# Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where $Hom(S^{1},X)$, with based point consideration, is equipped to compact open topology. See D.K. Biss Topology and its Applications 124 (2002) 355-371.

Is it true that every topological group is the topological fundamental group of a path connected topological space?

## 1 Answer

The fundamental group equipped with the natural quotient topology is not always a topological group. In fact, there are many errors in Biss' paper that you reference. Enough so that it has been retracted from Topology and its Applications. This object you describe is still useful but now is usually called the quasitopological fundamental group and denoted $\pi^{qtop}(X,x)$. It is not true that every quasitopological group is isomorphic to some fundamental group $\pi^{qtop}(X,x)$. For more on it, see

J. Brazas, P. Fabel, On fundamental groups with the quotient topology, J. Homotopy and Related Structures 10 (2015) 71-91. arXiv

There is a natural topology you can put on $\pi_1(X,x)$ which makes it a topological group and for which many classical algebraic topology theorems have topological group analogues. This alternative topology is characterized as the finest group topology such that the function $Hom((S^1,b),(X,x))\to \pi_1(X,x)$ identifying homotopy classes is continuous (but may not be a quotient map). The resulting topological group is usually denoted $\pi^{\tau}_{1}(X,x)$. It is true that for every topological group $G$, there is some path-connected space $X$ such that $\pi^{\tau}_{1}(X,x)\cong G$.

Along with generalized covering space theory, $\pi_{1}^{\tau}$ helped to solve some older questions on open subgroups of topological groups.

See:

J. Brazas, The fundamental group as a topological group, Topology Appl. 160 (2013) 170-188 arXiv