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Fundamental group

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space $$X$$ is denoted by $$\pi_1(X)$$.

Intuition
Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point—paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.

History
Henri Poincaré defined the fundamental group in 1895 in his paper "Analysis situs". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.

Definition
Throughout this article, X is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, $$x_0$$ is a point in X called the base-point. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on X can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

Homotopy of loops
Given a topological space X, a loop based at $$x_0$$ is defined to be a continuous function (also known as a continuous map)
 * $$\gamma \colon [0, 1] \to X$$

such that the starting point $$\gamma(0)$$ and the end point $$\gamma(1)$$ are both equal to $$x_0$$.

A homotopy is a continuous interpolation between two loops. More precisely, a homotopy between two loops $$\gamma, \gamma' \colon [0, 1] \to X$$ (based at the same point $$x_0$$) is a continuous map
 * $$h \colon [0, 1] \times [0, 1] \to X,$$

such that
 * $$h(0, t) = x_0$$ for all $$t \in [0, 1],$$ that is, the starting point of the homotopy is $$x_0$$ for all t (which is often thought of as a time parameter).
 * $$h(1, t) = x_0$$ for all $$t \in [0, 1],$$ that is, similarly the end point stays at $$x_0$$ for all t.
 * $$h(r, 0) = \gamma(r),\, h(r, 1) = \gamma'(r)$$ for all $$r \in [0, 1]$$.

If such a homotopy h exists, $$\gamma$$ and $$\gamma'$$ are said to be homotopic. The relation "$$\gamma$$ is homotopic to $$\gamma'$$" is an equivalence relation so that the set of equivalence classes can be considered:
 * $$\pi_1(X, x_0) := \{ \text{all loops }\gamma \text{ based at }x_0 \} / \text{homotopy}$$.

This set (with the group structure described below) is called the fundamental group of the topological space X at the base point $$x_0$$. The purpose of considering the equivalence classes of loops up to homotopy, as opposed to the set of all loops (the so-called loop space of X) is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above quotient is, in many cases, more manageable and computable.

Group structure
By the above definition, $$\pi_1(X, x_0)$$ is just a set. It becomes a group (and therefore deserves the name fundamental group) using the concatenation of loops. More precisely, given two loops $$\gamma_0, \gamma_1$$, their product is defined as the loop
 * $$\gamma_0 \cdot \gamma_1 \colon [0, 1] \to X$$
 * $$(\gamma_0 \cdot \gamma_1)(t) = \begin{cases}

\gamma_0(2t) & 0 \leq t \leq \tfrac{1}{2} \\ \gamma_1(2t - 1) & \tfrac{1}{2} \leq t \leq 1. \end{cases}$$ Thus the loop $$\gamma_0 \cdot \gamma_1$$ first follows the loop $$\gamma_0$$ with "twice the speed" and then follows $$\gamma_1$$ with "twice the speed".

The product of two homotopy classes of loops $$[\gamma_0]$$ and $$[\gamma_1]$$ is then defined as $$[\gamma_0 \cdot \gamma_1]$$. It can be shown that this product does not depend on the choice of representatives and therefore gives a well-defined operation on the set $$\pi_1(X, x_0)$$. This operation turns $$\pi_1(X, x_0)$$ into a group. Its neutral element is the constant loop, which stays at $$x_0$$ for all times t. The inverse of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction. More formally,
 * $$\gamma^{-1}(t) := \gamma(1-t)$$.

Given three based loops $$\gamma_0, \gamma_1, \gamma_2,$$ the product
 * $$(\gamma_0 \cdot \gamma_1) \cdot \gamma_2$$

is the concatenation of these loops, traversing $$\gamma_0$$ and then $$\gamma_1$$ with quadruple speed, and then $$\gamma_2$$ with double speed. By comparison,
 * $$\gamma_0 \cdot (\gamma_1 \cdot \gamma_2)$$

traverses the same paths (in the same order), but $$\gamma_0$$ with double speed, and $$\gamma_1, \gamma_2$$ with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The associativity axiom
 * $$[\gamma_0] \cdot \left([\gamma_1] \cdot [\gamma_2]\right) = \left([\gamma_0] \cdot [\gamma_1]\right) \cdot [\gamma_2]$$

therefore crucially depends on the fact that paths are considered up to homotopy. Indeed, both above composites are homotopic, for example, to the loop that traverses all three loops $$\gamma_0, \gamma_1, \gamma_2$$ with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn $$\pi_1(X, x_0)$$ into a group.

Dependence of the base point
Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism (actually, even up to inner isomorphism), this choice makes no difference as long as the space X is path-connected. For path-connected spaces, therefore, many authors write $$\pi_1(X)$$ instead of $$\pi_1(X, x_0).$$

Concrete examples
This section lists some basic examples of fundamental groups. To begin with, in Euclidean space ($$\R^n$$) or any convex subset of $$\R^n,$$ there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. More generally, any star domain – and yet more generally, any contractible space – has a trivial fundamental group. Thus, the fundamental group does not distinguish between such spaces.

The 2-sphere
A path-connected space whose fundamental group is trivial is called simply connected. For example, the 2-sphere $$S^2 = \left\{(x, y, z) \in \R^3 \mid x^2 + y^2 + z^2 = 1\right\}$$ depicted on the right, and also all the higher-dimensional spheres, are simply-connected. The figure illustrates a homotopy contracting one particular loop to the constant loop. This idea can be adapted to all loops $$\gamma$$ such that there is a point $$(x, y, z) \in S^2$$ that is in the image of $$\gamma.$$ However, since there are loops such that $$\gamma([0, 1]) = S^2$$ (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such as the Seifert–van Kampen theorem or the cellular approximation theorem.

The circle
The circle (also known as the 1-sphere)
 * $$S^1 = \left\{(x, y) \in \R^2 \mid x^2 + y^2 = 1\right\}$$

is not simply connected. Instead, each homotopy class consists of all loops that wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop that winds around m times and another that winds around n times is a loop that winds around m + n times. Therefore, the fundamental group of the circle is isomorphic to $$(\Z, +),$$ the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2.

The figure eight
The fundamental group of the figure eight is the free group on two letters. The idea to prove this is as follows: choosing the base point to be the point where the two circles meet (dotted in black in the picture at the right), any loop $$\gamma$$ can be decomposed as
 * $$\gamma = a^{n_1} b^{m_1} \cdots a^{n_k} b^{m_k}$$

where a and b are the two loops winding around each half of the figure as depicted, and the exponents $$n_1, \dots, n_k, m_1, \dots, m_k$$ are integers. Unlike $$\pi_1(S^1),$$ the fundamental group of the figure eight is not abelian: the two ways of composing a and b are not homotopic to each other:
 * $$[a] \cdot [b] \ne [b] \cdot [a].$$

More generally, the fundamental group of a bouquet of r circles is the free group on r letters.

The fundamental group of a wedge sum of two path connected spaces X and Y can be computed as the free product of the individual fundamental groups:
 * $$\pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y).$$

This generalizes the above observations since the figure eight is the wedge sum of two circles.

The fundamental group of the plane punctured at n points is also the free group with n generators. The i-th generator is the class of the loop that goes around the i-th puncture without going around any other punctures.

Graphs
The fundamental group can be defined for discrete structures too. In particular, consider a connected graph, with a designated vertex v0 in V. The loops in G are the cycles that start and end at v0. Let T be a spanning tree of G. Every simple loop in G contains exactly one edge in E \ T; every loop in G is a concatenation of such simple loops. Therefore, the fundamental group of a graph is a free group, in which the number of generators is exactly the number of edges in E \ T. This number equals E.

For example, suppose G has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then G has 24 edges overall, and the number of edges in each spanning tree is 16 − 1 = 15, so the fundamental group of G is the free group with 9 generators. Note that G has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

Knot groups
Knot groups are by definition the fundamental group of the complement of a knot K embedded in $$\R^3.$$ For example, the knot group of the trefoil knot is known to be the braid group $$B_3,$$ which gives another example of a non-abelian fundamental group. The Wirtinger presentation explicitly describes knot groups in terms of generators and relations based on a diagram of the knot. Therefore, knot groups have some usage in knot theory to distinguish between knots: if $$\pi_1(\R^3 \setminus K)$$ is not isomorphic to some other knot group $$\pi_1(\R^3 \setminus K')$$ of another knot K′, then K can not be transformed into K′. Thus the trefoil knot can not be continuously transformed into the circle (also known as the unknot), since the latter has knot group $$\Z$$. There are, however, knots that can not be deformed into each other, but have isomorphic knot groups.

Oriented surfaces
The fundamental group of a genus-n orientable surface can be computed in terms of generators and relations as
 * $$\left\langle A_1, B_1, \ldots, A_n, B_n \left| A_1 B_1 A_1^{-1} B_1^{-1} \cdots A_n B_n A_n^{-1} B_n^{-1} \right. \right\rangle.$$

This includes the torus, being the case of genus 1, whose fundamental group is
 * $$\left\langle A_1, B_1 \left| A_1 B_1 A_1^{-1} B_1^{-1} \right. \right\rangle \cong \Z^2.$$

Topological groups
The fundamental group of a topological group X (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a Lie group is commutative. In fact, the group structure on X endows $$\pi_1(X)$$ with another group structure: given two loops $$\gamma$$ and $$\gamma'$$ in X, another loop $$\gamma \star \gamma'$$ can defined by using the group multiplication in X:
 * $$(\gamma \star \gamma')(x) = \gamma(x) \cdot \gamma'(x).$$

This binary operation $$\star$$ on the set of all loops is a priori independent from the one described above. However, the Eckmann–Hilton argument shows that it does in fact agree with the above concatenation of loops, and moreover that the resulting group structure is abelian.

