A new proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings
A new proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings
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Heegaard分裂稳定等价的Reidemeister-Singer定理的新证明
DOI：
10.1090/s0002-9939-1976-0410749-9
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发表时间：
1976
影响因子：
0.8
通讯作者：
R. Craggs
中科院分区：
文献类型：
--
作者：
R. Craggs
给出了Heegaard分裂稳定等价的Reidemeister-Singer定理的证明。该证明利用了关于维数小于或等于三的复形细分保持单纯塌陷的奇林沃斯定理，并且基于细分和塌陷保持稳定等价性的观察。关于 3 流形 [Re]、[Si] 的 Heegaard 分裂的 Reidemeister-Singer 定理规定，3 流形的任何两个 Heegaard 分裂都是稳定等价的。尽管这个定理自三十年代以来就已经存在，但迄今为止它最重要的应用可能直到六十年代末才出现，当时 Waldhausen [Wd] 证明了 3 球体的所有 Heegaard 分裂都是标准的。不幸的是，[Re]和[Si]中的证明在同一点上似乎是模糊的：它们似乎在大约三十年之前就预见到了奇林沃斯定理[Ch]，即维度小于或等于三的直线单纯复形在细分下的单纯塌陷不变性。 Papakyriakopoulos 和 Waldhausen 几年前都向我们建议，稳定等价定理的完整证明应该出现在某个地方。在 Joe Martin 告诉我们威斯康星大学的拓扑研讨会使用 Chillingworth 定理重建了 Reidemeister-Singer 定理的证明之后，我们获得了这里给出的证明。证明的基本思想是细分和单纯折叠保持稳定的等价性。在另一篇论文（参见 [Cr]）中，我们对这一想法进行了一些相当复杂的改进，以证明连通 3 流形之间的简单连通 4 维配边是稳定分类的，即通过 3 流形的某些 Heegaard 表示，达到与副本 S2 x S2 的连通和。这里的一切要么属于普普洛范畴，要么属于单纯范畴。因此，特别是，映射和同胚总是 pwl 或单纯的。复形是某个欧几里得空间 En 中的有限直线单纯复形。多面体是复合体的载体，对于复合体 K，我们用 I KI 表示该载体。多面体 P 的三角剖分是复数 K，使得 |KI = P。如果 K 是复数且某个子复数，则三角剖分 a 编辑者于 1974 年 3 月 27 日接收，并以修订形式于 1975 年 8 月 26 日接收。AMS (MOS) 主题分类 (1970)。主要 57Cxx、57A10。
A proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings is given. This proof makes use of the Chillingworth theorem on the preservation of simplicial collapses for subdivisions of complexes of dimension less than or equal to three, and it is based on the observation that subdivision and collapsing preserve stable equivalence. The Reidemeister-Singer theorem on Heegaard splittings of 3-manifolds [Re], [Si] provides that any two Heegaard splittings of a 3-manifold are stably equivalent. Although this theorem has been around since the thirties, probably the most significant application of it thus far did not come until the late sixties when Waldhausen [Wd] proved that all Heegaard splittings of the 3-sphere are standard. Unfortunately the proofs in both [Re] and [Si] appear to be obscure at the same point: they seem to anticipate, by about thirty years, Chillingworth's theorem [Ch] on the invariance of simplicial collapses under subdivision for rectilinear simplicial complexes of dimension less than or equal to three. Both Papakyriakopoulos and Waldhausen suggested to us several years ago that a complete proof of the stable equivalence theorem ought to appear somewhere. We obtained the proof given here after Joe Martin told us that the Topology Seminar at The University of Wisconsin had used the Chillingworth theorem to reconstruct a proof of the Reidemeister-Singer theorem. The basic idea of the proof is that subdivision and simplicial collapsing preserve stable equivalence. In another paper (see [Cr]) we use some fairly intricate refinements of this idea to show that simply connected 4-dimensional cobordisms between connected 3-manifolds are stably classified, that is, up to connected sum with copies S2 x S2, by certain Heegaard representations for 3-manifolds. Everything here is either in the pwlor the simplicial category. Thus, in particular, maps and homeomorphisms are always pwl or simplicial. A complex is a finite rectilinear simplicial complex in some Euclidean space En. A polyhedron is the carrier of a complex, and for a complex K we denote this carrier by I KI. A triangulation of a polyhedron P is a complex K such that |KI = P. If K is a complex and some subcomplex either triangulates a Received by the editors March 27, 1974 and, in revised form, August 26, 1975. AMS (MOS) subject classifications (1970). Primary 57Cxx, 57A10.