PL Manifolds Research Articles

Hyperbolization of polyhedra

Hyperbolization is a process for converting a simplicial complex into a metric space with in the sense of Gromov. Several such processes are described in [19, §3.4]. One of the purposes of this paper is to elaborate this idea of Gromov. Another purpose is to use it to construct the three examples described below. Our approach to is based on the following construction of Williams [32]. Suppose that X is a space and that / : X -> σ is a map onto the standard λz-simplex. Suppose, also, that K is an ^-dimensional simplicial complex. To these data Williams associates a space XAK, constructed by replacing each n-simplex in the barycentric subdivision of K by a copy of X. The pair (X, /) is a ^-simplex if X is a nonpositively curved manifold with boundary and / has appropriate properties. (It is proved in §4 that hyperbolized simplices exist.) If (X, /) is a hyperbolized ^-simplex, then XAK is nonpositively curved; it is called a hyperbolization of K. In all three examples we begin with a polyhedral homology manifold having a desired feature; a then has the added feature of nonpositive curvature. The first example is a closed aspherical fourmanifold which cannot be triangulated. Taking the product of this example with a A2-torus, we obtain an aspherical manifold of any dimension > 4 which is not homotopy equivalent to a PL manifold. The second example is a closed smooth manifold of dimension n > 5 which carries a topological metric of nonpositive curvature, while its universal cover, though contractible, is not homeomorphic to a Euclidean n-space R. As we shall see, such a manifold cannot carry a PL or smooth metric of nonpositive curvature. The third example is a further refinement: M carries a topological metric of strict negative curvature and its universal

Analytic torsion for group actions

The Reidemeister torsion is a classical topological invariant for nonsimply-connected manifolds [14]. Let M be a closed oriented smooth manifold with the fundamental group πλ(M) nontrivial. Let p:πx{M) -• O(N) be an orthogonal representation of nx(M). If the twisted real cohomology H*(M, Ep) vanishes, then one can define the Reidemeister torsion τp GR , which is a homeomorphism invariant of M. The original interest of τp was that it is not a homotopy invariant, and so can distinguish spaces which are homotopy equivalent but are not homeomorphic. Ray and Singer asked whether, as for many other topological quantities, one can compute τp by analytic methods [17], Given a Riemannian metric g on M, they defined an analytic torsion T' e R as a certain combination of the eigenvalues of the Laplacian acting on twisted differential forms. They showed that under the above acyclicity condition, T is independent of the metric g, and they conjectured that the analytic expression T equals the combinatorial expression τ . This conjecture was proven to be true independently by Cheeger [4] and Mϋller [15]. One can look at the above situation in the following way. The group nχ{M) acts freely on the universal cover M, and so one has an invariant for free group actions. A natural question is whether the Reidemeister torsion can be extended to an invariant for more general group actions. For a finite group acting (not necessarily freely) on a closed oriented PL manifold X, a Reidemeister torsion was defined algebraically by Rothenberg [19]. One can then ask whether there is a corresponding analytic torsion when X is smooth, and whether the analytic torsion equals the combinatorial torsion. In §11 we define the analytic torsion Tp for a finite group action and show that if the relevant cohomology groups vanish, then it is independent of the G-invariant metric used in its definition. (This was shown previously in unpublished work by Cheeger [5].) The analysis involved to

Adaptive nonlinear digital filter with canonical piecewise-linear structure

A novel adaptive nonlinear filter with the least-mean-square (LMS) error criterion is presented. It is based on the so-called canonical piecewise-linear structure. As an alternative to approaches based on the Wiener-Volterra series which have so far been widely employed for adaptive nonlinear filtering, the proposed approach can exhibit adaptive performance, especially in strongly nonlinear cases, while saving computation and implementation cost. The performance of this adaptive nonlinear filter is illustrated by computer simulation results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Interpolation by piecewise-linear radial basis functions, I

In the two-dimensional plane, a set of points x 1, x 2, …, x n (called “nodes”) is given. It is desired to interpolate arbitrary data given on the nodes by continuous functions having piecewise-linear (“ PL ”) structure. For this purpose, one can employ the space of all PL -functions on a rectangular grid generated by the nodes. We study this space first. Next, we investigate the special PL -functions that are linear combinations of functions h i ( x) = ∥ x − x i ∥ 1, in which the l 1-norm on R 2 is employed. The “dual” case, involving the two-dimensional l ∞-norm, is included in our results, as are certain general interpolating functions of the form ( s, t) ↦ F( s − s i ) + G( t − t i ).