An inspection of the proof shows that, more generally, $$\pi_1(X)$$ is abelian for any H-space X, i.e., the multiplication need not have an inverse, nor does it have to be associative. For example, this shows that the fundamental group of a loop space of another topological space Y, $$X = \Omega(Y),$$ is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.

Functoriality
If $$f\colon X \to Y$$ is a continuous map, $$x_0 \in X$$ and $$y_0 \in Y$$ with $$f(x_0) = y_0,$$ then every loop in $$X$$ with base point $$x_0$$ can be composed with $$f$$ to yield a loop in $$Y$$ with base point $$y_0.$$ This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as $$\pi(f)$$ or, more commonly,

 * $$f_* \colon \pi_1(X, x_0) \to \pi_1(Y, y_0).$$

This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In the parlance of category theory, the formation of associating to a topological space its fundamental group is therefore a functor

 * $$\begin{align}

\pi_1 \colon \mathbf{Top}_* &\to \mathbf{Grp} \\ (X, x_0) &\mapsto \pi_1(X, x_0) \end{align}$$

from the category of topological spaces together with a base point to the category of groups. It turns out that this functor does not distinguish maps that are homotopic relative to the base point: if $$f,g:X\to Y$$ are continuous maps with $$f(x_0) = g(x_0) = y_0$$, and f and g are homotopic relative to {x0}, then f∗ = g∗. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:

 * $$X \simeq Y \implies \pi_1(X, x_0) \cong \pi_1(Y, y_0).$$

For example, the inclusion of the circle in the punctured plane
 * $$S^1 \subset \mathbb{R}^2 \setminus \{0\}$$

is a homotopy equivalence and therefore yields an isomorphism of their fundamental groups.

The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then

 * $$\pi_1 (X \times Y, (x_0, y_0)) \cong \pi_1(X, x_0) \times \pi_1(Y, y_0)$$

and if they are also locally contractible, then

 * $$\pi_1(X \vee Y) \cong \pi_1(X)*\pi_1(Y).$$

(In the latter formula, $$\vee$$ denotes the wedge sum of pointed topological spaces, and $$*$$ the free product of groups.) The latter formula is a special case of the Seifert–van Kampen theorem, which states that the fundamental group functor takes pushouts along inclusions to pushouts.

Abstract results
As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of algebraic topology.

Relationship to first homology group
The abelianization of the fundamental group can be identified with the first homology group of the space.

A special case of the Hurewicz theorem asserts that the first singular homology group $$H_1(X)$$ is, colloquially speaking, the closest approximation to the fundamental group by means of an abelian group. In more detail, mapping the homotopy class of each loop to the homology class of the loop gives a group homomorphism
 * $$\pi_1(X) \to H_1(X)$$

from the fundamental group of a topological space X to its first singular homology group $$H_1(X).$$ This homomorphism is not in general an isomorphism since the fundamental group may be non-abelian, but the homology group is, by definition, always abelian. This difference is, however, the only one: if X is path-connected, this homomorphism is surjective and its kernel is the commutator subgroup of the fundamental group, so that $$H_1(X)$$ is isomorphic to the abelianization of the fundamental group.

Gluing topological spaces
Generalizing the statement above, for a family of path connected spaces $$X_i,$$ the fundamental group $\pi_1 \left(\bigvee_{i \in I} X_i\right)$ is the free product of the fundamental groups of the $$X_i.$$ This fact is a special case of the Seifert–van Kampen theorem, which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces. For example, the 2-sphere $$S^2$$ can be obtained by gluing two copies of slightly overlapping half-spheres along a neighborhood of the equator. In this case the theorem yields $$\pi_1(S^2)$$ is trivial, since the two half-spheres are contractible and therefore have trivial fundamental group. The fundamental groups of surfaces, as mentioned above, can also be computed using this theorem.

In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes pushouts (in the category of topological spaces) along inclusions to pushouts (in the category of groups).

Coverings
Given a topological space B, a continuous map
 * $$f: E \to B$$

is called a covering or E is called a covering space of B if every point b in B admits an open neighborhood U such that there is a homeomorphism between the preimage of U and a disjoint union of copies of U (indexed by some set I),
 * $$\varphi: \bigsqcup_{i \in I} U \to f^{-1}(U)$$

in such a way that $$\pi \circ \varphi$$ is the standard projection map $$\bigsqcup_{i \in I} U \to U.$$

Universal covering
A covering is called a universal covering if E is, in addition to the preceding condition, simply connected. It is universal in the sense that all other coverings can be constructed by suitably identifying points in E. Knowing a universal covering
 * $$p: \widetilde{X} \to X$$

of a topological space X is helpful in understanding its fundamental group in several ways: first, $$\pi_1(X)$$ identifies with the group of deck transformations, i.e., the group of homeomorphisms $$\varphi : \widetilde{X} \to \widetilde{X}$$ that commute with the map to X, i.e., $$p \circ \varphi = p.$$ Another relation to the fundamental group is that $$\pi_1(X, x)$$ can be identified with the fiber $$p^{-1}(x).$$ For example, the map
 * $$p: \mathbb{R} \to S^1,\, t \mapsto \exp(2 \pi i t)$$

(or, equivalently, $$\pi: \mathbb{R} \to \mathbb{R} / \mathbb{Z},\ t \mapsto [t]$$) is a universal covering. The deck transformations are the maps $$t \mapsto t + n$$ for $$n \in \mathbb{Z}.$$ This is in line with the identification $$p^{-1}(1) = \mathbb{Z},$$ in particular this proves the above claim $$\pi_1(S^1) \cong \mathbb{Z}.$$

Any path connected, locally path connected and locally simply connected topological space X admits a universal covering. An abstract construction proceeds analogously to the fundamental group by taking pairs (x, γ), where x is a point in X and γ is a homotopy class of paths from x0 to x. The passage from a topological space to its universal covering can be used in understanding the geometry of X. For example, the uniformization theorem shows that any simply connected Riemann surface is (isomorphic to) either $$S^2,$$ $$\mathbb{C},$$ or the upper half plane. General Riemann surfaces then arise as quotients of group actions on these three surfaces.

The quotient of a free action of a discrete group G on a simply connected space Y has fundamental group
 * $$\pi_1(Y/G) \cong G.$$

As an example, the real n-dimensional real projective space $$\mathbb{R}\mathrm{P}^n$$ is obtained as the quotient of the n-dimensional unit sphere $$S^n$$ by the antipodal action of the group $$\mathbb{Z}/2$$ sending $$x \in S^n$$ to $$-x.$$ As $$S^n$$ is simply connected for n ≥ 2, it is a universal cover of $$\mathbb{R}\mathrm{P}^n$$ in these cases, which implies $$\pi_1(\mathbb{R}\mathrm{P}^n) \cong \mathbb{Z}/2$$ for n ≥ 2.

Lie groups
Let G be a connected, simply connected compact Lie group, for example, the special unitary group SU(n), and let Γ be a finite subgroup of G. Then the homogeneous space X = G/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space G. Among the many variants of this construction, one of the most important is given by locally symmetric spaces X = Γ \G/K, where

 * G is a non-compact simply connected, connected Lie group (often semisimple),
 * K is a maximal compact subgroup of G
 * Γ is a discrete countable torsion-free subgroup of G.

In this case the fundamental group is Γ and the universal covering space G/K is actually contractible (by the Cartan decomposition for Lie groups).

As an example take G = SL(2, R), K = SO(2) and Γ any torsion-free congruence subgroup of the modular group SL(2, Z).

From the explicit realization, it also follows that the universal covering space of a path connected topological group H is again a path connected topological group G. Moreover, the covering map is a continuous open homomorphism of G onto H with kernel Γ, a closed discrete normal subgroup of G:

 * $$1 \to \Gamma \to G \to H \to 1.$$

Since G is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the center of G. In particular π1(H) = Γ is an abelian group; this can also easily be seen directly without using covering spaces. The group G is called the universal covering group of H.

As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at Lattice of covering groups.

Fibrations
Fibrations provide a very powerful means to compute homotopy groups. A fibration f the so-called total space, and the base space B has, in particular, the property that all its fibers $$f^{-1}(b)$$ are homotopy equivalent and therefore can not be distinguished using fundamental groups (and higher homotopy groups), provided that B is path-connected. Therefore, the space E can be regarded as a "twisted product" of the base space B and the fiber $$F = f^{-1}(b).$$ The great importance of fibrations to the computation of homotopy groups stems from a long exact sequence
 * $$\dots \to \pi_2(B) \to \pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \pi_0(F) \to \pi_0(E)$$

provided that B is path-connected. The term $$\pi_2(B)$$ is the second homotopy group of B, which is defined to be the set of homotopy classes of maps from $$S^2$$ to B, in direct analogy with the definition of $$\pi_1.$$

If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism
 * $$\pi_1(B) \cong \pi_0(F)$$

which generalizes the above fact about the universal covering (which amounts to the case where the fiber F is also discrete). If instead F happens to be connected and simply connected, it reduces to an isomorphism
 * $$\pi_1(E) \cong \pi_1(B).$$

What is more, the sequence can be continued at the left with the higher homotopy groups $$\pi_n$$ of the three spaces, which gives some access to computing such groups in the same vein.

