For each Cantor set C in R^3, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in R^3 with the complement having the same fundamental group as the complement of C. This answers a question from "Open Problems in Topology" and has as anapplication a simple construction of nonhomeomorphic open -manifolds with the same fundamental group. The main techniques used are analysis of local genus of points of Cantor sets, a construction for producing rigid Cantor sets with simply connected complement, and manifold decomposition theory. The results presented give an argument that for certain groups G, there are uncountably many nonhomeomorphic open 3-manifolds with fundamental group G.
COBISS.SI-ID: 16636505
It is well-known that a paracompact space X is of covering dimension n if and only if any map f \colon X \to K from X to a simplicial complex K can be pushed into its n-skeleton K^(n). We use the same idea to define dimension in the coarse category. It turns out the analog of maps f \colon X \to K is related to asymptotically Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's Property A, and the dimension coincides with Gromov's asymptotic dimension.
COBISS.SI-ID: 16655193
It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of R^4. According to the theorem of Shelah [S. Shelah, Can the Fundamental (Homotopy) Group of a Space be the Rationals?, Proc. Amer. Math. Soc. 103, no. 2, (1988), 627-632] such space cannot be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X_G for every countable group G.
COBISS.SI-ID: 16654681
Let X be a connected CW complex and let K(G,n) be an Eilenberg-Mac Lane CW complex where G is abelian. As K(G,n) may be taken to be an abelian monoid, the weak homotopy type of the space of continuous functions X \to K(G,n) depends only upon the homology groups of X. The purpose of this note is to prove that this is true for the actual homotopy type. Precisely, the space \mathrm{map}_\ast \big(X, K(G,n)\big) of pointed continuous maps X \to K(G,n) is shown to be homotopy equivalent to the Cartesian product \prod_{i \leq n} \mathrm{map}_\ast \big(M_i, K(G,n)\big). Here, M_i is a Moore complex of type M\big(H_i(X), i\big). The spaces of functions are equipped with the compact open topology.
COBISS.SI-ID: 16643929
Parametrized homology is a variant of zigzag persistent homology that measures how the homology of the level sets of the space changes as we vary the parameter. This paper extends Alexander Duality to this setting. Let X \subset R^n \times R with n \geq 2 be a compact set satisfying certain conditions, let Y = (R^n \times R) \setminus X, and let p be the projection onto the second factor. Both X and Y are parametrized spaces with respect to the projection. We show that if (X, p|_X) has a well-defined parametrized homology, then the pair (Y, p|_Y) has a well-defined reduced parametrized homology. We also establish a relationship between the parametrized homology of (X, p|_X)and the reduced parametrized homology of (Y, p|_Y).
COBISS.SI-ID: 16804185