Monotone and open mappings on manifolds. I
Monotone and open mappings on manifolds. I
复制标题
流形上的单调和开放映射。
DOI：
10.1090/s0002-9947-1975-0375326-0
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发表时间：
1975
影响因子：
1.3
通讯作者：
J. Walsh
中科院分区：
文献类型：
--
作者：
J. Walsh
本文给出了一类开映射存在的充分条件。流形Mm，m > 3，到多面体Q上。此外，还证明了从MTM，m > 3到Q的映射f是M到Q的单调映射的同伦当且仅当f：irx（M）-irx（Q）到Q上.最后，证明了MTM（m > 3）到Q的单调映射可以用M到Q的单调开映射来近似.在最近的一篇论文[19]中，D. Wilson构造了任意紧连通p. l上的单调开映射。歧管MTM，m > 3，到任何细胞上。此外，他还构造了zz > 3时三维流形到n个胞元上的光开映射。存在这样的映射的问题出现在一个列表中的45个问题汇编的艾伦伯格在1949年在[3]。他问是否有一个开放的映射流形到一个空间的更高的层面，是否有一个轻开放映射流形与每一点逆康托集。第一个问题的答案是由R. D.安德森在[1]中提出了单调开映射的存在性。流形MTM，m>3，到任何细胞;然而，他从来没有发表证明。与安德森的结果类似的结果是由凯尔迪亚在大约相同的时间在[7]，[8]和[9]中获得的。威尔逊最近的工作[19]回答了第二个问题。相反，对于二维流形R. L.摩尔[13]和Roberts及Steenrod [14]证明了单调象的维数至多为2，Stoilow [16]和G. T. Whyburn（[4]是一个很好的调查，他的工作）表明，轻开映射是分支覆盖，因此，不提高维数。本文开始系统地研究了从流形到多面体的单调、单调开、开和轻开映射的存在性。特别是，我们感兴趣的是，当一个映射“同伦”或“可以近似”的上述四种类型的映射之一。在§2中，我们完全确定了同伦到单调映射的映射。在§3中，我们给出了开映射存在的一般判据，并利用它证明了单调映射可以用单调开映射来逼近，从而确定了单调开映射同伦的映射。编辑于1974年4月15日收到，修订版于1974年7月1日收到。AMS（MOS）主题分类（1970年）。小学54 C10;中学57 C99。
Sufficient conditions are given for the existence of open mappings from a p. 1. manifold Mm, m > 3, onto a polyhedron Q. In addition, it is shown that a mapping / from MTM, m > 3, to Q is homotopic to a monotone mapping of M onto Q iff ft : irx(M) —► irx(Q) is onto. Finally, it is shown that a monotone mapping of MTM, m > 3, onto Q can be approximated by a monotone open mapping of M onto Q. In a recent paper [19], D. Wilson constructs monotone open mappings of any compact, connected p.l. manifold MTM, m > 3, onto any cell. In addition, he constructs light open mappings of three dimensional manifolds onto n cells for zz > 3. The question of the existence of such mappings appears in a list of 45 problems compiled by Eilenberg in 1949 in [3]. He asks whether there is an open mapping of a manifold onto a space of higher dimension and whether there is a light open mapping of a manifold with each point inverse a Cantor set. The answer to the first question was given by R. D. Anderson in [1] where he announced the existence of monotone open mappings of any p.l. manifold MTM, m>3, onto any cell; however, he never published a proof. Results similar to Anderson's were obtained by Keldyä at about the same time in [7], [8], and [9]. The recent work of Wilson in [19] answers the second question. In contrast, for two dimensional manifolds R. L. Moore [13] and Roberts and Steenrod [14] show that monotone images have dimension at most two and Stoilow [16] and G. T. Whyburn ([4] is an excellent survey of his work) show that light open mappings are branched coverings and, hence, do not raise dimension. In this paper, we begin a systematic study of the existence of monotone, monotone open, open, and light open mappings from manifolds to polyhedra. In particular, we are interested in when a mapping "is homotopic to" or "can be approximated by" one of the above four types of mappings. In §2, we completely determine those mappings homotopic to monotone mappings. In §3, we give general criteria for the existence of open mappings and use this to show that monotone mappings can be approximated by monotone open mappings; this leads to a determination of those mappings homotopic to monotone open mappings. In a Received by the editors April 15, 1974 and, in revised form, July 1, 1974. AMS (MOS) subject classifications (1970). Primary 54C10; Secondary 57C99.