Construction of exotic homology manifolds and generalization of Quinn index

用SP^n（X）表示拓扑空间X的n重对称积。我们研究了什么样的n维紧度量空间可以嵌入到一维连续统X的n重对称积SP^n（X）中的问题。结果如下：定理1。n维球面S^n不能嵌入到任何一维连续统X的n重对称积中。为了证明定理，我们计算了1-球面S ^1的簇的n维上同调群。事实上，我们证明了1-球面S^1的n重对称乘积可以嵌入到1-球面S^1的n重对称乘积的笛卡尔乘积中。嵌入像是乘积空间的收缩核。因此，我们可以计算对称积的上同调群如下：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由Dydak-Koyama（Bull. Polish Academy of Sciences，2000，48（1），51-56）的定理1推出。我们有了另一个对称积的概念。设F_n（X）是X的基数不超过n的所有非空子集的集合。我们通常称赋Hausdorff度量的F_n（X）为X的n重对称积。一般来说，F_2n（X）等于SP^2（X），但当n > 2时，F_n（X）不同于SP^n（X）。然而，由于这些乘积具有相似性，我们面临着同样的问题：什么样的n维紧致度量空间可以嵌入到一维连续统X的n重对称乘积F_n（X）中。作为后续，我们知道Borsuk-Bott定理：F_3（S^1）同构于三维球面S^3。我们也研究了这个定理，并给出了一个现代证明和一个推广。

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.