Finding a Boundary for a Menger Manifold

LetMMbe either a topological manifold, a Hilbert cube manifold, or a Menger manifold and letDDbe an arbitrary countable dense subset ofMM. Consider the topological groupH(M,D)\mathcal {H}(M,D)which consists of all autohomeomorphisms ofMMthat mapDDonto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem forH(M,D)\mathcal {H}(M,D)as follows. IfMMis a one-dimensional topological manifold, thenH(M,D)\mathcal {H}(M,D)is homeomorphic toQ∞\mathbb {Q}^\infty, the countable power of the space of rational numbers. In all other cases we found thatH(M,D)\mathcal {H}(M,D)is homeomorphic to the famed Erdős spaceE\mathfrak E, which consists of the vectors in Hilbert spaceℓ2\ell ^2with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space.

Let M n + 3 {M^{n + 3}} be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K 3 {K^3} with an effective circle action of hyperbolic type. Assume that the fiber N n {N^n} in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set S TOP ( N n × D k , ∂ ) {S_{\text {TOP}}}\,({N^n} \times {D^k},\partial ) of topological surgery vanishes for sufficiently large k k . Then the projective class and Whitehead groups of M M vanish and S TOP ( M n + 3 × D k , ∂ ) = 0 {S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0 if n + k ⩾ 3 n + k \geqslant 3 or if K 3 {K^3} is closed and n = 2 n = 2 . The UNil \text {UNil} groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNil j {\text {UNil}}_j vanishes in spite of the failure of Cappell’s sufficient condition.

The Projective Class Group and the Whitehead Group

Chapter 26 - Wall's Finiteness Obstruction

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

The title statement is proved, and it is shown further that the subcontinua of a dendron actually form a Hilbert cube when (and only when) the branch points of the dendron are dense.Along the way, it is established that whenever a Hilbert cube manifold is compactified into a Hilbert cube factor by the addition of another Hilbert cube factor having Property Z in the compactification, then the resulting space is actually a Hilbert cube. It is known ([1], [14]) that the Cartesian product of any dendron with the Hilbert cube is homeomorphic to the Hilbert cube.In this paper, the hyperspace C(D) of all nonvoid subcontinua of a dendron Dis investigated and found to have the same property.In fact, it is shown that if the branch points of D are dense, then C(D) is a Hilbert cube.The method used presents in simple form the techniques employed elsewhere by the author and R. M. Schori ([11], [12]) to show that the hyperspace of all nonvoid, closed subsets of the unit interval is a Hilbert cube, and Theorem 1 is applied crucially in the extension of that result to finite, connected graphs in [13].The space X will be termed a Hilbert cube factor if for some space Y, Xx Y is homeomorphic to the Hilbert cube, Q.It is easy to see that this condition holds if and only if Xx Q is a Hilbert cube.(See [15, Lemma 2].)A dendron D is a uniquely arcwise connected Peano continuum.The branch points of D are those points which separate it into more than two

In this chapter, we prove the Torunczyk characterizations of Hilbert manifolds (non-separable Hilbert spaces are also considered as model spaces) and Hilbert cube manifolds. By using the characterization of Hilbert space, we show that every Frechet space is homeomorphic to Hilbert space with the same weight. Additionally, it is shown that if X is a non-discrete compactum and Y is a completely metrizable separable ANR Y without isolated points, then the space \(\operatorname {C}(X,Y)\) of maps from X to Y is an l2-manifold. By using the characterization of Q, we prove Keller’s Theorem, that is, every infinite-dimensional compact metrizable convex set in a locally convex topological linear space is homeomorphic to Q.

The title statement is proved, and it is shown further that the subcontinua of a dendron actually form a Hilbert cube when (and only when) the branch points of the dendron are dense. Along the way, it is established that whenever a Hilbert cube manifold is compactified into a Hilbert cube factor by the addition of another Hilbert cube factor having Property Z in the compactification, then the resulting space is actually a Hilbert cube.

