TITLE: Fundamental group as topological group
QUESTION [48 upvotes]: Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the subset of maps mapping $1 \to x$ with the subspace topology, and finally use the quotient topology on $\pi_1(X,x)$. This topology is relevant in some situations. A very interesting paper dealing with this topology is:

[1] Daniel K. Biss, A Generalized Approach to the Fundamental Group, The American Mathematical Monthly, Vol. 107

You can find this online. This is somehow an introduction to

[2] Daniel K. Biss, The topological fundamental group and generalized covering spaces , Topology and its Applications, Vol. 124

Question
How can we prove that $\pi_1(X,x)$ is a topological group? Clearly the inversion map $\pi_1(X,x) \to \pi_1(X,x)$ is continuous, since $S^1 \to S^1, z \mapsto \overline{z}$ is continuous and induces this map. But I don't know how to attack the continuity of the multiplication. It's not hard to see that the multiplication on $map((S^1,1),(X,x))$ is continuous, since it is induced by a fold map $S^1 \to S^1 + S^1$. In order to carry this over to $\pi_1(X,x)$, there are at least two problems which I encounter:

The quotient map $map((S^1,1),(X,x)) \to \pi_1(X,x)$ may be not open.
The product of the quotient maps $map((S^1,1),(X,x))^2 \to \pi_1(X,x)^2$ may be not a quotient map.

In [1] it is claimed that $\pi_1(X,x)$ is always a topological group, and this should be proven in [2], but I have no acecss to [2].
An example that products of quotient maps don't have to be quotient maps can be found here. Remark however that this is true in the category of compactly generated spaces.

REPLY [2 votes]: What has not been said is that if $X$ is path connected and has a universal cover, then the fundamental groupoid $\pi_1(X)$ can be given a topology so that the source map $s:\pi_1 (X) \to X$  is a bundle over $X$ with fibre over $x \in X$ the universal cover of $X$ based at $x$: see Chapter 10 of Topology and Groupoids. This has been known for a long time.