M = O(5)/O(2) X O(3), non-oriented real grassmannian of 2-planes through 0 in R. In (2) we note πx(M) = Γ = {Ίj^ ~ in particular the even dimensional spheres are the case m — 1. In (3) we note that the first case is the universal 3-fold covering of the second case. In (4) we have πλ{M) = Z2. Theorem 1 is based on a series of lemmas which can be pushed, with appropriate modification, to the case of a real cohomology w-sphere of dimension greater than n. Here we make the convention that a 0-sphere is a single point. By using a cohomology theory which satisfies the homotopy axiom (such as singular theory) we can also drop the requirement of compactness. Thus we push the method of proof of Theorem 1 and obtain Theorem 2. Let M be a connected riemannian symmetric space. Then M is a real cohomology n-sphere, 0 0 factors are euclidean spaces and irreducible symmetric spaces of noncompact type, and (β) M is one of the following spaces. (1) W = M/Γ, where M = S r i X X STM is the product of m > 0 spheres of positive even dimensions 2ru Γ=(Z2) m consists of all γλX Xγm such that γi is the identity or antipodal map on S s θ is any one of the 2 characters on Γ, and Γ is the kernel of θ. Express θ = θiχ θi$9 where