On the space of Lipschitz homeomorphisms of a compact polyhedron

Let X be a positive dimensional compact Euclidean polyhedron. Let H(X), HUP{X) and HPL(X) be respectively the space of homeomorphisms, the space of Lipschitz homeomorphisms and the space of piecewise-lin ear homeomorphisms of X onto itself. In this paper, we establish a homeomorphism taking the triple (H(X), HUp(X), H?L(X)) onto the triple (H(X) x s, HUp(X) x Σ, HFL(X) X σ), where 5 = (-1, l)ω, Σ = {(*,) e s\sup\Xi < 1} and σ = {(*,) e s\x t = 0 except for finitely many /}. As a consequence we prove that when X is a PL manifold with dim c Φ 4 and dX = 0, in case dimZ = 5, (H(X),HUP(X)) is an (s, Σ)-manifold pair if H(X) is an s-manifold. We also prove that if dim* = 1 or 2, then (H(X),HPL(X)) is an (5, σ)-manifold pair and (H(X),HUp(X)) is an (s, Σ)-manifold.

K -i obstructions to factoring an open manifold

Let M n+ k be an open PL manifold of dimension n+ k>5, let X be a finite polyhedron, and suppose ƒ: M n+k→ X× R K is a bounded homotopy equivalence. If k⩾1, we use radial engulfing and Siebenmann's twist-gluing (twist=id.) to construct a manifold M 1 with infinite cyclic cover M and a bounded homotopy equivalence ƒ 1: M 1→ X× R K−1 . By iterating this construction we obtain a manifold M k-1 and a bounded homotopy equivalence ƒ k−1: M k−1→ X×S 1×…× S 1× R . We show that the Siebenmann obstruction σ 1 in K ̃ 0( Zπ 1 M k−1) to factoring M k−1= N 1× R is an element of K ̃ −k+1 Zπ 1 X . If σ 1=0, let β be the automorphism of K̃ 0 induced by mapping the class of a projective P to the class of the projective P̄ consisting of antihomomorphisms from P into the coefficient ring Zπ( X× S 1×…× X 1) ( k-2 S 1 factors). Then subsequent obstructions lie in the quotients K̃ -1/image(1+β). That is, if σ 1 vanishes, there exists a sequence of obstructions σ i in K ̃ −k+1( Zπ1 X) /image(1+β), i=2,…, s, such that M= N s× R s , for some PL manifold N s if and only if each σ i vanishes.

End Behaviour and Ergodicity for Homeomorphisms of Manifolds with Finitely Many Ends

The recent paper of Berlanga and Epstein [5] demonstrated the significant role played by the “ends” of a noncompact manifold M in answering questions relating homeomorphisms of M to measures on M. In this paper we show that an analysis of the end behaviour of measure preserving homeomorphisms of a manifold also leads to an understanding of some of their ergodic properties, and allows results previously obtained for compact manifolds to be extended (with qualifications) to the noncompact case. We will show that ergodicity is typical (dense Gδ) with respect to various compact-open topology closed subsets of the space consisting of all homeomorphisms of a manifold M which preserve a measure μ. It may be interesting for topologists to note that we prove when M is a σ-compact connected n-manifold, n≧ 2, then M is the countable union of an increasing family of compact connected manifolds. If M is a PL or smooth manifold, this is well known and easy. If M is just, however, a topological n-manifold then we apply the recent results [9] and [12] to prove the result. The Borel measure μ, is taken to be nonatomic, locally finite, positive on open sets, and zero for the manifold boundary of M.

Average Intersection and Pivoting Densities

For the purposes of quantifying efficiency of more-than-one codimension piecewise-linear continuation algorithms, higher dimensional analogues of the average directional density are defined and studied. Two such are explicitly considered—the average intersection density, which measures the local density of intersections of an implicitly defined piecewise-linear manifold with complementing faces of a decomposition of Euclidean space, and average pivoting density, which considers the number of pivots such an intersection entails. General results are established which relate the densities to geometric properties of the decomposition. Explicit, closed formulae are presented for the most interesting case of the Freudenthal–Kuhn triangulation and asymptotic expressions are developed.