Classical Lie groups
Such fiber sequences can be used to inductively compute fundamental groups of compact classical Lie groups such as the special unitary group $$\mathrm{SU}(n),$$ with $$n \geq 2.$$ This group acts transitively on the unit sphere $$S^{2n-1}$$ inside $$\mathbb C^n = \mathbb R^{2n}.$$ The stabilizer of a point in the sphere is isomorphic to $$\mathrm{SU}(n-1).$$ It then can be shown that this yields a fiber sequence
 * $$\mathrm{SU}(n-1) \to \mathrm{SU}(n) \to S^{2n-1}.$$

Since $$n \geq 2,$$ the sphere $$S^{2n-1}$$ has dimension at least 3, which implies
 * $$\pi_1(S^{2n-1}) \cong \pi_2(S^{2n-1}) = 1.$$

The long exact sequence then shows an isomorphism
 * $$\pi_1(\mathrm{SU}(n)) \cong \pi_1(\mathrm{SU}(n - 1)).$$

Since $$\mathrm{SU}(1)$$ is a single point, so that $$\pi_1(\mathrm{SU}(1))$$ is trivial, this shows that $$\mathrm{SU}(n)$$ is simply connected for all $$n.$$

The fundamental group of noncompact Lie groups can be reduced to the compact case, since such a group is homotopic to its maximal compact subgroup. These methods give the following results:

A second method of computing fundamental groups applies to all connected compact Lie groups and uses the machinery of the maximal torus and the associated root system. Specifically, let $$T$$ be a maximal torus in a connected compact Lie group $$K,$$ and let $$\mathfrak t$$ be the Lie algebra of $$T.$$ The exponential map
 * $$\exp : \mathfrak t \to T$$

is a fibration and therefore its kernel $$\Gamma \subset \mathfrak t$$ identifies with $$\pi_1(T).$$ The map
 * $$\pi_1(T) \to \pi_1(K)$$

can be shown to be surjective with kernel given by the set I of integer linear combination of coroots. This leads to the computation
 * $$\pi_1(K) \cong \Gamma / I.$$

This method shows, for example, that any connected compact Lie group for which the associated root system is of type $G_2$ is simply connected. Thus, there is (up to isomorphism) only one connected compact Lie group having Lie algebra of type $$G_2$$; this group is simply connected and has trivial center.

Edge-path group of a simplicial complex
When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.

If X is a connected simplicial complex, an edge-path in X is defined to be a chain of vertices connected by edges in X. Two edge-paths are said to be edge-equivalent if one can be obtained from the other by successively switching between an edge and the two opposite edges of a triangle in X. If v is a fixed vertex in X, an edge-loop at v is an edge-path starting and ending at v. The edge-path group E(X, v) is defined to be the set of edge-equivalence classes of edge-loops at v, with product and inverse defined by concatenation and reversal of edge-loops.

The edge-path group is naturally isomorphic to π1(|X |, v), the fundamental group of the geometric realisation |X | of X. Since it depends only on the 2-skeleton X 2 of X (that is, the vertices, edges, and triangles of X), the groups π1(|X |,v) and π1(|X 2|, v) are isomorphic.

The edge-path group can be described explicitly in terms of generators and relations. If T is a maximal spanning tree in the 1-skeleton of X, then E(X, v) is canonically isomorphic to the group with generators (the oriented edge-paths of X not occurring in T) and relations (the edge-equivalences corresponding to triangles in X). A similar result holds if T is replaced by any simply connected—in particular contractible—subcomplex of X. This often gives a practical way of computing fundamental groups and can be used to show that every finitely presented group arises as the fundamental group of a finite simplicial complex. It is also one of the classical methods used for topological surfaces, which are classified by their fundamental groups.

The universal covering space of a finite connected simplicial complex X can also be described directly as a simplicial complex using edge-paths. Its vertices are pairs (w,γ) where w is a vertex of X and γ is an edge-equivalence class of paths from v to w. The k-simplices containing (w,γ) correspond naturally to the k-simplices containing w. Each new vertex u of the k-simplex gives an edge wu and hence, by concatenation, a new path γu from v to u. The points (w,γ) and (u, γu) are the vertices of the "transported" simplex in the universal covering space. The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just X.

It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space. This was doubtless known to Eduard Čech and Jean Leray and explicitly appeared as a remark in a paper by André Weil; various other authors such as Lorenzo Calabi, Wu Wen-tsün, and Nodar Berikashvili have also published proofs. In the simplest case of a compact space X with a finite open covering in which all non-empty finite intersections of open sets in the covering are contractible, the fundamental group can be identified with the edge-path group of the simplicial complex corresponding to the nerve of the covering.

Realizability

 * Every group can be realized as the fundamental group of a connected CW-complex of dimension 2 (or higher). As noted above, though, only free groups can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs).
 * Every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). But there are severe restrictions on which groups occur as fundamental groups of low-dimensional manifolds.  For example, no free abelian group of rank 4 or higher can be realized as the fundamental group of a manifold of dimension 3 or less. It can be proved that every group can be realized as the fundamental group of a compact Hausdorff space if and only if there is no measurable cardinal.

Higher homotopy groups
Roughly speaking, the fundamental group detects the 1-dimensional hole structure of a space, but not higher-dimensional holes such as for the 2-sphere. Such "higher-dimensional holes" can be detected using the higher homotopy groups $$\pi_n(X)$$, which are defined to consist of homotopy classes of (basepoint-preserving) maps from $$S^n$$ to X. For example, the Hurewicz theorem implies that for all $$n \ge 1$$ the n-th homotopy group of the n-sphere is
 * $$\pi_n(S^n) = \Z.$$

As was mentioned in the above computation of $$\pi_1$$ of classical Lie groups, higher homotopy groups can be relevant even for computing fundamental groups.

Loop space
The set of based loops (as is, i.e. not taken up to homotopy) in a pointed space X, endowed with the compact open topology, is known as the loop space, denoted $$\Omega X.$$ The fundamental group of X is in bijection with the set of path components of its loop space:
 * $$\pi_1(X) \cong \pi_0(\Omega X).$$

Fundamental groupoid
The fundamental groupoid is a variant of the fundamental group that is useful in situations where the choice of a base point $$x_0 \in X$$ is undesirable. It is defined by first considering the category of paths in $$X,$$ i.e., continuous functions
 * $$\gamma \colon [0, r] \to X$$,

where r is an arbitrary non-negative real number. Since the length r is variable in this approach, such paths can be concatenated as is (i.e., not up to homotopy) and therefore yield a category. Two such paths $$\gamma, \gamma'$$ with the same endpoints and length r, resp. r' are considered equivalent if there exist real numbers $$u,v \geqslant 0$$ such that $$r + u = r' + v$$ and $$ \gamma_u, \gamma'_v \colon [0, r + u] \to X$$ are homotopic relative to their end points, where $$ \gamma_u (t) = \begin{cases} \gamma(t), & t \in [0, r] \\ \gamma(r), & t \in [r, r + u]. \end{cases} $$

The category of paths up to this equivalence relation is denoted $$\Pi (X).$$ Each morphism in $$\Pi (X)$$ is an isomorphism, with inverse given by the same path traversed in the opposite direction. Such a category is called a groupoid. It reproduces the fundamental group since
 * $$\pi_1(X, x_0) = \mathrm{Hom}_{\Pi(X)}(x_0, x_0)$$.

More generally, one can consider the fundamental groupoid on a set A of base points, chosen according to the geometry of the situation; for example, in the case of the circle, which can be represented as the union of two connected open sets whose intersection has two components, one can choose one base point in each component. The van Kampen theorem admits a version for fundamental groupoids which gives, for example, another way to compute the fundamental group(oid) of $$S^1.$$

Local systems
Generally speaking, representations may serve to exhibit features of a group by its actions on other mathematical objects, often vector spaces. Representations of the fundamental group have a very geometric significance: any local system (i.e., a sheaf $$\mathcal F$$ on X with the property that locally in a sufficiently small neighborhood U of any point on X, the restriction of F is a constant sheaf of the form $$\mathcal F|_U = \Q^n$$) gives rise to the so-called monodromy representation, a representation of the fundamental group on an n-dimensional $$\Q$$-vector space. Conversely, any such representation on a path-connected space X arises in this manner. This equivalence of categories between representations of $$\pi_1(X)$$ and local systems is used, for example, in the study of differential equations, such as the Knizhnik–Zamolodchikov equations.

