Compact Polyhedron Research Articles

Delooping controlled pseudo-isotopies of Hilbert cube manifolds

This paper is concerned with the controlled simple homotopy theory and controlled pseudo-isotopy theory of a Hurewicz fibration p: E→ B from a compact Hilbert cube manifold E to a compact polyhedron B. The main result is that the controlled pseudo-isotopy space is homotopy equivalent to the loop space of the controlled Whitehead space.

Nielsen zeta-function

In this paper we introduce a new zeta-function in the theory of dynamical systems. We find a sharp bound for the radius of convergence of the Nielsen zeta-function in terms of the topological entropy of the map. It follows from this that the Nielsen zeta-function has a positive radius of convergence. We prove that for an orientation-preserving homeomorphism of a compact surface the Nielsen zeta-function is either a rational function or the radical of a rational function. We calculate the Nielsen zeta-function for maps of circles, spheres, tori, protective spaces, for expanding maps of an orientable smooth compact manifold, for a homotopy periodic map of a connected compact polyhedron having no locally separating point.

Fixed points and braids. II

A basic problem in fixed point theory is to find the least number of fixed points for a homotopy class of maps. That is, given a map f : X ~ X of a connected compact polyhedron X into itself, to find M F [ f ] :=Min{ #Fix (g) lg~f :X,X} . The invariant central to this theory is the Nielson number N(f) , defined as the number of essential fixed point classes (see [1] ~)r [4]). N( f ) is always a lower bound to MF[f] and it has been a long-standing problem to prove or to disprove that N ( f ) = MFEf ]. We know [3] that if X has no local cut points and X is not a surface of negative Euler characteristic, then N ( f ) = M F [ f ] for every map f : X , X . The aim of the present paper is to generalize the result of [5] and to show

Nielsen numbers of mappings of surfaces

With each continuous map f of a compact polyhedron into itself there is associated a certain natural number, its Nielsen number N(f). The Nielsen number N(f) is bounded below by the number of fixed points of any map homotopic to f. The question of the sharpness of this estimate is classical: can one find for a given map f a map homotopic to it having exactly N(f) fixed points? It is known that this estimate is not sharp in general and that it is sharp for maps of compact polyhedra, which do not have locally separating points and which are not surfaces. The main result of the paper shows that this estimate is sharp for homotopy self-equivalence of compact surfaces. Its proof is based on Thurston's theory of diffeomorphisms of surfaces. In addition examples of maps of compact surfaces into themselves are discussed in the paper for which it seems that this estimate is not sharp.

Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p: M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ⩾ 0.

Invariance of Torsion and the Borsuk Conjecture

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.

Eine verallgemeinerung eines satzes von A. A. feldbaum

An upper bound of the number of switch-over points of extremal controls is given for certain linear controllable systems with stationary equations of motion and a set of feasible controls in form of any compact convex polyhedron. For these investigations results of the linear parametric programming are used.

Ein seitenalgorithmus zur lösung linear er optimierungsprobleme

An upper bound of the number of switch-over points of extremal controls is given for certain linear controllable systems with stationary equations of motion and a set of feasible controls in form of any compact convex polyhedron, For these investigations results of the linear parametric programming are used.

Some Nowhere Equicontinuous Homeomorphisms

It is shown that a nowhere equicontinuous homeomorphism can be defined on a compact polyhedron X if and only if X does not have cell decomposition which contains a principal 1-cell. It is also shown that for each locally connected contractible continuum C in the plane, there is a nowhere equicontinuous homeomorphism ${h_c}$ on a disk in the plane such that the fixed point set of ${h_c}$ is C.

Some nowhere equicontinuous homeomorphisms

It is shown that a nowhere equicontinuous homeomorphism can be defined on a compact polyhedron X if and only if X does not have cell decomposition which contains a principal 1-cell. It is also shown that for each locally connected contractible continuum C in the plane, there is a nowhere equicontinuous homeomorphism ${h_c}$ on a disk in the plane such that the fixed point set of ${h_c}$ is C.

The fixed point index for local condensing maps

We define below a fixed point index for local condensing maps f defined on open subset of «nice» metricANR’s. We prove that all the properties of classical fixed point index for continuous maps defined in compact polyhedra have appropriate generalizations. If our map is compact (a special case of a condensing map) and defined on an open subset of a Banach space, we prove that our fixed point index agrees with Leray-Schauder degree.

Desuspending collapses

Suppose X and Y are compact polyhedra and suppose the suspension of X collapses to the suspension of Y. This paper investigates the conditions under which it follows that X collapses to Y. If X is a piecewise linear manifold and the codimension of Y is greater than two, the only difficulty is with the fundamental group. The hypothesis in this case that the inclusion of Y into X induces an isomorphism of fundamental groups is sufficient to show that X collapses to Y. Codimension one and two require additional hypotheses. The major tool used is the piecewise linear s-cobordism theorem.

