Construction of exotic homology manifolds and generalization of Quinn index

我们用SP^n(X)表示拓扑空间X的n次对称积，研究了一维连续体X的n次对称积SP^n(X)中可以嵌入什么样的n维紧化度量空间的问题。n维球面S^n不能嵌入到任何一维连续统x的n折对称积中。为了证明定理，我们计算了1维球面S^1束的n维上同群。事实上，我们证明了1球S^1的束的n次对称积可以嵌入到1球S^1的n次对称积的笛卡尔积中。嵌入图像是产品空间的缩回。因此，我们可以计算对称积的上同群：定理2。H^n（SP^n(vS^1)）同构于n维环面的n维上同调群的直和。定理1是从dyak - koyama的定理1推导出来的。波兰科学院，2000,48(1),51-56。我们有另一个对称积的概念。即，对于一个拓扑空间X，设F_n(X)为X的所有非空子集的集合，其个数不超过n。我们通常称F_n(X)为赋予Hausdorff度规的X的n倍对称积。一般情况下，F_n(X)等于SP^2(X)，但如果n > 2, F_n(X)不同于SP^n(X)。然而，由于这些乘积具有相似性，我们又遇到了同样的问题，即一维连续体X的n次对称乘积F_n(X)中可以嵌入什么样的n维紧化度量空间。由此我们知道Borsuk-Bott定理：F_3（S^1）与三维球面S^3同构。我们也研究了这个定理，并给出了一个现代的证明和推广。

By SP^n(X) we denote the n-fold symmetric product of a topological space X. We have investigated the problem what kind of n-dimensional compact metric spaces can be embedded into the n-fold symmetric product SP^n(X) of a one-dimensional continuum X. Our result is the following :Theorem 1. The n-dimensional sphere S^n cannot be embedded into the n-fold symmetric product of any one-dimensional continuum X.In order to prove Theorem we have calculated the n-dimensional cohomology group of the bouquet of the 1-sphere S^1. In fact, we showed that the n-fold symmetric product of the bouquet of the 1-sphere S^1 can be embedded into the Cartesian product of the n-fold symmetric products of 1-sphere S^1. Moreover the embedding image is the retract of the product space. Therefore we can calculate cohomology group of the symmetric product as follows :Theorem 2. H^n(SP^n(vS^1)) is isomorphic to the direct sum of the n-dimensional cohomology groups of the n-dimensional tori.Theorem 1 follows from Theorem 1 by Dydak-Koyama(Bull. Polish Academy of Sciences, 2000, 48(1), 51-56).We have another notion of symmetric products. Namely, for a topological space X let F_n(X) be the set of all nonempty subsets of X whose cardinalities are at most n. We often call F_n(X) endowed the Hausdorff metric the n-fold symmetric product of X. In general, F_2n(X) is equal to SP^2(X), but if n > 2, F_n(X) is different from SP^n(X). However, as those product have similarities, we have the same problem what kind of n-dimensional compact metric spaces can be embedded into the n-fold symmetric product F_n(X) of a one-dimensional continuum X. As a folhlore we know Borsuk-Bott Theorem: F_3(S^1) is isomorphic to the 3-dimensional sphere S^3. We also investigate this theorem and give a modern proof and a generalizations.