Étale fundamental group
In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group. Since the Zariski topology on an algebraic variety or scheme X is much coarser than, say, the topology of open subsets in $$\R^n,$$ it is no longer meaningful to consider continuous maps from an interval to X. Instead, the approach developed by Grothendieck consists in constructing $$\pi_1^\text{et}$$ by considering all finite étale covers of X. These serve as an algebro-geometric analogue of coverings with finite fibers.

This yields a theory applicable in situation where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a finite field. Also, the étale fundamental group of a field is its (absolute) Galois group. On the other hand, for smooth varieties X over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.

Fundamental group of algebraic groups
The fundamental group of a root system is defined, in analogy to the computation for Lie groups. This allows to define and use the fundamental group of a semisimple linear algebraic group G, which is a useful basic tool in the classification of linear algebraic groups.

Fundamental group of simplicial sets
The homotopy relation between 1-simplices of a simplicial set X is an equivalence relation if X is a Kan complex but not necessarily so in general. Thus, $$\pi_1$$ of a Kan complex can be defined as the set of homotopy classes of 1-simplices. The fundamental group of an arbitrary simplicial set X are defined to be the homotopy group of its topological realization, $$|X|,$$ i.e., the topological space obtained by gluing topological simplices as prescribed by the simplicial set structure of X.

title: Fundamental group


text:
February 19

Pre-1600

 * 197 – Emperor Septimius Severus defeats usurper Clodius Albinus in the Battle of Lugdunum, the bloodiest battle between Roman armies.
 * 356 – The anti-paganism policy of Constantius II forbids the worship of pagan idols in the Roman Empire.
 * 1594 – Having already been elected to the throne of the Polish–Lithuanian Commonwealth in 1587, Sigismund III of the House of Vasa is crowned King of Sweden, having succeeded his father John III of Sweden in 1592.
 * 1600 – The Peruvian stratovolcano Huaynaputina explodes in the most violent eruption in the recorded history of South America.

1601–1900

 * 1649 – The Second Battle of Guararapes takes place, effectively ending Dutch colonization efforts in Brazil.
 * 1674 – England and the Netherlands sign the Treaty of Westminster, ending the Third Anglo-Dutch War. A provision of the agreement transfers the Dutch colony of New Amsterdam to England.
 * 1714 – Great Northern War: The battle of Napue between Sweden and Russia is fought in Isokyrö, Ostrobothnia.
 * 1726 – The Supreme Privy Council is established in Russia.
 * 1807 – Former Vice President of the United States Aaron Burr is arrested for treason in Wakefield, Alabama, and confined to Fort Stoddert.
 * 1819 – British explorer William Smith discovers the South Shetland Islands and claims them in the name of King George III.
 * 1836 – King William IV signs Letters Patent establishing the Province of South Australia.
 * 1846 – In Austin, Texas, the newly formed Texas state government is officially installed. The Republic of Texas government officially transfers power to the State of Texas government following the annexation of Texas by the United States.
 * 1847 – The first group of rescuers reaches the Donner Party.
 * 1878 – Thomas Edison patents the phonograph.
 * 1884 – More than sixty tornadoes strike the Southern United States, one of the largest tornado outbreaks in U.S. history.

1901–present

 * 1913 – Pedro Lascuráin becomes President of Mexico for 45 minutes; this is the shortest term to date of any person as president of any country.
 * 1915 – World War I: The first naval attack on the Dardanelles begins when a strong Anglo-French task force bombards Ottoman artillery along the coast of Gallipoli.
 * 1937 – Yekatit 12: During a public ceremony at the Viceregal Palace (the former Imperial residence) in Addis Ababa, Ethiopia, two Ethiopian nationalists of Eritrean origin attempt to kill viceroy Rodolfo Graziani with a number of grenades.
 * 1942 – World War II: Nearly 250 Japanese warplanes attack the northern Australian city of Darwin, killing 243 people.
 * 1942  – World War II: United States President Franklin D. Roosevelt signs executive order 9066, allowing the United States military to relocate Japanese Americans to internment camps.
 * 1943 – World War II: Battle of Kasserine Pass in Tunisia begins.
 * 1945 – World War II: Battle of Iwo Jima: About 30,000 United States Marines land on the island of Iwo Jima.
 * 1948 – The Conference of Youth and Students of Southeast Asia Fighting for Freedom and Independence convenes in Calcutta.
 * 1949 – Ezra Pound is awarded the first Bollingen Prize in poetry by the Bollingen Foundation and Yale University.
 * 1953 – Book censorship in the United States: The Georgia Literature Commission is established.
 * 1954 – Transfer of Crimea: The Soviet Politburo of the Soviet Union orders the transfer of the Crimean Oblast from the Russian SFSR to the Ukrainian SSR.
 * 1959 – The United Kingdom grants Cyprus independence, which is formally proclaimed on August 16, 1960.
 * 1960 – China successfully launches the T-7, its first sounding rocket.
 * 1963 – The publication of Betty Friedan's The Feminine Mystique reawakens the feminist movement in the United States as women's organizations and consciousness raising groups spread.
 * 1965 – Colonel Phạm Ngọc Thảo of the Army of the Republic of Vietnam, and a communist spy of the North Vietnamese Viet Minh, along with Generals Lâm Văn Phát and Trần Thiện Khiêm, all Catholics, attempt a coup against the military junta of the Buddhist Nguyễn Khánh.
 * 1976 – Executive Order 9066, which led to the relocation of Japanese Americans to internment camps, is rescinded by President Gerald Ford's Proclamation 4417.
 * 1978 – Egyptian forces raid Larnaca International Airport in an attempt to intervene in a hijacking, without authorisation from the Republic of Cyprus authorities. The Cypriot National Guard and Police forces kill 15 Egyptian commandos and destroy the Egyptian C-130 transport plane in open combat.
 * 1985 – William J. Schroeder becomes the first recipient of an artificial heart to leave the hospital.
 * 1985  – A Boeing 727 operating as Iberia Flight 610 crashes Mount Oiz in Spain, killing 148; it is the deadliest accident to occur in Iberia's history and the deadliest in Basque County to occur.
 * 1986 – Akkaraipattu massacre: the Sri Lankan Army massacres 80 Tamil farm workers in eastern Sri Lanka.
 * 1989 – Flying Tiger Line flight 66 crashes into a hill near Sultan Abdul Aziz Shah Airport in Malaysia, killing four.
 * 2002 – NASA's Mars Odyssey space probe begins to map the surface of Mars using its thermal emission imaging system.
 * 2003 – An Ilyushin Il-76 military aircraft crashes near Kerman, Iran, killing 275.
 * 2006 – A methane explosion in a coal mine near Nueva Rosita, Mexico, kills 65 miners.
 * 2011 – The debut exhibition of the Belitung shipwreck, containing the largest collection of Tang dynasty artifacts found in one location, begins in Singapore.
 * 2012 – Forty-four people are killed in a prison brawl in Apodaca, Nuevo León, Mexico.
 * 2021 – Mya Thwe Thwe Khine, a 19-year-old protester, becomes the first known casualty of anti-coup protests that formed in response to the 2021 Myanmar coup d'état.

Pre-1600

 * 1461 – Domenico Grimani, Italian cardinal (d. 1523)
 * 1473 – Nicolaus Copernicus, Prussian mathematician and astronomer (d. 1543)
 * 1497 – Matthäus Schwarz, German fashion writer (d. 1574)
 * 1519 – Froben Christoph of Zimmern, German author of the Zimmern Chronicle (d. 1566)
 * 1526 – Carolus Clusius, Flemish botanist and academic (d. 1609)
 * 1532 – Jean-Antoine de Baïf, French poet (d. 1589)
 * 1552 – Melchior Klesl, Austrian cardinal (d. 1630)
 * 1594 – Henry Frederick, Prince of Wales (d. 1612)

1601–1900

 * 1611 – Andries de Graeff, Dutch politician (d. 1678)
 * 1630 – Shivaji, Indian warrior-king and the founder of Maratha Empire (d. 1680)
 * 1660 – Friedrich Hoffmann, German physician and chemist (d. 1742)
 * 1717 – David Garrick, English actor, playwright, and producer (d. 1779)
 * 1743 – Luigi Boccherini, Italian cellist and composer (d. 1805)
 * 1798 – Allan MacNab, Canadian soldier, lawyer, and politician, Premier of Canada West (d. 1862)
 * 1800 – Émilie Gamelin, Canadian nun and social worker, founded the Sisters of Providence (d. 1851)
 * 1804 – Carl von Rokitansky, German physician, pathologist, and philosopher (d. 1878)
 * 1821 – August Schleicher, German linguist and academic (d. 1868)
 * 1833 – Élie Ducommun, Swiss journalist and activist, Nobel Prize laureate (d. 1906)
 * 1838 – Lydia Thompson, British burlesque performer (d. 1908)
 * 1841 – Elfrida Andrée, Swedish organist, composer, and conductor (d. 1929)
 * 1855 – Nishinoumi Kajirō I, Japanese sumo wrestler, the 16th Yokozuna (d. 1908)
 * 1859 – Svante Arrhenius, Swedish physicist and chemist, Nobel Prize laureate (d. 1927)
 * 1865 – Sven Hedin, Swedish geographer and explorer (d. 1952)
 * 1869 – Hovhannes Tumanyan, Armenian-Russian poet and author (d. 1923)
 * 1872 – Johan Pitka, Estonian admiral (d. 1944)
 * 1876 – Constantin Brâncuși, Romanian-French sculptor, painter, and photographer (d. 1957)
 * 1877 – Gabriele Münter, German painter (d. 1962)
 * 1878 – Harriet Bosse, Swedish–Norwegian actress (d. 1961)
 * 1880 – Álvaro Obregón, Mexican general and politician, 39th President of Mexico (d. 1928)
 * 1886 – José Abad Santos, Filipino lawyer and jurist, 5th Chief Justice of the Supreme Court of the Philippines (d. 1942)
 * 1888 – José Eustasio Rivera, Colombian lawyer and poet (d. 1928)
 * 1893 – Cedric Hardwicke, English actor and director (d. 1964)
 * 1895 – Louis Calhern, American actor (d. 1956)
 * 1896 – André Breton, French poet and author (d. 1966)
 * 1897 – Alma Rubens, American actress (d. 1931)
 * 1899 – Lucio Fontana, Argentinian-Italian painter and sculptor (d. 1968)