Pregeometric quantum lattice: A general discussion

We put forward an idea that the fundamental, i.e. pregeometric, structure of spacetime is given by an abstract set, so called abstract simplicial complex ASC. Thus, at the pregeometric level there is no (smooth) spacetime manifold. However, we argue that the structure described by an abstract simplicial complex is dynamical. This dynamics is then assumed to ensure that ASC can be realized as a lattice on a four-dimensional manifold with the simplest topologies dominating. We rewrite the pregeometric model, which is quantized using euclidean path-integral formalism, in an exact way so that as a four-dimensional manifold with the simples topologies dominating. is done by definition. The first step in bringing the continuum into the arena is to build up a lattice on a four-dimensional manifold from a given ASC. In fact, we choose a specific lattice: The Regge calculus lattice, i.e. a piecewise linear (flat) metric spacetime manifold. Secondly, we introduce a smooth (C ∞) manifold (described by a metric tensor g μν ) to approximate the Regge calculus manifold (described by a metric tensor g μν RC). It turns out that after integrating (and summing) out all other degrees of freedom than the metric tensor field g μν , the resulting continuum theory is nonlocal (as would be expected). However, it is our main point to show that the nonlocality is not very severe since it is only of finite range. We argue that the points in the introduced continuum which represent lattice points have so great quantum fluctuations that they are in a high temperature phase with no long-range correlations. In other words, although the effective action for the continuum formulation is not totally local, it is effectively so because it has only finite range nonlocalities. We can prove this kind of weak locality of the effective action by means of a general high-temperature theorem. Then we claim that the resulting local (or rather almost local) model with reparametrization invariance and g μν as a field gives essentially the ordinary Einstein's gravity theory in the long wavelength limit.

On the homology of metacyclic coverings

In this paper the authors study irregular metacyclic branched covering spaces. These arise as follows: Suppose G is a Z/m extension of Z/n. Then G contains a cyclic subgroup of order m, Cm, which we suppose is not normal. Suppose G acts on a PL manifold X. Then there are maps X→X/Cm→X/G. The map M=X/Cm→X/G=S is an irregular metacyclic covering. (It is not induced by the action of a group on M because Cm is not normal.) In the most interesting case S is a simply connected manifold, often a sphere. Then M→S is a covering space in the usual sense away from a codimension 2 subset of S, called the branch set. Usually the branch set is taken to be a knot or link in S. Regular branched coverings have been widely studied in many contexts (knot theory, algebraic geometry, etc.), but irregular coverings have been much less studied although they are important also. For example, every closed oriented 3-manifold is a 3-fold irregular covering of S3 with branch set a knot. (This is the case G equals the dihedral group of order 6.). That this result does not generalize to dihedral groups of order 2p, p an odd prime, follows from Theorem 2 of this paper, which is too technical to state here. But specializing Theorem 2 to the context of rational homology and dihedral groups of order 2p, p an odd prime, the authors obtain the following theorem: dimH1(M;Q)≡0 modulo 1/2(p−1). The methods of proof in the paper are algebraic.

Computing Volumes of Polyhedra

In this note we give two simple methods for calculating the volume of any closed bounded polyhedron in R' having an orientable boundary which is triangulated into a set of (n - 1)-dimensional simplices. The formulas given require only coordinates of the vertices of the polyhedron. 1. Introduction. The purpose of this note is to give two simple methods for calculating the volume of any closed bounded polyhedron P in R' having an orientable boundary aP which is triangulated into a set T of (n - 1)-dimensional simplices. Following Hadwiger (2), we define a polyhedron to be the union of pairwise disjoint convex polyhedra, each of which is the convex hull of a finite number of points. In (1) we have described an algorithm for obtaining a piecewise linear manifold which closely approximates an implicitly defined manifold. If P has been given in such a way, then the affine pieces of aP are in general easy to triangulate with an inherited orientation. For polyhedra P which are determined by a given system of inequalities, methods and programs for triangulating P have been given in (5), (6). For such polyhedra, a triangulation of the boundary is not easily available, so our method is inappropriate. Of course, our approach would also be unnecessary for computing the volume of a parallelotope. Practical applications of the methods given here may be made to the approxima- tion of an area bounded by an implicitly defined curve or to approximation of the volume of a solid which is bounded by an implicitly defined surface. Although some formulas for the volume of convex polyhedra in Rn appear in the literature (e.g., (3)), these formulas generally require the computation of the (n - 1)- volume of the facets and additionally they involve extra computations of certain distances. The volume formulas we give here involve only the coordinates of the vertices of P.

An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold

A simplicial method is used to approximate the solution manifold to a system of nonlinear equations, $H(x) = \theta $, where $H:\mathbb{R}^{N + K} \to \mathbb{R}^N $ The method begins at a point $x_0 $ in the solution set where the derivative $DH(x_0 )$ is of full rank. Given any $\varepsilon > 0$, a piecewise linear manifold is constructed along which $\| {H(x)} \|_\infty < \varepsilon $. An algorithm is presented to carry out this construction in an efficient fashion.

