The homotopy dimension of codiscrete subsets of the 2-sphere S 2
نویسندگان
چکیده
Andreas Zastrow conjectured, and Cannon-Conner-Zastrow proved, (see [3,pp. 44-45]) that filling one hole in the Sierpinski curve with a disk results in a planar Peano continuum that is not homotopy equivalent to a 1-dimensional set. Zastrow's example is the motivation for this paper, where we characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following question: Is a planar Peano continuum homotopically 1-dimensional if its fundamental group is isomorphic with the fundamental group of a 1-dimensional planar Peano continuum? 1. Introduction. We say that a subset X of the 2-sphere S 2 is codiscrete if and only if its complement D(X), as subspace of S 2 , is discrete. The set B(X) of limit points of D(X) in S 2 , which is necessarily a closed subset of X having dimension ≤ 1, is called the bad set of X. Our main theorem characterizes the homotopy dimension of X in terms of the interplay between D(X) and B(X):
منابع مشابه
On the Topology of Weakly and Strongly Separated Set Complexes
We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set [n] = {1, 2, . . . , n}, which, after deleting all cone points, we denote by ∆̂ws(n) and ∆̂ss(n), respectively. In particular, we find that ∆̂ws(n) is contractible for n ≥ 4, while ∆̂ss(n) is homotopy equivalent to a sphere of dimension n− 3. We also show that our homotopy equivalences ...
متن کاملFinite subset spaces of S 1
Given a topological space X denote by expkX the space of non-empty subsets of X of size at most k , topologised as a quotient of X . This space may be regarded as a union over 1 ≤ l ≤ k of configuration spaces of l distinct unordered points in X . In the special case X = S we show that: (1) expkS 1 has the homotopy type of an odd dimensional sphere of dimension k or k − 1; (2) the natural inclu...
متن کاملFramed Bordism and Lagrangian Embeddings of Exotic Spheres
Contents 1. Introduction 1 2. Construction of the bounding manifold 3 3. Transversality 21 4. Preliminaries for gluing 28 5. Construction of an extended gluing map 37 6. Local injectivity of the gluing map 52 7. Gluing near higher codimension strata 61 8. Construction of a smooth manifold with corners 74 9. Triviality of the tangent space of the cobordism 80 Appendix A. Pointwise estimates 88 A...
متن کاملSome Cyclic Group Actions on Homotopy Spheres
In [4J Orlik defined a free cyclic group action on a homotopy sphere constructed as a Brieskorn manifold and proved the following theorem: THEOREM. Every odd-dimensional homotopy sphere that bounds a para-llelizable manifold admits a free Zp-action for each prime p. On the other hand, it was shown ([3J) that there exists a free Zp-action on a 2n-1 dimensional homotopy sphere so that its orbit s...
متن کاملBalanced complexes and complexes without large missing faces
The face numbers of simplicial complexes without missing faces of dimension larger than i are studied. It is shown that among all such (d−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal f -vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal h-vector. It is also ve...
متن کامل