1901–present

 * 1902 – Kay Boyle, American novelist, short story writer, and educator (d. 1992)
 * 1904 – Havank, Dutch journalist and author (d. 1964)
 * 1911 – Merle Oberon, Indian-American actress (d. 1979)
 * 1912 – Saul Chaplin, American composer (d. 1997)
 * 1912  – Dorothy Janis, American actress (d. 2010)
 * 1913 – Prince Pedro Gastão of Orléans-Braganza (d. 2007)
 * 1913  – Frank Tashlin, American animator and screenwriter (d. 1972)
 * 1914 – Thelma Kench, New Zealand Olympic sprinter (d. 1985)
 * 1915 – Dick Emery, English actor and comedian (d. 1983)
 * 1915 – John Freeman, English lawyer, politician, and diplomat, British Ambassador to the United States (d. 2014)
 * 1916 – Eddie Arcaro, American jockey and sportscaster (d. 1997)
 * 1917 – Carson McCullers, American novelist, short story writer, playwright, and essayist (d. 1967)
 * 1918 – Fay McKenzie, American actress (d. 2019)
 * 1920 – C. Z. Guest, American actress, fashion designer, and author (d. 2003)
 * 1920  – Jaan Kross, Estonian author and poet (d. 2007)
 * 1920  – George Rose, English actor and singer (d. 1988)
 * 1922 – Władysław Bartoszewski, Polish journalist and politician, Polish Minister of Foreign Affairs (d. 2015)
 * 1924 – David Bronstein, Ukrainian chess player and theoretician (d. 2006)
 * 1924  – Lee Marvin, American actor (d. 1987)
 * 1926 – György Kurtág, Hungarian composer and academic
 * 1927 – Philippe Boiry, French journalist (d. 2014)
 * 1929 – Jacques Deray, French director and screenwriter (d. 2003)
 * 1930 – John Frankenheimer, American director and producer (d. 2002)
 * 1930  – K. Viswanath, Indian actor, director, and screenwriter (d. 2023)
 * 1932 – Joseph P. Kerwin, American captain, physician, and astronaut
 * 1935 – Chung-Yun Hse, Wood scientist (d. 2021)
 * 1935  – Dave Niehaus, American sportscaster (d. 2010)
 * 1935  – Russ Nixon, American MLB catcher and coach (d. 2016)
 * 1936 – Sam Myers, American singer-songwriter (d. 2006)
 * 1936  – Frederick Seidel, American poet
 * 1937 – Terry Carr, American author and educator (d. 1987)
 * 1937  – Norm O'Neill, Australian cricketer and sportscaster (d. 2008)
 * 1938 – Choekyi Gyaltsen, 10th Panchen Lama (d. 1989)
 * 1939 – Erin Pizzey, English activist and author, founded Refuge
 * 1940 – Saparmurat Niyazov, Turkmen engineer and politician, 1st President of Turkmenistan (d. 2006)
 * 1940  – Smokey Robinson, American singer-songwriter and producer
 * 1940  – Bobby Rogers, American singer-songwriter (d. 2013)
 * 1941 – David Gross, American physicist and academic, Nobel Prize laureate
 * 1941  – Jenny Tonge, Baroness Tonge, English politician
 * 1942 – Cyrus Chothia, English biochemist and emeritus scientist at the Laboratory of Molecular Biology (d. 2019)
 * 1942  – Paul Krause, American football player and politician
 * 1942  – Will Provine, American biologist, historian, and academic (d. 2015)
 * 1942  – Howard Stringer, Welsh businessman
 * 1943 – Lou Christie, American singer-songwriter
 * 1943  – Homer Hickam, American author and engineer
 * 1943  – Tim Hunt, English biochemist and academic, Nobel laureate
 * 1944 – Les Hinton, English-American journalist and businessman
 * 1945 – Yuri Antonov, Uzbek-Russian singer-songwriter
 * 1946 – Paul Dean, Canadian guitarist
 * 1946  – Peter Hudson, Australian footballer and coach
 * 1946  – Karen Silkwood, American technician and activist (d. 1974)
 * 1947 – Jackie Curtis, American actress and playwright (d. 1985)
 * 1947  – Tim Shadbolt, New Zealand businessman and politician, 42nd Mayor of Invercargill
 * 1948 – Mark Andes, American singer-songwriter and bass player
 * 1948  – Pim Fortuyn, Dutch sociologist, academic, and politician (d. 2002)
 * 1948  – Tony Iommi, English guitarist and songwriter
 * 1949 – Danielle Bunten Berry, American game designer and programmer (d. 1998)
 * 1949  – Eddie Hardin, English singer-songwriter and pianist (d. 2015)
 * 1949  – Barry Lloyd, English footballer and manager
 * 1949  – William Messner-Loebs, American author and illustrator
 * 1950 – Juice Leskinen, Finnish singer-songwriter (d. 2006)
 * 1950  – Andy Powell, English singer-songwriter and guitarist
 * 1951 – Muhammad Tahir-ul-Qadri, Pakistani scholar and politician, founder of Minhaj-ul-Quran
 * 1952 – Ryū Murakami, Japanese novelist and filmmaker
 * 1952  – Rodolfo Neri Vela, Mexican engineer and astronaut
 * 1952  – Amy Tan, American novelist, essayist, and short story writer
 * 1952  – Danilo Türk, Slovene academic and politician, 3rd President of Slovenia
 * 1953 – Corrado Barazzutti, Italian tennis player
 * 1953  – Cristina Fernández de Kirchner, Argentine lawyer and politician, President of Argentina and Vice President of Argentina
 * 1953  – Massimo Troisi, Italian actor, director, and screenwriter (d. 1994)
 * 1954 – Francis Buchholz, German bass player
 * 1954  – Michael Gira, American singer-songwriter, guitarist, and producer
 * 1954  – Sócrates, Brazilian footballer and manager (d. 2011)
 * 1955 – Jeff Daniels, American actor and playwright
 * 1956 – Peter Holsapple, American singer-songwriter and guitarist
 * 1956  – Roderick MacKinnon, American biologist and academic, Nobel Prize laureate
 * 1956  – Dave Wakeling, English singer-songwriter and guitarist
 * 1957 – Falco, Austrian singer-songwriter, rapper, and musician (d. 1998)
 * 1957  – Dave Stewart, American baseball player, coach, and executive
 * 1957  – Ray Winstone, English actor
 * 1958 – Leslie David Baker, American actor
 * 1958  – Helen Fielding, English author and screenwriter
 * 1958  – Steve Nieve, English keyboard player and composer
 * 1959 – Roger Goodell, American businessman, 6th National Football League Commissioner
 * 1960 – Prince Andrew, Duke of York
 * 1960  – John Paul Jr., American race car driver (d. 2020)
 * 1961 – Justin Fashanu, English footballer (d. 1998)
 * 1961  – Ernie Gonzalez, American golfer (d. 2020)
 * 1962 – Hana Mandlíková, Czech-Australian tennis player and coach
 * 1963 – Seal, English singer-songwriter
 * 1963  – Jessica Tuck, American actress
 * 1964 – Doug Aldrich, American singer-songwriter and guitarist
 * 1964  – Jennifer Doudna, American biochemist
 * 1964  – Jonathan Lethem, American novelist, essayist, and short story writer
 * 1965 – Jon Fishman, American drummer
 * 1965  – Clark Hunt, American businessman
 * 1965  – Leroy, American singer-songwriter, guitarist, and producer
 * 1966 – Justine Bateman, American actress and producer
 * 1966  – Paul Haarhuis, Dutch tennis player and coach
 * 1966  – Eduardo Xol, American designer and author
 * 1967 – Benicio del Toro, Puerto Rican actor, director, and producer
 * 1968 – Prince Markie Dee, American rapper and actor (d. 2021)
 * 1968  – Frank Watkins, American bass player (d. 2015)
 * 1969 – Burton C. Bell, American singer-songwriter and guitarist
 * 1969  – Helena Guergis, Canadian businesswoman and politician
 * 1970 – Joacim Cans, Swedish singer-songwriter
 * 1970  – Bellamy Young, American actress
 * 1971 – Miguel Batista, Dominican baseball player and poet
 * 1971  – Richard Green, Australian golfer
 * 1971  – Jeff Kinney, American author and illustrator
 * 1972 – Francine Fournier, American wrestler and manager
 * 1972  – Sunset Thomas, American pornographic actress
 * 1973 – Eric Lange, American actor
 * 1975 – Daniel Adair, Canadian drummer and producer
 * 1975  – Daewon Song, South Korean-American skateboarder, co-founded Almost Skateboards
 * 1977 – Ola Salo, Swedish singer-songwriter and keyboard player
 * 1977  – Andrew Ross Sorkin, American journalist and author
 * 1977  – Gianluca Zambrotta, Italian footballer and manager
 * 1978 – Ben Gummer, English scholar and politician
 * 1978  – Immortal Technique, Peruvian-American rapper
 * 1979 – Steve Cherundolo, American soccer player and manager
 * 1980 – Dwight Freeney, American football player
 * 1980  – Ma Lin, Chinese table tennis player
 * 1980  – Mike Miller, American basketball player
 * 1981 – Beth Ditto, American singer
 * 1981  – Shawn Spears, Canadian wrestler
 * 1983 – Kotoōshū Katsunori, Bulgarian sumo wrestler
 * 1983  – Mika Nakashima, Japanese singer and actress
 * 1983  – Reynhard Sinaga, Indonesian sex offender
 * 1983  – Ryan Whitney, American ice hockey player
 * 1983  – Jawad Williams, American basketball player
 * 1984 – Chris Richardson, American singer-songwriter
 * 1985 – Haylie Duff, American actress and singer
 * 1985  – Arielle Kebbel, American actress and model
 * 1985  – Kosta Perović, Serbian basketball player
 * 1986 – Kyle Chipchura, Canadian ice hockey player
 * 1986  – Linus Klasen, Swedish ice hockey player
 * 1986  – Marta, Brazilian footballer
 * 1986  – Maria Mena, Norwegian singer-songwriter
 * 1987 – Anna Cappellini, Italian ice dancer
 * 1987  – Josh Reddick, American baseball player
 * 1988 – Shawn Matthias, Canadian ice hockey player
 * 1988  – Seth Morrison, American guitarist
 * 1989 – Sone Aluko, English-Nigerian footballer
 * 1991 – Trevor Bayne, American race car driver
 * 1991  – Christoph Kramer, German footballer
 * 1991  – Adreian Payne, American basketball player (d. 2022)
 * 1992 – Camille Kostek, American model
 * 1993 – Mauro Icardi, Argentine footballer
 * 1993  – Victoria Justice, American actress and singer
 * 1994 – Tiina Trutsi, Estonian footballer
 * 1995 – Nikola Jokić, Serbian basketball player
 * 1996 – Mabel, British-Swedish singer
 * 1996  – D. J. Wilson, American basketball player
 * 1998 – Katharina Gerlach, German tennis player
 * 2001 – Lee Kang-in, South Korean footballer
 * 2001  – David Mazouz, American actor
 * 2003 – Jakub Ojrzyński, Polish footballer
 * 2004 – Millie Bobby Brown, English actress, model and producer