Flat Hilbert cube manifold pairs

The purpose of this paper is to study embeddings of g-manifolds into g-manifolds. Mainly, we relate flat ζ)-manifold pairs with PL manifold pairs by using a relative version of the Chapman Splitting Theorem. The concepts of βPL embedding and βPL homeomorphism are introduced.

A characterization of piecewise linear homology manifolds

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

Resolving homologies in BPL and PL characteristic numbers

In (1) it was shown that PL characteristic numbers determine (unoriented) PL bordism. In §§ 1 and 2 of this paper we use D. Sullivan's obstruction theory (9), (5) to provide some geometric insight into this result, as follows. We take an unoriented PL manifold M and let νM: M → BPL classify its stable normal bundle, and assume that all its PL characteristic numbers vanish, i.e. that (νM)* [M] = 0, where [M] is the fundamental class of M and . To show that M bounds, we first ‘resolve the homology to zero’ given by (νM)* [M] = 0. This is done by defining obstructions in whose vanishing ensures that the homology can be ‘resolved’, i.e. can be replaced by a bordism F: W → BPL between νM and a constant map, g: N → BPL, say. Then it is shown that these obstructions all vanish in this case.

Convergence to mode form solutions in impulsively loaded piecewise linear rigid-plastic structures

Models in load and velocity space which illustrate the behaviour of impulsively loaded piecewise linear rigid-plastic structures are considered. Attention is focused on the question of determining onto which mode a particular initial velocity field will converge, or conversely, what initial velocity fields converge onto a particular mode. Some answers to this question are provided, and links with the determination of the optimal time bound and with a recently presented algorithm for determining mode shapes are established.

Modelling of the space of paths in problems of constructing optimal trajectories

The basic problem of constructing optimal paths in non-simply connected domains is formulated, and the structure of its solution, based on a discrete-continual model of the space of paths, is considered. The special case of the basic problem, connected with the construction of extremals in a piecewise linear three-dimensional manifold, is studied. The proposed model is extended to problems of constructing, in non-simply-connected domain, extremals with moving end and connecting networks. The result of solving one of the problems is quoted.

An approximation theorem in shape theory

In this paper it is shown that if X is a compactum in the interior of a PL manifold M and if U is a neighborhood of X in M, then there is a compactum X′ in U such that X and X′ have the same relative shape in U and the embedding dimension of X′ equals the fundamental dimension of X. Whenever the dimension of M is not equal to three, the relative shape equivalence from X′ to X can be realized by an infinite isotopy of M.

Structures of topological type

In this paper we continue the study of structures of various types initiated by the author in the earlier paper “Structures of extensions” (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure β on the set X×X, and a bitopological space is a pair (X,β). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures τ1 and τ2-these structures define a structure β=τ1×τ2 on the set X×X. A settopological structure on the set X is any topological structureζ on the setρχ={A¦A⊂χ. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.

The Topology of Real Algebraic Sets with Isolated Singularities

In this paper we give a topological classification of real algebraic sets with isolated singularities, showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points. The question of which topological spaces are homeomorphic to real algebraic sets (solutions of polynomial equations in Euclidean space) has been long studied. In 1936 Seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set [121 and in 1952 Nash extended this result to all smooth compact manifolds [11]. In 1973 Tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set [13], so at least compact nonsingular algebraic sets are classified. Little has been done with singular algebraic sets however, since the transversality arguments used by Seifert, Nash and Tognoli no longer apply except in some special cases. One could use stability of singularities such as Kuiper [71 and Akbulut [1] used to show certain nonsmoothable PL manifolds are algebraic sets or one could use the projective version of Seifert-Nash-Tognoli as King did [6], but one could still not hope these techniques would allow even a characterization of isolated singularities. To get around this problem we take a cue from Hironaka's resolution of singularities [4]. The idea is to take a 'topological resolution' of a space if it exists. We can apply transversality techniques (SeifertNash-Tognoli) to the resolved space and then blow down algebraically and end up with the original space as an algebraic set. It seems likely that this technique allows one to classify all algebraic sets but in any case, we show that it classifies all algebraic sets with isolated singularities. In future papers we will use this technique to show, for instance, that all compact PL manifolds are homeomorphic to real algebraic sets [17] and that 2-dimensional real algebraic sets are topologically characterized as polyhedra satisfying Sullivan's even local Euler characteristic condition [16].