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Math.GT] 29 Nov 2017 Embedded in Skeletons (Or Scaffoldings) of the Canonical Cubical Honeycombs of an Euclidean Or Hyperbolic Space

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Math.GT] 29 Nov 2017 Embedded in Skeletons (Or Scaffoldings) of the Canonical Cubical Honeycombs of an Euclidean Or Hyperbolic Space Topological surfaces as gridded surfaces in geometrical spaces. Juan Pablo D´ıaz,∗ Gabriela Hinojosa, Alberto Verjosvky, y November 27, 2017 Abstract In this paper we study topological surfaces as gridded surfaces in the 2-dimensional scaffolding of cubic honeycombs in Euclidean and hy- perbolic spaces. Keywords: Cubulated surfaces, gridded surfaces, euclidean and hyperbolic honeycombs. AMS subject classification: Primary 57Q15, Secondary 57Q25, 57Q05. 1 Introduction The category of cubic complexes and cubic maps is similar to the simplicial category. The only difference consists in considering cubes of different di- mensions instead of simplexes. In this context, a cubulation of a manifold is a cubical complex which is PL homeomorphic to the manifold (see [4], [7], [12]). In this paper we study the realizations of cubulations of manifolds arXiv:1703.06944v2 [math.GT] 29 Nov 2017 embedded in skeletons (or scaffoldings) of the canonical cubical honeycombs of an euclidean or hyperbolic space. In [1] it was shown the following theorem: ∗This work was supported by FORDECYT 265667 (M´exico) yThis work was partially supported by CONACyT (M´exico),CB-2009-129280 and PA- PIIT (Universidad Nacional Aut´onomade M´exico)#IN106817. 1 Theorem 1.1. Let M; N ⊂ Rn+2, N ⊂ M, be closed and smooth subman- ifolds of Rn+2 such that dimension (M) = n + 1 and dimension (N) = n. Suppose that N has a trivial normal bundle in M (i.e., N is a two-sided hy- persurface of M). Then there exists an ambient isotopy of Rn+2 which takes M into the (n+1)-skeleton of the canonical cubulation C of Rn+2 and N into the n-skeleton of C. In particular, N can be deformed by an ambient isotopy into a cubical manifold contained in the canonical scaffolding of Rn+2. In particular, the previous theorem establishes that smooth knotted surfaces in R4 are isotopic to cubulated 2-knots in the 4-dimensional cubic honeycomb. Orientable smooth closed surfaces are cubulated manifolds. Definition 1.2. Let S be a topological 2-manifold embedded in either Rn or Hn. We say that S is a geometrically gridded surface if it is contained in the 2-skeleton of either the canonical cubulation f4; 3n−2; 4g of Rn, the hyper- bolic cubic honeycomb f4; 3; 5g of the hyperbolic space H3 or the hyperbolic cubic honeycomb f4; 3; 3; 5g of the hyperbolic space H4. Here geometrically means that the surface is gridded by isometric pieces of regular euclidean or hyperbolic squares and it is placed in the corresponding skeleton in the scaffolding S2 of a regular cubic euclidean or hyperbolic honeycomb C. If it is understood the type of squares we simply say that S is a gridded surface. A gridded surface S on S2 of C in Rn is a piecewise linear surface such that each linear piece is a unit square with its vertices in the Zn-lattice of Rn. n However in the case of hyperbolic spaces H the description of the vertices is more complicated since the vertices belong to an orbit of a non-abelian n discrete group acting by isometries on hyperbolic space H as we will see in the next sections. We remark that our gridded surfaces are in a natural way length spaces [2]. Hilbert (1901, [9]) proved that there is no regular smooth isometric immer- sion X : H2 ! R3. Efimov (1961, [5]) generalizad this nonexistence theorem of Hilbert to the case of complete surfaces of nonpositive curvature; more precisely, he showed that there is no C2-isometric immersion of a complete, two dimensional, Riemannian manifold M ⊂ R3 whose curvature satisfies K ≤ c < 0. However, J. Nash (1956, [18]) proved that any Ck-manifold n k n 3 (M ; g) can be C -isometrically immersed into R where q ≥ 2 n(n+1)(n+9) and k ≥ 3. This implies that the hyperbolic space can be embedded into a high dimensional Euclidean space; for instance, we can find a C3-isometric 2 embedding of H2 into R99 (see [8], [18]). In this paper we study topological surfaces as geometrically gridded surfaces in the scaffolding of cubic honeycombs in Euclidean and hyperbolic spaces. We prove that connected orientable surfaces can be gridded in R3 or H3 and all the surfaces, orientable or not, can be gridded in R4 or H4. 