Pre-1600

 * 197 – Clodius Albinus, Roman usurper (b. 150)
 * 446 – Leontius of Trier, Bishop of Trier
 * 1133 – Irene Doukaina, Byzantine wife of Alexios I Komnenos (b. 1066)
 * 1275 – Lal Shahbaz Qalandar, Sufi philosopher and poet (b. 1177)
 * 1300 – Munio of Zamora, General of the Dominican Order
 * 1408 – Thomas Bardolf, 5th Baron Bardolf, English rebel
 * 1414 – Thomas Arundel, Archbishop of Canterbury (b. 1353)
 * 1445 – Eleanor of Aragon, queen of Portugal (b. 1402)
 * 1491 – Enno I, Count of East Frisia, German noble (b. 1460)
 * 1553 – Erasmus Reinhold, German astronomer and mathematician (b. 1511)

1601–1900

 * 1602 – Philippe Emmanuel, Duke of Mercœur (b. 1558)
 * 1605 – Orazio Vecchi, Italian composer (b. 1550)
 * 1622 – Henry Savile, English scholar and politician (b. 1549)
 * 1672 – Charles Chauncy, English-American minister, theologian, and academic (b. 1592)
 * 1709 – Tokugawa Tsunayoshi, Japanese shōgun (b. 1646)
 * 1716 – Dorothe Engelbretsdatter, Norwegian author and poet (b. 1634)
 * 1785 – Mary, Countess of Harold, English aristocrat and philanthropist (b. 1701)
 * 1789 – Nicholas Van Dyke, American lawyer and politician, 7th Governor of Delaware (b. 1738)
 * 1799 – Jean-Charles de Borda, French mathematician, physicist, and sailor (b. 1733)
 * 1806 – Elizabeth Carter, English poet and translator (b. 1717)
 * 1837 – Georg Büchner, German-Swiss poet and playwright (b. 1813)
 * 1837  – Thomas Burgess, English bishop and philosopher (b. 1756)
 * 1887 – Multatuli, Dutch-German author and civil servant (b. 1820)
 * 1897 – Karl Weierstrass, German mathematician and academic (b. 1815)

1901–present

 * 1915 – Gopal Krishna Gokhale, Indian philosopher and politician (b. 1866)
 * 1916 – Ernst Mach, Austrian-Czech physicist and philosopher (b. 1838)
 * 1927 – Robert Fuchs, Austrian composer and educator (b. 1847)
 * 1928 – George Howard Earle Jr., American lawyer and businessman (b. 1856)
 * 1936 – Billy Mitchell, American general and pilot (b. 1879)
 * 1945 – John Basilone, American sergeant, Medal of Honor recipient (b. 1916)
 * 1951 – André Gide, French novelist, essayist, and dramatist, Nobel Prize laureate (b. 1869)
 * 1952 – Knut Hamsun, Norwegian novelist, poet, and playwright, Nobel Prize laureate (b. 1859)
 * 1953 – Richard Rushall, British businessman (b. 1864)
 * 1957 – Maurice Garin, Italian-French cyclist (b. 1871)
 * 1959 – Willard Miller, American sailor, Medal of Honor recipient (b. 1877)
 * 1962 – Georgios Papanikolaou, Greek-American pathologist, invented the Pap smear (b. 1883)
 * 1969 – Madge Blake, American actress (b. 1899)
 * 1970 – Ralph Edward Flanders, US Senator from Vermont (b. 1890)
 * 1972 – John Grierson, Scottish director and producer (b. 1898)
 * 1972  – Lee Morgan, American trumpet player and composer (b. 1938)
 * 1973 – Joseph Szigeti, Hungarian violinist (b. 1892)
 * 1977 – Anthony Crosland, English author and politician, Secretary of State for Foreign and Commonwealth Affairs (b. 1918)
 * 1977  – Mike González, Cuban baseball player, coach, and manager (b. 1890)
 * 1980 – Bon Scott, Scottish-Australian singer-songwriter (b. 1946)
 * 1983 – Alice White, American actress (b. 1904)
 * 1988 – André Frédéric Cournand, French-American physician and physiologist, Nobel Prize laureate (b. 1895)
 * 1992 – Tojo Yamamoto, American wrestler and manager (b. 1927)
 * 1994 – Derek Jarman, English director and set designer (b. 1942)
 * 1996 – Charlie Finley, American businessman (b. 1918)
 * 1997 – Leo Rosten, Polish-American author and academic (b. 1908)
 * 1997  – Deng Xiaoping, Chinese politician, 1st Vice Premier of the People's Republic of China (b. 1904)
 * 1998 – Grandpa Jones, American singer-songwriter and banjo player (b. 1913)
 * 1999 – Mohammad Mohammad Sadeq al-Sadr, Iraqi cleric (b. 1943)
 * 2000 Friedensreich Hundertwasser, Austrian-New Zealand painter and illustrator (b. 1928)
 * 2001 – Liza 'N' Eliaz, Belgian, transgender, hardcore DJ (b. 1958)
 * 2001 – Stanley Kramer, American director and producer (b. 1913)
 * 2001  – Charles Trenet, French singer-songwriter (b. 1913)
 * 2002 – Sylvia Rivera, American transgender LGBT activist (b. 1951)
 * 2003 – Johnny Paycheck, American singer-songwriter and guitarist (b. 1938)
 * 2007 – Janet Blair, American actress and singer (b. 1921)
 * 2007  – Celia Franca, English-Canadian dancer and director, founded the National Ballet of Canada (b. 1921)
 * 2008 – Yegor Letov, Russian singer-songwriter (b. 1964)
 * 2008  – Lydia Shum, Chinese-Hong Kong actress and singer (b. 1945)
 * 2009 – Kelly Groucutt, English singer and bass player (b. 1945)
 * 2011 – Ollie Matson, American sprinter and football player (b. 1930)
 * 2012 – Ruth Barcan Marcus, American philosopher and logician (b. 1921)
 * 2012  – Jaroslav Velinský, Czech author and songwriter (b. 1932)
 * 2012  – Vitaly Vorotnikov, Russian politician, 27th Prime Minister of Russia (b. 1926)
 * 2013 – Armen Alchian, American economist and academic (b. 1914)
 * 2013  – Park Chul-soo, South Korean director, producer, and screenwriter (b. 1948)
 * 2013  – Robert Coleman Richardson, American physicist and academic, Nobel Prize laureate (b. 1937)
 * 2013  – Donald Richie, American-Japanese author and critic (b. 1924)
 * 2013  – Eugene Whelan, Canadian farmer and politician, 22nd Canadian Minister of Agriculture (b. 1924)
 * 2014 – Kresten Bjerre, Danish footballer and manager (b. 1946)
 * 2014  – Dale Gardner, American captain and astronaut (b. 1948)
 * 2014  – Valeri Kubasov, Russian engineer and astronaut (b. 1935)
 * 2015 – Harold Johnson, American boxer (b. 1928)
 * 2015  – Nirad Mohapatra, Indian director, producer, and screenwriter (b. 1947)
 * 2015  – Harris Wittels, American actor, producer, and screenwriter (b. 1984)
 * 2016 – Umberto Eco, Italian novelist, literary critic, and philosopher (b. 1932)
 * 2016  – Harper Lee, American author (b. 1926)
 * 2016  – Chiaki Morosawa, Japanese anime screenwriter (b. 1959)
 * 2016  – Samuel Willenberg, Polish-Israeli sculptor and painter (b. 1923)
 * 2017 – Larry Coryell, American jazz guitarist (b. 1943)
 * 2019  – Karl Lagerfeld, German fashion designer (b. 1933)
 * 2020 – José Mojica Marins, Brazilian filmmaker, actor, composer, screenwriter, and television horror host (b. 1936)
 * 2020  – Pop Smoke, American rapper (b. 1999)