Mapping cylinders of Hilbert cube factors II. the relative case

Projective modules and class groups

In this paper, we develop a geometric procedure for producing a “reverse” to Quillen’s plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudocollars, which are stackings of 1-sided h-cobordisms. Each of our pseudocollars has the same boundary and prohomology systems at infinity and similar group-theoretic properties for their profundamental group systems at infinity. In particular, the kernel group of each group extension for each 1-sided h-cobordism in the pseudocollars is the same group. Nevertheless, the profundamental group systems at infinity are all distinct. A good deal of combinatorial group theory is needed to verify this fact, including an application of Thompson’s group V. The notion of pseudocollars originated in Hilbert cube manifold theory, where it was part of a necessary and sufficient condition for placing a Z-set as the boundary of an open Hilbert cube manifold.

Homeomorphism groups of Hilbert cube manifolds

0. Introduction. In this note we shall describe how to adapt the methods of [1] to studying the group of pseudo-isotopy classes of diffeomorphisms of 1-connected manifolds. We shall use the notation and results of [1] freely. In the first four sections we shall discuss smooth manifolds and orientation preserving diffeomorphisms, but the results may be proved for piecewise linear (p.l.) manifolds and p.l. equivalences using the technique of [3] to adapt these proofs to the p.l. case. I am indebted to W.-C. Hsiang and J. Milnor for helpful comments. In ?1 we consider the problem of when a given homotopy equivalence is homotopic to a diffeomorphism. We get a condition in terms of the normal bundle. Our theorem refines a result of Novikov [12]. In ?2 we consider the uniqueness question, and get results on pseudo-isotopy of diffeomorphisms. In ?3 we use the results of ?1 to construct diffeomorphisms which are homotopic to the identity but are not pseudo-isotopic to the identity, even piecewise linearly or topologically. The lowest dimensional example is on a 5-manifold, e.g., S2 X S3. Some similar examples have been obtained by Hodgson [9]. In ?4 we consider the group of diffeomorphisms 92(M) (up to pseudo-isotopy) and deduce an exact sequence relating it to the group of tangential equivalences $5(M). It follows that the kernel of the homomorphism of 92(M) into Y(M) is finite up to conjugacy. In ?5 we deduce some facts about automorphisms of Sn X S1. We show in particular that the group of pseudo-isotopy classes of p.l. automorphisms of Sn X S1 is Z2+Z2+Z2 for n _ 5 (cf. [6]). If ek is a k-plane bundle we denote by T(6) the Thom complex of 6, T(6) =E(6)/E0(6), where E(f) is the total space of the closed disk bundle of 6, EO(6) is its boundary sphere bundle. We recall that if Mn is a smooth manifold embedded in Sn + k with normal bundle vk, k very large, then the natural collapsing map of Sn+ k onto the Thom complex T(v) = E(v)/Eo(v) defines an element in 7rn + k(T(v)). Any bundle automorphism sends this element into another one obtained in this way. We will call the set of such elements the normal invariants of Mn, and denote the set by

In chapter I we define the oritical points of a continuous function ϕ: X → IR on a compact metric space X, and the Morse polynomial for an isolated critical point. We reproduce the proof of the Morse relations for a function with isolated critical points for this more general set-up. In chapter II we define polynomial, absolute and alternating curvature measures of certain maps or embeddings of X into EN in terms of critical points and we obtain Morse relations for these curvature measures. In chapter III we discuss maps (and sets) in EN of minimal total absolute curvature (tight maps) and we recall some results on tight embeddings of smooth, piecewise linear and topological manifolds.

Jack Milnor has recently given this account of his unexpected encounter with exoticity: …I was trying to study 3-connected 8-manifolds. The case H4=0 seemed too hard. For \(H_{4} = \mathbb{Z}\), one can assume that the 4-skeleton is a 4-sphere. To build an 8-manifold, one can try to fatten it up by taking a tubular normal bundle neighborhood, and then adjoin an 8-cell. This worked so beautifully that I came up with many manifolds which couldn’t possibly exist… This article begins with a detailed elementary exposition of his revolutionary 1956 article that resolved the mentioned paradoxes, by unveiling exotic smooth structures on the 7-sphere. It continues by extensively describing Milnor’s introduction of ‘surgery’ to classify smooth structures on spheres. It concludes by sketching (all too briefly) the subsequent evolution of surgery theory—always confined to dimensions ≥5. In the hands of several following generations of topologists, it has become and remains the dominant tool for classification of compact, smooth (or piecewise linear or topological) manifolds.