[This is a guest post written by Purnima Tiwari.]

Henri Poincaré defined the fundamental group in 1895 in his paper, titled Analysis situs, with the objective of formalizing the notion of 'holes' using continuous loops. This delves into the problem of distinguishing topological spaces up to homotopy equivalence, something that lies at the heart of algebraic topology.

Given a topological space $X$ with a basepoint $x_0$, we consider the set of continuous maps $\gamma_1, \gamma_2 : [0,1] \to X$ such that $\gamma_i (0)= \gamma_i (1)= x_0$. Under the homotopy equivalence relation, these loops form a group under loop concatenation: $[\gamma_1] \cdot [\gamma_2] = [\gamma_1 \star \gamma_2]$.

The fundamnetal group $\pi_1 (X, x_0)$ is defined as the set of loops (based at $x_0$), modulo path homotopy, where two loops are homotopic if there exists a continuous family $F: [0,1] \times [0,1] \to X$, continuously deforming $\gamma_0$ into $\gamma_1$ while keeping the basepoint $x_0$ fixed throughout: $F(s,0)=F(s,1)=x_0$. The group operation associated to the set is concatenation, which is associative, admits an identity and provides inverses modulo homotopy, denoted as: $$ \pi_1(X,x_0) := \{ \gamma: [0,1] \to X \mid \gamma(0)=\gamma(1)=x_0\} / \sim.$$ Unlike homology groups, $\pi_1(X, x_0)$ is generally non-abelian, hence storing non-commutative path configurations. This non-abelian structure makes the fundamental group a much richer invariant than the first homology group $H_1(X, \mathbb{Z})$ which by the Hurewicz theorem, is its abelianization:$$H_1(X, \mathbb{Z}) \cong \pi_1(X, x_0)^{\text{ab}} = \pi_1(X, x_0) / [\pi_1(X, x_0), \pi_1(X, x_0)]$$The Seifert–van Kampen theorem provides a way to compute $\pi_1$ by expressing the fundamental group of a union of open sets $U \cup V$ as a free product with amalgamation $\pi_1(U) *_{\pi_1(U \cap V)} \pi_1(V)$, showing that local loop structures can combine algebraically to give global loops.

The fundamental group detects 1-dimensional loop structures, and the higher-dimensional extensions were introduced by Witold Hurewicz in the 1930s by replacing the 1-sphere $S^1$ with higher-dimensional spheres $S^n$. For $n \ge 1$, the $n$-th homotopy group $\pi_n(X, x_0)$ is defined as the set of basepoint-preserving maps from $S^n$ to $X$ modulo basepoint-preserving homotopy:$$\pi_n(X, x_0) := [(S^n, s_0), (X, x_0)] = \{ f: S^n \to X \mid f(s_0) = x_0 \} \;/\; \sim$$The homotopy group $\pi_n(X, x_0)$ is always abelian for $n \ge 2$. This is proven via the Eckmann–Hilton argument, where having two commuting operations with a shared identity in dimensions $n \ge 2$ allows one map to 'slide around' the other, forcing commutativity:$$[f] + [g] = [g] + [f] \quad \text{for } n \ge 2.$$Despite being abelian, higher homotopy groups are notoriously difficult to compute compared to homology groups ($H_n(X)$). In homology, tools like the excision theorem and Mayer–Vietoris sequences allow for straightforward local-to-global calculations. Homotopy groups, however, lack general excision. An illustration of this subtle behavior is found in the homotopy groups of spheres $\pi_{n+k}(S^n)$. Here $H_k(S^n) = 0$ for all $k > n$, but higher homotopy groups $\pi_{n+k}(S^n)$ can be wildly non-trivial.The earliest hint of this was Heinz Hopf's 1931 discovery of the Hopf fibration $S^1 \hookrightarrow S^3 \to S^2$, which demonstrated that $\pi_3(S^2) \cong \mathbb{Z}$, generated by the map measuring the linking of fibers in $S^3$. Later work by Jean-Pierre Serre using spectral sequences showed that for any sphere $S^n$, the stable stems $\pi_{n+k}(S^n)$ (for large $n$) are finite abelian groups for $k > 0$, except when $k = 0$ or when $n$ is even with $k = n - 1$.

One may refer to this (https://arxiv.org/abs/1309.1198) for the advancements in the field of pro-étale topology for schemes, by Bhargav Bhatt and Peter Scholze.