Fredric Ancel - Academia.edu (original) (raw)

Papers by Fredric Ancel

arXiv (Cornell University), Dec 27, 2021

A knot is a possibly wild simple closed curve in S 3. A knot J is semi-isotopic to a knot K if th... more A knot is a possibly wild simple closed curve in S 3. A knot J is semi-isotopic to a knot K if there is an annulus A in S 3 × [0, 1] such that A ∩ (S 3 × {0, 1}) = ∂A = (J × {0}) ∪ (K × {1}) and there is a homeomorphism e : S 1 × [0, 1) → A − (K × {1}) such that e(S 1 × {t}) ⊂ S 3 × {t} for every t ∈ [0, 1). Theorem. Every knot is semi-isotopic to an unknot.

Topology and its Applications, Mar 1, 1983

A collection 9 of proper maps into a locally compact Hausdorff space Xfixes the ropo[ogy of X if ... more A collection 9 of proper maps into a locally compact Hausdorff space Xfixes the ropo[ogy of X if the only locally compact Hausdorff topology on X which makes each element of 9 continuous and proper is the given topology. In I* = [-1, 11 x [-1, 11, neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in 1', both the collection of C" arcs and the collection of regular C' arcs fix the topology. In W2, the collection of polynomial rays together with any collection of paths does not fix the topology. However, in II', the collection of regular injective entire rays together with either the collection of C" arcs or the collection of regular C' arcs fixes the topology. AMS(MOS) Subj. Class. (1980): Primary 54AlO; Secondary 26899,30D99. fixes the topology passes through a sequence analytic path regular arc Cm arc polynomial ray entire ray

arXiv (Cornell University), May 14, 2023

Lebesgue measurable subsets A and B of parallel or identical kdimensional affine subspaces of Euc... more Lebesgue measurable subsets A and B of parallel or identical kdimensional affine subspaces of Euclidean n-space E n satisfy The Product Formula for Volume: V ol k (A)V ol k (B) = J ∈S(n,k) V ol k (π J (A))V ol k (π J (B)). Here V ol k denotes k-dimensional Lebesgue measure; S(n, k) denotes the set of all k-element subsets of {1, 2, • • • , n}; and for J ∈ S(n, k), E J = {(x 1 , x 2 , • • • , xn) ∈ E n : x i = 0 for all i / ∈ J} and π J : E n → E J is the projection that sends the i th coordinate of a point of E n to 0 whenever i / ∈ J. Setting B = A, we obtain the corollary: The Pythagorean Theorem for Volume: V ol k (A) 2 = J ∈S(n,k) (V ol k (π J (A))) 2 .

arXiv (Cornell University), Jun 16, 2016

This paper presents some partial answers to the following question. Question. If a normal space X... more This paper presents some partial answers to the following question. Question. If a normal space X is the union of an increasing sequence of open sets U 1 ⊂ U 2 ⊂ U 3 ⊂. .. such that each U n contracts to a point in X, must X be contractible? The main results of the paper are: Theorem 1. If a normal space X is the union of a sequence of open subsets {U n } such that cl(U n) ⊂ U n+1 and U n contracts to a point in U n+1 for each n ≥ 1, then X is contractible. Corollary 2. If a locally compact σ-compact normal space X is the union of an increasing sequence of open sets U 1 ⊂ U 2 ⊂ U 3 ⊂. .. such that each U n contracts to a point in X, then X is contractible.

Michigan Mathematical Journal, 1993

Contemporary mathematics, 1984

Topology and its Applications, Nov 1, 1999

Let Y be a compact metric space that is not an (n − 1)-sphere. If the cone over Y is an n-cell, t... more Let Y be a compact metric space that is not an (n − 1)-sphere. If the cone over Y is an n-cell, then Y × [0, 1] is an n-cell; if n 4, then Y is an (n − 1)-cell. Examples are given to show that the converse of the first part is false (for n 5) and that the second part does not extend beyond n = 4. An application concerning when hyperspaces of simple nods are cones over unique compacta is given, which answers a question of Charatonik.

