Homotopy equivalence and hemeomorphism of 3-manifolds
Homotopy equivalence and hemeomorphism of 3-manifolds
复制标题
3-流形的同伦等价和同胚
DOI：
10.1016/0040-9383(92)90046-k
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
Peter Scott
中科院分区：
文献类型：
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作者：
J. Hass;Peter Scott
本文将由其基本群确定到同胚的3-流形类推广到包含满足两个性质的奇异曲面的闭可定向不可约3-流形类，l-线交性质和4-平面性质.维流形是决定在何种程度上同伦类型的闭流形决定流形同胚。在具有有限基本群的3-流形的情况下，已知存在非同胚的同伦等价流形，但没有已知的具有同构的非同胚的无限基本群的闭可定向不可约3-流形的例子。Waldhausen [30]和Heil Cl [23]证明了如果M和M '是具有同构基本群的Haken 3-流形，则它们是同态的。如果M和M '是双曲的，则Mostow刚性定理得到相同的结果[19]。但如果仅假设M是双曲的，则它是未知的。Boehme [3]将Waldhausen定理推广到某些非Haken Seifert纤维空间。Scott [27]证明了一个同伦等价于具有无限基本群的Seifert纤维空间的闭定向不可约3-流形与该Seifert纤维空间同胚。这些Scifert纤维空间中的许多是非Haken的。然而，到目前为止，塞弗特纤维空间提供了唯一的例子，非哈肯3-流形，这是已知的，以同胚由他们的基本群。对于有边界的流形，有一些非同伦同伦等价哈肯流形的简单例子，例如与三穿孔球面的圆的乘积和与一次穿孔环面的圆的乘积。这样的例子可以通过3-流形的特征分解来很好地理解[17][16]。另一个麻烦例子的可能来源是取一个3-流形与一个假同伦3-球面的连通和，得到一个同伦等价的非同胚3-流形。这种可能性将被排除在一个成功的解决庞加莱猜想。为了解决这个潜在的问题，我们使用不可约的3-流形，其中任何2-球面都有一个边界。由于不可定向的P2-不可约3-流形总是Haken的，我们将注意力限制在可定向的情况。为了简化符号，我们有时遵循不区分曲面到流形M的映射和映射的像的惯例。时
IN THIS paper we extend the class of 3-manifolds which are determined up to homeomorphism by their fundamental groups to the class of closed orientable irreducible 3-manifolds containing a singular surface satisfying two properties, the l-line-intersection property and the 4-plane property.A basic problem in the classification of 3-dimensional manifolds is to decide to what extent the homotopy type of a closed manifold determines the manifold up to homeomorphism. In the case of 3-manifolds with finite fundamental groups, it is known that there are homotopy equivalent manifolds which are not homeomorphic, but there are no known examples of closed orientable irreducible 3-manifolds with isomorphic infinite fundamental groups which are not homcomorphic. Waldhausen [30] and Heil Cl23 proved that if M and M’are Haken 3-manifolds which have isomorphic fundamental groups then they are homcomorphic. If M and M’arc hyperbolic then the Mostow rigidity theorem implies the same result [19]. but if only M is assumed to be hyperbolic then it is unknown. Boehme [3] extended Waldhausen’s theorem to certain non-Haken Seifert fiber spaces. Scott [27] showed that a closed orientable irreducible 3-manifold which is homotopy equivalent to a Seifcrt fiber space with infinite fundamental group is homeomorphic to that Seifert fiber space. Many of these Scifert fiber spaces are non-Haken. To date however, Seifert fiber spaces have provided the only examples of non-Haken 3-manifolds which are known to be determined up to homeomorphism by their fundamental groups. For manifolds with boundary there are simple examples of non-homemorphic homotopy equivalent Haken manifolds, such as the product with the circle of a thricepunctured sphere and the product with the circle of a once-punctured torus. Such examples are well understood in terms of the characteristic decomposition of the 3-manifold [17][16]. Another possible source of troublesome examples comes by taking the connected sum of a 3-manifold with a fake homotopy 3-sphere, resulting in a homotopy equivalent non-homeomorphic 3-manifold. This possibility would be ruled out by a successful solution to the Poincari conjecture. To get around this potential problem we work with irreducible 3-manifolds, in which any 2-sphere bounds a bail. Since non-orientable P2-irreducible 3-manifolds are always Haken, we restrict our attention to the orientable case. For simplicity of notation, we sometimes follow the convention of not distinguishing between a map of a surface into a manifold M and the image of the map. When it is