Group structure on arbitrary topological spaces

Question

Definition. Let $(G,\ast)$ be any group. Then $G$ will be said to be a topological group if there exists a topology on $G$ such that the map $f:G\times G\to G$ defined by $f(x,y)=xy^{-1}$ for all $(x,y)\in G\times G$ is continuous.

Now observe that in this definition it is important to note that we first have a group and then the topology which makes $f$ cotinuous. In other words, we impose topological stricture on a group.

My question

1. Is it possible to impose group structure on a topological space? I don't think that it is possible always. So my question is, if $(X,\tau)$ be a topological space then under which condition(s) does there exists a binary operation $\ast:X\times X\to X$ such that $(X,\ast)$ is a group?

2. If there is any related research in mathematics literature then can you please let me know about some of those?

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It has been pointed out (see below) that this question is a duplicate of this question. However, I don't think that they are for the reason as explained in this comment. It is also not clear to me that imposing a group structure on a topological space (if it's possible of course) is expected to result in a topological group.

Answer 1

There is a gigantic literature on topological groups, with specialized literature on many special kinds of topological groups. You might start with Lie groups (topological groups which are manifolds). If you like number theory, you will discover that the $p$-adic numbers $\mathbb Q_p$ are a topological group which is locally homeomorphic to the Cantor set.

No, it is not always possible to impose a group structure on a topological space.

For example, a topological group $G$ must be homogeneous which means that for any two points $x,y \in G$ there exists a homeomorphism $L : G \to G$ such that $L(x)=y$: simply use the map $L(g)=f(g,y^{-1}x)$. But there are many nonhomogeneous topological spaces. The first one that comes to mind is $[0,1]$, which has no homeomorphism taking $0$ to $1/2$.

There are even homogeneous topological spaces on which one cannot impose a group structure, in particular many manifolds. You learn in the theory of Lie groups that if a topological group $G$ is a compact manifold then the Euler characteristic of $G$ equals zero. However, there are plenty of compact manifolds with nonzero Euler characterestic, starting with the 2-sphere $S^2$ which has Euler characteristic equal to $2$.