Bulletin of the Polish Academy of Sciences. Mathematics, 2010

We explore the interior geometry of the CAT(0) spaces {Xα : 0 < α ≤ π/2}, constructed by Croke an... more We explore the interior geometry of the CAT(0) spaces {Xα : 0 < α ≤ π/2}, constructed by Croke and Kleiner [Topology 39 (2000)]. In particular, we describe a diffraction effect experienced by the family of geodesic rays that emanate from a basepoint and pass through a certain singular point called a triple point, and we describe the shadow this family casts on the boundary. This diffraction effect is codified in the Transformation Rules stated in Section 3 of this paper. The Transformation Rules have various applications. The earliest of these, described in Section 4, establishes a topological invariant of the boundaries of all the Xα's for which α lies in the interval [π/2(n + 1), π/2n), where n is a positive integer. Since the invariant changes when n changes, it provides a partition of the topological types of the boundaries of Croke-Kleiner spaces into a countable infinity of distinct classes. This countably infinite partition extends the original result of Croke and Kleiner which partitioned the topological types of the Croke-Kleiner boundaries into two distinct classes. After this countably infinite partition was proved, a finer partition of the topological types of the Croke-Kleiner boundaries into uncountably many distinct classes was established by the second author [J. Group Theory 8 (2005)], together with other applications of the Transformation Rules.

Topology and its Applications, Oct 1, 1997

An inconclusive proof in a 1937 paper by G. Chogoshvili spawned an interesting dimensiontheoretic... more An inconclusive proof in a 1937 paper by G. Chogoshvili spawned an interesting dimensiontheoretic conjecture which we call the Chogoshvili-Pontrjagin Conjecture. In 1991, Y. Stemfeld found an ingenious counterexample to this conjecture which he and M. Levin greatly generalized in 1995. In this note we point out a previously unobserved property of the Stemfeld-Levin examples, and we reinterpret their significance in light of this property. Also, we present a version of the Levin-Stemfeld proof which is more "topological" and less "lattice-theoretic" than the original.

Michigan Mathematical Journal, 1987

Annals of Mathematics, 1979

By means of M.A. Stan'ko's clever unknotting technique we show that every embedding f: Mn... more By means of M.A. Stan'ko's clever unknotting technique we show that every embedding f: Mn' -> N" (n > 5) from a topological (n-l)-manif old Mn-l into a topological n-manifold N" can be approximated by locally flat embeddings. A serious technical difficulty forces us to work in the category of cell-like embedding relations rather than single-valued embeddings. The bonus of this enforced generality is that the results obtained will surely have application to the study of cell-like decompositions and generalized manifolds.

Topology, 1989

THIS PAPER is a study of a special class of toroidal decompositions of 3-manifolds called Bing-Wh... more THIS PAPER is a study of a special class of toroidal decompositions of 3-manifolds called Bing-Whitehead decompositions. It is well known that a Bing-Whitehead decomposition of a 3-manifold is shrinkable if all successive stages are Bing nested; but it is not shrinkable if all successive stages are Whitehead nested. (See Figs 1 and 2.) Consider a Bing-Whitehead decomposition of a 3-manifold which is defined by h, successive Bing nested stages. followed by 1 Whitehead nested stage, followed by h, successive Bing nested stages, followed by 1 Whitehead nested stage,. . The principal result of this paper is that this decomposition is shrinkable if and only if This result clarifies an issue raised in the proof of M. H. Freedman's Disk Theorem for Four Dimensional Manifolds ([4], p. 652). We recall the relevant definitions. A decomposition G of a 3-manifold M is a collection of pairwise disjoint non-empty subsets of M whose union is M. G is upper semicontinuous if each element of G is compact and if the quotient map M-+ M/G is a closed map. A d&iny sequence for G is a sequence {Xi: i 20) of compact 3-manifolds in int(M) such that Xi c int (Xi _ t) for i L: 1 and such that the non-singleton elements of G coincide with the non-singleton components of n {Xi: i 2 0). A defining sequence f Xi > is toroidal if each component of each Xi is a solid torus. If G has a toroidal defining sequence then G is called a toroidal decomposition. G is cell-like if for each element C of G, the inclusion of C into each of its neighborhoods in M is null-homotopic. Toroidal decompositions need not be cell-like; however. the special class of toroidal decompositions of concern here-the Bing-Whitehead decompositions-are, in fact, cell-like. Let T be a solid torus. A ram(ficution of T is a finite family { T, ,. .. , T, } of solid tori in T such that the (k + I)-tuple (T. T,,. .. , TI) is homeomorphic to the (k + I)-tuple (S' x B,S' x B,,. . , S' x Bk) where S' is a circle and {Br,.. , I&) is a pairwise disjoint family of disks in the interior of the disk B. Suppose G is a toroidal decomposition of a 3-manifold M, and {Xi) is a toroidal defining sequence for G. Let i 2 1, Xi is Bing nested in Xi _ 1 if for each component T of Xi _ 1, there is a ramification (T,,. . , T,}ofTsuchthatTnXicu{int(q):l<j<k} and for eachj, 1 <j I k, the pair (T,. Tin Xi) is homeomorphic to the pair (U, Vu W)

