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Articles 1 - 12 of 12
Full-Text Articles in Geometry and Topology
The Mathematics Of T. Benny Rushing, James Keesling, Luis Montejano, Gerard A. Venema
The Mathematics Of T. Benny Rushing, James Keesling, Luis Montejano, Gerard A. Venema
University Faculty Publications and Creative Works
No abstract provided.
Representing Homology Classes Of Simply Connected 4-Manifolds, Vo Thanh Liem, Gerard A. Venema
Representing Homology Classes Of Simply Connected 4-Manifolds, Vo Thanh Liem, Gerard A. Venema
University Faculty Publications and Creative Works
The main theorem asserts that every 2-dimensional homology class of a compact simply connected PL 4-manifold can be represented by a codimension-0 submanifold consisting of a contractible manifold with a single 2-handle attached. One consequence of the theorem is the fact that every map of S2 into a simply connected, compact PL 4-manifold is homotopic to an embedding if and only if the same is true for every homotopy equivalence. The theorem is also the main ingredient in the proof of the following result: If W is a compact, simply connected, PL submanifold of S4, then each element of H2(W;ℤ) …
A 4-Dimensional 1-Lcc Shrinking Theorem, Mladen Bestvina, Robert J. Daverman, Gerard A. Venema
A 4-Dimensional 1-Lcc Shrinking Theorem, Mladen Bestvina, Robert J. Daverman, Gerard A. Venema
University Faculty Publications and Creative Works
This paper contains several shrinking theorems for decompositions of 4-dimensional manifolds. Let f : M → X be a closed, cell-like mapping of a 4-manifold M onto a metric space X and let Y be a closed subset of X such that X - Y is a 4-manifold and Y is locally simply co-connected in X. The main result states that f can be approximated by homeomorphisms if Y is a 1-dimensional ANR. The techniques of the proof also show that f can be approximated by homeomorphisms in case Y is an arbitrary 0-dimensional closed subset. Combining the two results …
A Manifold That Does Not Contain A Compact Core, Gerard A. Venema
A Manifold That Does Not Contain A Compact Core, Gerard A. Venema
University Faculty Publications and Creative Works
A core of a (noncompact) manifold is a submanifold with the property that the inclusion of the submanifold into the manifold is a homotopy equivalence. It is shown by example that a manifold may fail to contain a compact core even though the manifold has the homotopy type of a finite complex.
A Homotopy Equivalence That Is Not Homotopic To A Topological Embedding, Vo Thanh Liem, Yukio Matsumoto, Gerard A. Venema
A Homotopy Equivalence That Is Not Homotopic To A Topological Embedding, Vo Thanh Liem, Yukio Matsumoto, Gerard A. Venema
University Faculty Publications and Creative Works
An open subset W of Sn, n ≥ 6 or n = 4, and a homotopy equivalence f : S2 × Sn-4 → W are constructed having the property that f is not homotopic to any topological embedding.
Gn.3(C) Is Prime If N Is Odd, John Ferdinands
Gn.3(C) Is Prime If N Is Odd, John Ferdinands
University Faculty Publications and Creative Works
A finite CW complex X is said to be prime if for every Hurewicz fibration F → E → B with E homotopy equivalent to X, and B and F homotopically equivalent to finite CW complexes, either B or F is contractible. We show that the 3-plane complex Grassmannian Gn.3(C) is prime if n is odd.
Cell-Like Images And UvM Groups, Gerard A. Venema
Cell-Like Images And UvM Groups, Gerard A. Venema
University Faculty Publications and Creative Works
Let X and A be compact metric spaces. The main problem studied in this paper is that of finding conditions under which a map f{hook}:A→X can be lifted to a cell-like space; i.e., conditions are sought under which there exist a cell-like continuum Z and continuous maps g:Z→X and f{hook}′:A→Z such that g{ring operator}f{hook}′=f{hook}. A theorem is proved which spells out technical conditions on the embedding of A into X and on the homotopy pro-groups of X under which such a lifting exists. The main corollary asserts the following: If X is UVk+1, dim A≤k, and f{hook}′:A→X is continuous, then …
Characterization Of Knot Complements In The 4-Sphere, Vo Thanh Liem, Gerard A. Venema
Characterization Of Knot Complements In The 4-Sphere, Vo Thanh Liem, Gerard A. Venema
University Faculty Publications and Creative Works
Knot complements in S4 are characterized as follows: A connected open set W ⊂ S4 is homeomorphic to the complement of some locally flat 2-sphere in S4 if and only if H1(W) is infinite cyclic, W has one end, and the fundamental group of that end is infinite cyclic. Applications include a characterization of weakly flat 2-spheres in S4 and a complement theorem for 2-spheres in S4.
Some Complex Grassmannian Manifolds That Do Not Fibre Nontrivially, John Ferdinands
Some Complex Grassmannian Manifolds That Do Not Fibre Nontrivially, John Ferdinands
University Faculty Publications and Creative Works
A finite CW complex X is said to be prime if, given a Hurewicz fibration F→E→B with E homotopy equivalent to X, and B and F homotopy equivalent to finite CW complexes, either B or F is contractible. We show that certain 3- and 4-plane complex Grassmanian manifolds are prime. © 1991.
Neighborhoods Of Compacta In 4-Manifolds, Gerard A. Venema
Neighborhoods Of Compacta In 4-Manifolds, Gerard A. Venema
University Faculty Publications and Creative Works
In this paper we investigate conditions under which a compact set in a piecewise linear 4-manifold has close neighborhoods with 1-dimensional spines. We prove that such neighborhoods exist in case the compactum satisfies the inessential loops condition and either has fundamental dimension 0 or has the shape of S1. Corollaries include two complement theorems and a weak flatness theorem for compacta in S4.
Ce Equivalence And Shape Equivalence Of 1-Dimensional Compacta, R. J. Daverman, Gerard A. Venema
Ce Equivalence And Shape Equivalence Of 1-Dimensional Compacta, R. J. Daverman, Gerard A. Venema
University Faculty Publications and Creative Works
In this paper the relationship between CE equivalence and shape equivalence for locally connected, 1-dimensional compacta is investigated. Two theorems are proved. The first asserts that every path connected planar continuum is CE equivalent either to a bouquet of circles or to the Hawaiian earring. The second asserts that for every locally connected, 1-dimensional continuum X there is a cell-like map of X onto a planar continuum. It follows that CE equivalence and shape equivalence are the same for the class of all locally connected, 1-dimensional compacta. In addition, an example of Ferry is generalized to show that for every …
An Approximation Theorem In Shape Theory, Gerard A. Venema
An Approximation Theorem In Shape Theory, Gerard A. Venema
University Faculty Publications and Creative Works
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.