2 Preliminaries This section consists of two topics. The first studies the type of “scaffolding" (i.e.. the 2-skeleton of an Euclidean or hyperbolic honeycomb) in which we can embed a surface in order to make it a gridded surface. For this purpose we start studying the regular cubic honeycombs of dimensions 3 and 4 in the Euclidean and hyperbolic cases. In the second topic we will revise the theorem of topological classification of non-compact surfaces, in particular non-compact surfaces with Cantor sets of ends of planar and nonplanar type, and also with Cantor sets of non- orientable ends. 2.1 Regular cubic honeycombs We are interested in geometrically regular cubic honeycombs which are ge- ometric spaces filled with hypercubes which are euclidean or hyperbolic hy- percubes. We denote these honeycombs by their Schl¨afflisymbols. For a cube the symbol is f4; 3g. This means that the faces of the regular cube are squares with Schl¨afflisymbol f4g and that there are 3 squares around each vertex. These cubic honeycombs have Schl¨afflisymbols which describe their geometry and start by f4; 3; :::g: 2.1.1 Euclidean cubic honeycombs f4; 3n−2; 4g The canonical cubulation Cn of Rn is its decomposition into n-dimensional cubes which are the images of the unit n-cube In = [0; 1]n by translations by vectors with integer coefficients. Then all vertices of Cn have integers in their coordinates. 3 Any cubulation of Rn is obtained by applying a conformal transformation to the canonical cubulation. Remember that a conformal transformation is of the form x 7! λA(x) + a; where λ 6= 0; a 2 Rn;A 2 SO(n). Any n−cubulation has the same combinatorial structure as honeycomb. The regular hypercubic honeycomb whose Schl¨afflisymbol is f4; 3n−2; 4g is a cubu- lation of Rn which is its decomposition into a collection Cn of right-angled n-dimensional hypercubes f4; 3n−2g called the cells such that any two are either disjoint or meet in one common k−face of some dimension k. This provides Rn with the structure of a cubic complex whose category is similar to the simplicial category PL. The combinatorial structure of the regular honeycomb f4; 3; 4g is as follows: there are 6 edges, 12 squares and 8 cubes which are incident for each vertex and there are 4 squares and 4 cubes which are incident for each edge. The combinatorial structure of the regular honeycomb f4; 3; 3; 4g is as fol- lows: there are 8 edges, 24 squares, 32 cubes and 16 hypercubes which are incident for each vertex; there are 6 squares, 32 cubes and 16 hypercubes which are incident for each edge and there are 4 cubes and 4 hypercubes which are incident for each square. Figure 1: The 3-dimensional cubic kaleidoscopic honeycomb f4; 3; 4g. This Figure is courtesy of Roice Nelson [13]. 4 Definition 2.1. The k-skeleton of the canonical cubulation Cn of Rn, de- noted by Ck, consists of the union of the k-skeletons of the hypercubes in Cn, i.e., the union of all cubes of dimension k contained in the n-cubes in Cn. We will call the 2-skeleton C2 of Cn the canonical scaffolding of Rn. 2.1.2 Hyperbolic cubic honeycombs f4; 3; 5g and f4; 3; 3; 5g A gridded surface is a surface made of congruent squares contained in the corresponding scaffoldings which have disjoint interiors and are glued only in their edges, two squares are either disjoint or share one edge or a ver- tex. Around each vertex there is a circular circuit of squares (the squared star of the vertex). The condition of regularity of platonic solids in the 3- dimensional case implies that the number of squares around each vertex is a constant k > 2. If k = 3 then the regular gridded surface is the cube f4; 3g which is homeomorphic to a sphere S2. If k = 4 then the regular surface is the gridded plane f4; 4g which is isometric to the Euclidean plane R2. If k > 4 then the regular surfaces are the gridded hyperbolic planes which are denoted by f4; 5g; f4; 6g; :::. These gridded hyperbolic planes are length spaces [2] and are isometric, as length metric spaces to the hyperbolic plane H2. Analogously, we consider regular gridded 3-manifolds made of congruent cubes in scaffoldings which are disjoint and glued only in their boundary squares, two cubes are either disjoint or meet at a vertex an edge or a square. There is a circular circuit of cubes around each edge and one spherical cir- cuit of cubes around each vertex as PL 3-manifolds. For each edge there is the same number of cubes k > 2. If k = 3 then the regular gridded 3- manifold is the hypercube f4; 3; 3g which is homeomorphic to the 3-sphere S3. If k = 4 then the regular gridded 3-manifold is the cubic space f4; 3; 4g which is isometric to the Euclidean space R3. If k = 5 then the regular grid- ded 3-manifold is the cubic hyperbolic 3-space f4; 3; 5g which is isometric as a length space to the hyperbolic 3-space H3 [2]. Finally we construct regular gridded 4-manifolds made of congruent hyper- cubes contained in the corresponding scaffoldings which are disjoint and glued only in their boundary cubes, two hypercubes are either disjoint, or meet along a boundary cube of some dimension.
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