Does the fundamental group provide information about the topological space?

Question

Today I started to study about fundamental group by my own free will. I would like to know if it is possible to extract information about a topological space $(X, \tau_X)$ based on its fundamental group. This information I talk about is for example: The topological space is Hausdorff, connected or compact.

Answer 1

Of course the fundamental group provides information about the topological space, but not that mentioned in your question:

This information I talk about is for example: The topological space is Hausdorff, connected or compact.

The fundamental group does not distinguish between homotopy equivalent spaces, and Hausdorffness and compactness are not invariant under homotopy equivalence. As an example consider a one-point space $X$ (which is compact Hausdorff), a two-point space $Y$ with the trivial topology (which is compact and non-Hausdorff) and $Z = \mathbb R$ (which is Hausdorff and non-compact). The fundamental group of all three spaces is trivial (i.e. has only one element).

Also connectedness cannot be detected via fundamental group. For example, a discrete space with more than one point is not connected and has trivial fundamental group for each basepoint.

The fundamental group $\pi_1(X,x_0)$ only depends on the path-component $P(x_0)$ of $x_0$ in $X$ ( i.e. $\pi_1(X,x_0) = \pi_1(P(x_0),x_0)$). In a pathwise connected space $P$ the groups $\pi_1(P,p)$ are isomorphic for all basepoints $p \in P$, thus if you have good luck you can verify that a space is not pathwise connected if you find to points $x_1, x_2 \in X$ such $\pi_1(X,x_i)$ are not isomorphic. But the disjoint union of two copies of a pathwise connected spaces is not pathwise connected and has the same fundamental group for each basepoint.