Holidays and observances

 * Armed Forces Day (Mexico)
 * Brâncuși Day (Romania)
 * Christian feast day:
 * Barbatus of Benevento
 * Boniface of Brussels
 * Conrad of Piacenza
 * Lucy Yi Zhenmei (one of Martyrs of Guizhou)
 * February 19 (Eastern Orthodox liturgics)
 * Commemoration of Vasil Levski (Bulgaria)
 * Flag Day (Turkmenistan)
 * Shivaji Jayanti (Maharashtra, India)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: February 19


text:
February 24

Pre-1600

 * 484 – King Huneric of the Vandals replaces Nicene bishops with Arian ones, and banishes some to Corsica.
 * 1303 – The English are defeated at the Battle of Roslin, in the First War of Scottish Independence.
 * 1386 – King Charles III of Naples and Hungary is assassinated at Buda.
 * 1525 – A Spanish-Austrian army defeats a French army at the Battle of Pavia.
 * 1527 – Coronation of Ferdinand I as the king of Bohemia in Prague.
 * 1538 – Treaty of Nagyvárad between Holy Roman Emperor Ferdinand I and King John Zápolya of Hungary and Croatia.
 * 1582 – With the papal bull Inter gravissimas, Pope Gregory XIII announces the Gregorian calendar.
 * 1597 – The last battle of the Cudgel War takes place on the Santavuori Hill in Ilmajoki, Ostrobothnia.

1601–1900

 * 1607 – L'Orfeo by Claudio Monteverdi, one of the first works recognized as an opera, receives its première performance.
 * 1711 – Rinaldo by George Frideric Handel, the first Italian opera written for the London stage, is premièred.
 * 1739 – Battle of Karnal: The army of Iranian ruler Nader Shah defeats the forces of the Mughal emperor of India, Muhammad Shah.
 * 1803 – In Marbury v. Madison, the Supreme Court of the United States establishes the principle of judicial review.
 * 1809 – London's Drury Lane Theatre burns to the ground, leaving its owner, Irish writer and politician Richard Brinsley Sheridan, destitute.
 * 1821 – Final stage of the Mexican War of Independence from Spain with Plan of Iguala.
 * 1822 – The first Swaminarayan temple in the world, Shri Swaminarayan Mandir, Ahmedabad, is inaugurated.
 * 1826 – The signing of the Treaty of Yandabo marks the end of the First Anglo-Burmese War.
 * 1831 – The Treaty of Dancing Rabbit Creek, the first removal treaty in accordance with the Indian Removal Act, is proclaimed. The Choctaws in Mississippi cede land east of the river in exchange for payment and land in the West.
 * 1848 – King Louis-Philippe of France abdicates the throne.
 * 1854 – A Penny Red with perforations becomes the first perforated postage stamp to be officially issued for distribution.
 * 1863 – Arizona is organized as a United States territory.
 * 1868 – Andrew Johnson becomes the first President of the United States to be impeached by the United States House of Representatives. He is later acquitted in the Senate.
 * 1875 – The SS Gothenburg hits the Great Barrier Reef and sinks off the Australian east coast, killing approximately 100, including a number of high-profile civil servants and dignitaries.
 * 1876 – The stage première of Peer Gynt, a play by Henrik Ibsen with incidental music by Edvard Grieg, takes place in Christiania (Oslo), Norway.
 * 1881 – China and Russia sign the Sino-Russian Ili Treaty.
 * 1895 – Revolution breaks out in Baire, a town near Santiago de Cuba, beginning the Cuban War of Independence; the war ends along with the Spanish–American War in 1898.

1901–present

 * 1916 – The Governor-General of Korea establishes a clinic called Jahyewon in Sorokdo to segregate Hansen's disease patients.
 * 1917 – World War I: The U.S. ambassador Walter Hines Page to the United Kingdom is given the Zimmermann Telegram, in which Germany pledges to ensure the return of New Mexico, Texas, and Arizona to Mexico if Mexico declares war on the United States.
 * 1918 – Estonian Declaration of Independence.
 * 1920 – Nancy Astor becomes the first woman to speak in the House of Commons of the United Kingdom following her election as a Member of Parliament (MP) three months earlier.
 * 1920  – The Nazi Party (NSDAP) was founded by Adolf Hitler in the Hofbräuhaus beer hall in Munich, Germany
 * 1942 – Seven hundred ninety-one Romanian Jewish refugees and crew members are killed after the MV Struma is torpedoed by the Soviet Navy.
 * 1942  – The Battle of Los Angeles: A false alarm led to an anti-aircraft barrage that lasted into the early hours of February 25.
 * 1945 – Egyptian Premier Ahmad Mahir Pasha is killed in Parliament after reading a decree.
 * 1946 – Colonel Juan Perón, founder of the political movement that became known as Peronism, is elected to his first term as President of Argentina.
 * 1949 – The Armistice Agreements are signed, to formally end the hostilities of the 1948 Arab-Israeli War.
 * 1967 – Cultural Revolution: Zhang Chunqiao announces the dissolution of the Shanghai People's Commune, replacing its local government with a revolutionary committee.
 * 1968 – Vietnam War: The Tet Offensive is halted; South Vietnamese forces led by Ngo Quang Truong recapture the citadel of Hué.
 * 1971 – The All India Forward Bloc holds an emergency central committee meeting after its chairman, Hemantha Kumar Bose, is killed three days earlier. P.K. Mookiah Thevar is appointed as the new chairman.
 * 1976 – The 1976 constitution of Cuba is formally proclaimed.
 * 1978 – The Yuba County Five disappear in California. Four of their bodies are found four months later.
 * 1981 – The 6.7 Gulf of Corinth earthquake affected Central Greece with a maximum Mercalli intensity of VIII (Severe). Twenty-two people were killed, 400 were injured, and damage totaled $812 million.
 * 1983 – A special commission of the United States Congress condemns the Japanese American internment during World War II.
 * 1984 – Tyrone Mitchell perpetrates the 49th Street Elementary School shooting in Los Angeles, killing two children and injuring 12 more.
 * 1989 – United Airlines Flight 811, bound for New Zealand from Honolulu, rips open during flight, blowing nine passengers out of the business-class section.
 * 1991 – Gulf War: Ground troops cross the Saudi Arabian border and enter Iraq, thus beginning the ground phase of the war.
 * 1996 – Two civilian airplanes operated by the Miami-based group Brothers to the Rescue are shot down in international waters by the Cuban Air Force.
 * 1999 – China Southwest Airlines Flight 4509, a Tupolev Tu-154 aircraft, crashes in Rui'an, Zhejiang, China. All 61 people on board are killed.
 * 2004 – The 6.3 Al Hoceima earthquake strikes northern Morocco with a maximum Mercalli intensity of IX (Violent). At least 628 people are killed, 926 are injured, and up to 15,000 are displaced.
 * 2006 – Philippine President Gloria Macapagal Arroyo declares Proclamation 1017 placing the country in a state of emergency in attempt to subdue a possible military coup.
 * 2007 – Japan launches its fourth spy satellite, stepping up its ability to monitor potential threats such as North Korea.
 * 2008 – Fidel Castro retires as the President of Cuba and the Council of Ministers after 32 years. He remains as head of the Communist Party for another three years.
 * 2015 – A Metrolink train derails in Oxnard, California following a collision with a truck, leaving more than 30 injured.
 * 2016 – Tara Air Flight 193, a de Havilland Canada DHC-6 Twin Otter aircraft, crashed, with 23 fatalities, in Solighopte, Myagdi District, Dhaulagiri Zone, while en route from Pokhara Airport to Jomsom Airport.
 * 2020 – Mahathir Mohamad resigns as Prime Minister of Malaysia following an attempt to replace the Pakatan Harapan government, which triggered the 2020-2022 Malaysian political crisis.
 * 2022 – Russo-Ukrainian War: Days after recognising Donetsk and Luhansk as independent states, Russian president Vladimir Putin orders a full scale invasion of Ukraine.

