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1.
Motivated by the concept of A-category of a manifold introduced by Clapp and Puppe, we give a different proof of a (slightly generalized) Theorem of Hempel and McMillan: If M is a closed 3-manifold that is a union of three open punctured balls then M is a connected sum of S3 and S2-bundles over S1.  相似文献   

2.
A closed topological n-manifold M n is of S 1-category 2 if it can be covered by two open subsets W 1, W 2 such that the inclusions W i M n factor homotopically through maps W i S 1. We show that for n > 3 the fundamental group of such an n-manifold is either trivial or infinite cyclic.  相似文献   

3.
A closed topological n-manifold M n is of S 1-category 2 if it can be covered by two open subsets W 1,W 2 such that the inclusions W i M n factor homotopically through maps W i S 1M n . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.   相似文献   

4.
We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ2 which is not presentable as a countable union of convex sets satisfies the following dichotomy:
(1)  There is a perfect nonemptyPS so that |CP|<3 for every convexCS. In this case coveringS by convex subsets ofS is equivalent to coveringP by finite subsets, hence no nontrivial convex covers ofS can exist.
(2)  There exists a continuous pair coloringf: [N]2→{0, 1} of the spaceN of irrational numbers so that coveringS by convex subsets is equivalent to coveringN byf-monochromatic sets. In this case it is consistent thatS has a convex cover of cardinality strictly smaller than the continuumc in some forcing extension of the universe.
We also show that iff: [N]2→{0, 1} is a continuous coloring of pairs, and no open subset ofN isf-monochromatic, then the least numberκ off-monochromatic sets required to coverN satisfiesK +>-c. Consequently, a closed subset of ℝ2 that cannot be covered by countably many convex subsets, cannot be covered by any number of convex subsets other than the continuum or the immediate predecessor of the continuum. The analogous fact is false for closed subsets of ℝ3.  相似文献   

5.
 By solid balls, we mean a set of balls in R 3 no two of which can penetrate each other. Every finite graph G can be represented by arranging solid balls in the following way: Put red balls in R 3, one for each vertex of G, and connect two red balls by a chain when they correspond to a pair of adjacent vertices of G, where a chain means a finite sequence of blue solid balls in which each consecutive balls are tangent. (We may omit the chain if the two red balls are already tangent.) The ball number b(G) of G is the minimum number of balls (red and blue) necessary to represent G. If we put the balls and chains on a table so that all balls sit on the table, then the minimum number of balls for G is denoted by b T (G). Among other things, we prove that b(K 6)=8,b(K 7)=13 and b T (K 5)=8,b T (K 6)=14. We also prove that c 1 n 3<b(K n )<c 2 n 3 log n, c 3 n 4 log n<b T (K n )<c 4 n 4. Received: March 29, 1999 Final version received: January 17, 2000  相似文献   

6.
We characterize C *-algebras as those complete normed associative complex algebras having approximate units bounded by one and whose open unit balls are bounded symmetric domains. Such a characterization follows from the more general fact, also proved in the paper, that non-commutative JB *-algebras coincide with complete normed (possibly non-associative) complex algebras having approximate units bounded by one and whose open unit balls are bounded symmetric domains. Received: 3 October 2000 / Revised version: 26 January 2001  相似文献   

7.
Let T denote a binding component of an open book (S, f){(\Sigma, \phi)} compatible with a closed contact 3-manifold (M, ξ). We describe an explicit open book (S¢, f¢){(\Sigma', \phi')} compatible with (M, ζ), where ζ is the contact structure obtained from ξ by performing a full Lutz twist along T. Here, (S¢, f¢){(\Sigma', \phi')} is obtained from (S, f){(\Sigma, \phi)} by a local modification near the binding.  相似文献   

8.
A function f is continuous iff the pre-image f-1[V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical open mapping theorems in analysis provide a variety of sufficient conditions for openness.By the main theorem of recursive analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping Vf-1[V] being effective: given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f-1[V]. Analogously, effective openness requires the mapping Uf[U] on open real subsets to be effective.The present work combines real analysis with algebraic topology and Tarski's quantifier elimination to effectivize classical open mapping theorems and to establish several rich classes of real functions as effectively open.  相似文献   