Proceedings of the American Mathematical Society, Apr 1, 1988

For n > 4, let M be an (n-l)-manifold embedded in an nmanifold N. For each point p of M, there is... more For n > 4, let M be an (n-l)-manifold embedded in an nmanifold N. For each point p of M, there is an (n-l)-sphere T, in N such that E n M is a neighborhood of p in M.

Topology and its Applications, Mar 1, 1992

Ancel, F.D., Topologies on Iw" induced by smooth subsets, Topology and its Applications 43 (1992)... more Ancel, F.D., Topologies on Iw" induced by smooth subsets, Topology and its Applications 43 (1992) 189-201. If Y is a collection of subsets of iw", let Y!? denote the largest topology on Iw" which restricts to the standard topology on each element of Y, and let ?t$ denote the homeomorphism group of Iw" wtth the topology &. Let Y_, denote the standard topology on Iw" and let %?_d denote the homeomorphism group of [w" with the standard topology. Theorem 1. If 9 is any collection of subsets of R" which contains all Cl regular 1-manifolds, then Jq = Tqcd A natural collection of subsets of [w" called smooth sets is defined which includes the zero set of every nonconstant polynomial and every C2 regular submanifold of iw" of dimension <n. Theorem 2. If Y is the collection qf all smooth subsets of R", then F:, is strictly larger than F',,, and X9 is strictly smaller than X,t,, Theorem 3. There is an injectioe function f: R"+R" which is discontinuous at each point of a countable dense subset of R", and whose restriction to each smooth subset of R" is continuous.

Illinois Journal of Mathematics, Dec 1, 1976

Proceedings of the American Mathematical Society, Feb 1, 1983

A space is rigid if its only self-homeomorphism is the identity. In response to a question of Jan... more A space is rigid if its only self-homeomorphism is the identity. In response to a question of Jan van Mill, Ancel and Singh have given examples of rigid «-dimensional compacta, for each n > 4, whose squares are manifolds. We construct a rigid 3-dimensional compactum whose square is the manifold S3 X S3. In fact, we construct uncountably many topologically distinct compacta with these properties.

Topology and its Applications, Dec 1, 1986

For ns4, every embedding of an (n-1)-manifold in an n-manifold has a S-resolution for each S > 0.... more For ns4, every embedding of an (n-1)-manifold in an n-manifold has a S-resolution for each S > 0. Consequently, for n z 4, every embedding of an (n-I)-manifold in an n-manifold can be approximated by tame embeddings.

Topology, Nov 1, 1999

Suppose an open n-manifold ML may be compacti"ed to an ANR MY L so that MY L!ML is a Z-set in MY ... more Suppose an open n-manifold ML may be compacti"ed to an ANR MY L so that MY L!ML is a Z-set in MY L. It is shown that (when n*5) the double of MY L along its &&Z-boundary'' is an n-manifold. More generally, if ML and NL each admit compacti"cations with homeomorphic Z-boundaries, then their union along this common boundary is an n-manifold. This result is used to show that in many cases Z-compacti"able manifolds are determined by their Z-boundaries. For example, contractible open n-manifolds with homeomorphic Z-boundaries are homeomorphic. As an application, some special cases of a weak Borel conjecture are veri"ed. Speci"cally, it is shown that closed aspherical n-manifolds (nO4) having isomorphic fundamental groups which are either word hyperbolic or CA¹(0) have homeomorphic universal covers.

Topology and its Applications, Feb 1, 1985

In a recent paper [6], van Mill and Mogilski prove that a proper hereditary shape equivalence pre... more In a recent paper [6], van Mill and Mogilski prove that a proper hereditary shape equivalence preserves property C, if its domain is a-compact. In this note, the same result is established without the hypothesis of u-compactness.