Pre-1600

 * 1103 – Emperor Toba of Japan (d. 1156)
 * 1304 – Ibn Battuta, Moroccan jurist
 * 1413 – Louis, Duke of Savoy (d. 1465)
 * 1463 – Giovanni Pico della Mirandola, Italian philosopher (d. 1494)
 * 1494 – Johan Friis, Danish statesman (d. 1570)
 * 1500 – Charles V, Holy Roman Emperor (d. 1558)
 * 1536 – Pope Clement VIII (d. 1605)
 * 1545 – John of Austria (d. 1578)
 * 1553 – Cherubino Alberti, Italian engraver and painter (d. 1615)
 * 1557 – Matthias, Holy Roman Emperor (d. 1619)
 * 1593 – Henry de Vere, 18th Earl of Oxford, English soldier and courtier (d. 1625)
 * 1595 – Maciej Kazimierz Sarbiewski, Polish author and poet (d. 1640)

1601–1900

 * 1604 – Arcangela Tarabotti, Venetian nun and feminist (d. 1652)
 * 1619 – Charles Le Brun, French painter and theorist (d. 1690)
 * 1622 – Johannes Clauberg, German theologian and philosopher (d. 1665)
 * 1709 – Jacques de Vaucanson, French engineer (d. 1782)
 * 1721 – John McKinly, Irish-American physician and politician, 1st Governor of Delaware (d. 1796)
 * 1723 – John Burgoyne, English general and politician (d. 1792)
 * 1736 – Charles Alexander, Margrave of Brandenburg-Ansbach (d. 1806)
 * 1743 – Joseph Banks, English botanist and explorer (d. 1820)
 * 1762 – Charles Frederick Horn, German-English composer and educator (d. 1830)
 * 1767 – Rama II of Siam (d. 1824)
 * 1774 – Prince Adolphus, Duke of Cambridge (d. 1850)
 * 1786 – Martin W. Bates, American lawyer and politician (d. 1869)
 * 1786  – Wilhelm Grimm, German anthropologist, author, and academic (d. 1859)
 * 1788 – Johan Christian Dahl, Norwegian-German painter (d. 1857)
 * 1827 – Lydia Becker, English-French activist (d. 1890)
 * 1831 – Leo von Caprivi, German general and politician, Chancellor of Germany (d. 1899)
 * 1835 – Julius Vogel, English-New Zealand journalist and politician, 8th Prime Minister of New Zealand (d. 1899)
 * 1836 – Winslow Homer, American painter and illustrator (d. 1910)
 * 1837 – Rosalía de Castro, Spanish poet (d. 1885)
 * 1842 – Arrigo Boito, Italian journalist, author, and composer (d. 1918)
 * 1848 – Andrew Inglis Clark, Australian engineer, lawyer, and politician (d. 1907)
 * 1852 – George Moore, Irish author, poet, and playwright (d. 1933)
 * 1868 – Édouard Alphonse James de Rothschild, French financier and polo player (d. 1949)
 * 1869 – Zara DuPont, American suffragist (d. 1946)
 * 1874 – Honus Wagner, American baseball player, coach, and manager (d. 1955)
 * 1877 – Rudolph Ganz, Swiss pianist, composer, and conductor (d. 1972)
 * 1877  – Ettie Rout, Australian-New Zealand educator and activist (d. 1936)
 * 1885 – Chester W. Nimitz, American admiral (d. 1966)
 * 1885  – Stanisław Ignacy Witkiewicz, Polish author, poet, and painter (d. 1939)
 * 1890 – Marjorie Main, American actress (d. 1975)
 * 1896 – Richard Thorpe, American director and screenwriter (d. 1991)
 * 1898 – Kurt Tank, German pilot and engineer (d. 1983)
 * 1900 – Irmgard Bartenieff, German-American dancer and physical therapist, leading pioneer of dance therapy (d. 1981)

1901–present

 * 1903 – Vladimir Bartol, Italian-Slovene author and playwright (d. 1967)
 * 1908 – Telford Taylor, American general, lawyer, and historian (d. 1998)
 * 1909 – August Derleth, American anthologist and author (d. 1971)
 * 1914 – Ralph Erskine, English-Swedish architect, designed The Ark and Byker Wall (d. 2005)
 * 1914  – Weldon Kees, American author, poet, painter, and pianist (d. 1955)
 * 1915 – Jim Ferrier, Australian golfer (d. 1986)
 * 1919 – John Carl Warnecke, American architect (d. 2010)
 * 1921 – Abe Vigoda, American actor (d. 2016)
 * 1922 – Richard Hamilton, English painter and academic (d. 2011)
 * 1922  – Steven Hill, American actor (d. 2016)
 * 1924 – Hal Herring, American football player and coach (d. 2014)
 * 1924  – Erik Nielsen, Canadian lawyer and politician, 3rd Deputy Prime Minister of Canada (d. 2008)
 * 1925 – Bud Day, American colonel and pilot, Medal of Honor recipient (d. 2013)
 * 1926  – Dave Sands, Australian boxer (d. 1952)
 * 1927 – Emmanuelle Riva, French actress (d. 2017)
 * 1929 – Kintarō Ōki, South Korean wrestler (d. 2006)
 * 1930 – Barbara Lawrence, American model and actress (d. 2013)
 * 1931 – Dominic Chianese, American actor and singer
 * 1931  – Brian Close, English cricketer and coach (d. 2015)
 * 1932 – Michel Legrand, French pianist, composer, and conductor (d. 2019)
 * 1932  – Zell Miller, American sergeant and politician, 79th Governor of Georgia (d. 2018)
 * 1932  – John Vernon, Canadian-American actor (d. 2005)
 * 1933 – Judah Folkman, American physician and biologist (d. 2008)
 * 1933  – Ali Mazrui, Kenyan-American political scientist, philosopher, and academic (d. 2014)
 * 1933  – David "Fathead" Newman, American saxophonist and composer (d. 2009)
 * 1934 – Bettino Craxi, Italian lawyer and politician, 45th Prime Minister of Italy (d. 2000)
 * 1934  – Johnny Hills, English footballer (d. 2021)
 * 1934  – Renata Scotto, Italian soprano (d. 2023)
 * 1935 – Ryhor Baradulin, Belarusian poet, essayist, and translator (d. 2014)
 * 1936 – Carol D'Onofrio, American public health researcher (d. 2020)
 * 1936  – Guillermo O'Donnell, Argentine political scientist (d. 2011)
 * 1938 – James Farentino, American actor (d. 2012)
 * 1938  – Phil Knight, American businessman and philanthropist, co-founded Nike, Inc.
 * 1939 – Jamal Nazrul Islam, Bangladeshi physicist and cosmologist (d. 2013)
 * 1940 – Pete Duel, American actor (d. 1971)
 * 1940  – Jimmy Ellis, American boxer (d. 2014)
 * 1940  – Denis Law, Scottish footballer and sportscaster
 * 1941 – Joanie Sommers, American singer and actress
 * 1942 – Paul Jones, English singer, harmonica player, and actor
 * 1942  – Celia Kaye, American actress
 * 1942  – Joe Lieberman, American lawyer and politician (d. 2024)
 * 1942  – Jenny O'Hara, American actress
 * 1942  – Gayatri Chakravorty Spivak, Indian philosopher, theorist, and academic
 * 1943 – Kent Haruf, American novelist (d. 2014)
 * 1943  – Gigi Meroni, Italian footballer (d. 1967)
 * 1943  – Pablo Milanés, Cuban singer-songwriter and guitarist (d. 2022)
 * 1944 – Nicky Hopkins, English keyboard player (d. 1994)
 * 1944  – Ivica Račan, Croatian lawyer and politician, 7th Prime Minister of Croatia (d. 2007)
 * 1945 – Barry Bostwick, American actor and singer
 * 1946 – Grigory Margulis, Russian mathematician and academic
 * 1947 – Rupert Holmes, English-American singer-songwriter and playwright
 * 1947  – Edward James Olmos, American actor and director
 * 1948 – Jayalalithaa, Indian actress and politician, 16th Chief Minister of Tamil Nadu (d. 2016)
 * 1948  – Dennis Waterman, English actor (d. 2022)
 * 1950 – George Thorogood, American musician
 * 1951 – David Ford, Northern Irish social worker and politician
 * 1951  – Derek Randall, English cricketer
 * 1951  – Debra Jo Rupp, American actress
 * 1951  – Helen Shaver, Canadian actress and director
 * 1951  – Laimdota Straujuma, Latvian economist and politician, 12th Prime Minister of Latvia
 * 1952 – Tommy Burleson, American basketball player
 * 1953 – Anatoli Kozhemyakin, Soviet footballer (d. 1974)
 * 1954 – Plastic Bertrand, Belgian singer-songwriter and producer
 * 1954  – