In his paper "Symmetric products of the circle" (1966) H. R. Morton proves among other things that the usual multiplication
\begin{align\*}
\text{SP}^n(S^1)&\to S^1 \\
[x\_1,\dots,x\_n]&\mapsto x\_1\cdots x\_n
\end{align\*}
is a fibration with fiber $\Delta^{n-1}$. Such a map can still be definied if one replaces $S^1$ by an arbitrary abelian topological group, so I was wondering under which conditions this continues to be a fibration and whether it still has such a nice fiber in this case.



Fibration from symmetric product to initial space
Let $A$ be a topological abelian group. One condition that guarantees that the map $\mu\_n\colon\operatorname{SP}^n(A)\to A$ is a fiber bundle is that $A$ is "locally divisible by $n$". More precisely, suppose there exists an open neighborhood $U$ of the identity, together with a continuous function $U\to A$ that behaves like $x\mapsto \frac{x}{n}$. Then for every $x\in A$, there is a homeomorphism $\mu^{-1}(x+U)\xrightarrow{\cong} (x+U)\times \mu^{-1}(x)$, which sends an unordered $n$-tuple $[y\_1, \ldots, y\_n]$ to the following $$\left(\mu\_n(y\_1,\ldots,y\_n), \left[y\_1-\frac{\mu\_n(y\_1,\ldots,y\_n)-x}{n}, \ldots, y\_n-\frac{\mu\_n(y\_1,\ldots,y\_n)-x}{n}\right]\right).$$


(I am using additive notation for $A$. In particular $\mu\_n(y\_1, \ldots, y\_n)=y\_1+\cdots+y\_n$)


If I am not mistaken, this condition holds, in particular, for all locally compact abelian groups.


Regarding the fiber of the map $\mu$, there is at least one other case when it has a very nice description. Namely, if $A=T^2$ is the torus, the fiber of the map $\operatorname{SP}^n(T^2)\to T^2$ is $\mathbb CP^{n-1}$. This is a special case of a fibration sequence that exists when $M\_g$ is the orientable surface of genus $g$ (and $n>2g-2$, as per Dan's comment)
$$\mathbb CP^{n-g}\to \operatorname{SP}^n(M\_g)\to T^{2g}.$$
This result appears to be due to Mattuck "Picard bundles" (1961).


However when $\dim(A)>2$ I suspect it is unlikely that there is such a nice description of the fiber. For example, consider the case $A=\mathbb R^m$. In this case the fiber of $\mu\_n$ is contractible, but it is homeomorphic to the cone of the space $\Sigma^{-(m-1)}S^{mn}{/\_{\Sigma\_n}}$. This is a well-studied space (For example its homology is well understood, at least with rational or mod $p$ coefficients). But it does not admit a simpler description than what I just wrote.


One more remark is that the homotopy type of fiber of the map $\mu\_\infty\colon \operatorname{SP}^\infty(A)\to A$ is easily described by the Dold-Thom theorem. The fibers of $\mu\_n$ define a filtration of the fiber of $\mu\_\infty$ by a sequence of subspaces, and it is conceivable that one could, taking this as a starting point, work out the homotopy type of the fiber of $\mu\_n$, or at least the homology of the fiber. Indeed, the homology of the space $\operatorname{SP}^n(A)$ is known, as a functor of $A$, and the Serre/Eilenberg-Moore spectral sequence of the map $\mu\_n$ might be tractable.


