Akira Koyama*
Shizuoka University, Suruga, Shizuoka, Japan
The symmetric products of the circle
By X(n) we denote the space of non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric and call the n-fold symmetric product of X. This was introduced by Borsul-Ulam (Bull. A. M. S. 37 (1931)).
In this talk we shall describe the symmetric product $\mathbb{S}^1(n)$ as
a compactification of an open cone over $\Sigma D^{n-2}$.
Then we shall determine the homotopy type of $\mathbb{S}^1(n)$ and
detect several topological properties of $\mathbb{S}^1(n)$.
As its consequence we determine the homotopy type of $\mathbb{S}^1(n)$ and
give an alternative proof of Borsuk-Bott theorem ``$\mathbb{S}^1(3) = \mathbb{S}^3$''.
* This is a joint work with Naotsugu Chinen.