9.
Let S be an open set in R 3 such that: (1) whenever three points x, y, z in S see each other via S, then conv{x, y, z} S, and (2) every seven points in S see via S a common point. Then S is finitely starlike. The proof uses the topological version of Helly's theorem.  相似文献   

10.
It is shown that if a closed setS in the plane is (n+1)-convex, then it has no more thann 4 holes. As a consequence, it can be covered by≤n 6 convex subsets. This is an improvement on the known bound of 2 n ·n 3. The author would like to thank the BSF for partially supporting this research. Publication no. 354.  相似文献   

11.
A packing (resp. covering) ? of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ?. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real‐valued functions that rests on so‐called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non‐compact spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

12.
In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold M diffeomorphic to the standard two-dimensional sphere. The first result is that the length of a shortest closed geodesic l(M) is bounded from above by 4r , where r is the radius of M . (In particular that means that l(M) is bounded from above by 2d, when M can be covered by a ball of radius d/2, where d is the diameter of M.) The second result is that l(M) is bounded from above by 2( max{r1,r2}+r1+r2), when M can be covered by two closed metric balls of radii r1,r2 respectively. For example, if r1 = r2= d/2 , thenl(M) 3d. The third result is that l(M) 2(max{r1,r2r3}+r1+r2+r3), when M can be covered by three closed metric balls of radii r1,r2,r3. Finally, we present an estimate for l(M) in terms of radii of k metric balls covering M, where k 3, when these balls have a special configuration.  相似文献   

13.
By attaching cables to the centers of the balls and certain intersections of the boundaries of the balls of a ball covering ofE d with unit balls, we can associate to any ball covering a collection of cabled frameworks. It turns out that a finite subset of balls can be moved, maintaining the covering property, if and only if the corresponding finite subframework in one of the cabled frameworks is not rigid. As an application of this cabling technique we show that the thinnest cubic lattice sphere covering ofE d is not finitely stable. The first two authors were partially supported by the Hungarian National Science Foundation under Grant No. 326-0413.  相似文献   

14.
In the euclidean planeE 2 letS 1,S 2, ... be a sequence of strips of widthsw 1,w 2, .... It is shown thatE 2 can be covered by translates of the stripsS i if w 1 3/2 = . Further results concern conditions in order that a compact convex domain inE 2 can be covered by translates ofS 1,S 2, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.  相似文献   

15.
Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS 3, connected sums of handlesS 1 × S 2 , euclidean and spherical space forms, and other examples with non-trivial fundamental group.  相似文献   

16.
Summary The aim of this paper is to prove that every open (i.e. noncompaet without boundary) manifold of dimensionn can be covered with exactlyn open disks. This is a generalization of a theorem of E. Luft [3] concerning the case of any open 2-dimensional manifold. It is then proved that every compact manifold of dimensionn with nonempty boundary can also be covered with exactlyn open disks. The proofs of the theorems are in the spirit of Morse theory [1].  相似文献   

17.
Let S be a nonempty proper subset of R 3 . A point y in cl, S is clearly R -visible from a point x via S if and only if there exists a neighborhood N of y such that S contains all closed half-lines emanating from x through points of . The following Krasnosel'skii-type theorem is proved—if every two boundary points of S are clearly R -visible via S from a common point, then S is a cone. This improves an earlier result with the number three and answers an open question. Received March 18, 1998, and in revised form June 2, 1999.  相似文献   

18.
In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?n if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?n if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

19.
Let X be a Peano continuum having a free arc. If X admits a sensitive open map, then X either is homeomorphic to the closed interval [0, 1], or is homeomorphic to the unit circle S1.  相似文献   

20.
We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S 1-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, ℂP 2, 2,S 2×S 2and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4,ℂP 2,S 2×S 2,ℂP 2#  2 or ℂP 2# ℂP 2. Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5,S 3×S 2, then on trivial S 3-bundle over S 2 or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.  相似文献   

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