Embeddings and compressions of polyhedra and smooth manifolds

Topology Vol. 4 pp. 361-369. EMBEDDINGS Pergamon Press, 1966. Printed in Great Britain AND COMPRESSIONS OF POLYHEDRA SMOOTH MANIFOLDS? AND MORRIS W. HIRSZH (Receiued 23 July 1965) $1. INTRODUCTION IT IS OFTEN desirable to compress a subset X of a manifold V into a submanifold Y’ by an isotopy of V. For example, V might be a Euclidean space R” and V’ might be p, with k much smaller than n. If X has some special properties ensuring that such a compression is always possible, then we conclude that X embeds in ti. In this article the problem of compressing X into the boundary of an s-cell is studied, where s = dim V. As applications several theorems are proved about embedding polyhedra and smooth manifolds in Eucli- dean space. The main theorems say that a compact polyhedron or smooth submanifold XC V compresses into the boundary of an s-cell provided (1) there exists a “Dehn cone” on X, and (2) V-X is highly connected. A Dehn cone on XC V is an embedding X x I c V with X x 0 = X together with a null homotopy of X x 1 in Y - X x 0. If V-X is sufhci- ently connected, a regular neighborhood theorem of Hudson and Zeeman, together with an engulfing theorem due to Zeeman and the author, provides an s-cell E c V with X c i?E. If V = R’, this implies the compressibility of X into R* -I or Ss -‘. First the piecewise linear (=PL) theory is developed, then the smooth case is reduced to the PL case. In the applications we take X c R4+’ and try to compress X into R4. There are three steps : (1) extend X to X x I c RQ+’ ; (2) choose the extension so that X x 1 3: 0 in RQ+’ - X; (3) prove P+’ - X is sufficiently connected for the Enguhing Theorem to apply. Step (1) is difficult in the PL case, so in the applications it is assumed as part of the hypothesis. In the smooth case, however, (1) is equivalent to the existence of a normal vector field on the smooth submanifold X. Step (2) is accomplished through algebraic topology, sometimes by luck-as when X is a smooth homology sphere-more usually by just assuming that the obstructions to a t This work was supported by the National Science Foundation grant GP-4035

On Regular Neighbourhoods

IN (11) J. H. C. Whitehead introduced the theory of regular neighbourhoods, which has become a basic tool in combinatorial topology. We extend the theory in three ways. First we relativize the concept, and introduce the regular neighbourhood N of X mod Y in M, where X and Y are two compact polyhedra in the manifold M, satisfying a certain condition called link-collapsibility. We prove existence and uniqueness theorems. The idea is that N should be a neighbourhood of X— Y, but should avoid Y as much as possible. The notion is extremely useful in practice, and is illustrated by the following examples. We assume M to be closed for the examples. (i) If Y = 0 then N is a, regular neighbourhood of X. Therefore the relative theory is a generalization of the absolute theory. (ii) If X is a manifold with boundary Y, then the interior of X lies in the interior of N and the boundary of X lies in the boundary of N; in other words X is properly embedded in N. (iii) Let Ibe a cone and suppose that X n Y is contained in the base of the cone. Then N is a ball containing X— Y in its interior, In Y in its boundary, and Y — X in its exterior. The last example was used in ((12) Lemma 6), and was one of the examples which suggested the need for a relative theory. Other illustrations of the use are to be found in the proofs of Theorem 2, Corollary 8, and Lemmas 7, 8, and 9 below, in the proof of Theorem 3 of (4), and in forthcoming papers by us on isotopy. Secondly, Whitehead proved a uniqueness theorem that said that any two regular neighbourhoods were (piecewise linearly) homeomorphic. We strengthen this result by showing them to be isotopic, keeping a smaller regular neighbourhood fixed (Theorem 2). In fact they are ambient isotopic provided that they meet the boundary regularly (Theorem 3), which is always the case if M is unbounded. Thirdly, Whitehead wrote the theory in the combinatorial category, and we rewrite it in the polyhedral category. The difference is that the combinatorial category consists of simplicial complexes and piecewiselinear maps, whereas the polyhedral category consists of polyhedra and piecewise-linear maps. In this paper by a polyhedron we mean a topological

Open 3-manifolds which are simply connected at infinity

A triangulated open manifold M will be called 1-connected at infinity if each compact subset C of M is contained in a compact polyhedron P in M such that M-P is connected and simply connected. Stallings has shown that, if M is a contractible open combinatorial manifold which is 1-connected at infinity and is of dimension n> 5, then M is piecewise-linearly homeomorphic to Euclidean n-space En [5].

The Poincare Polynomial of a Symmetric Product

Let X be a compact polyhedron, Xn the topological product of n factors equal to X. The symmetric group Sn operates on Xn by permuting the factors, and hence if G is any subgroup of Sn we have an orbit space Xn/G obtained by identifying each point of Xn with its images under G. In particular Xn/Sn is the nth symmetric product of X, and if G is a cyclic subgroup of order n then Xn/G is the nth cyclic product of X.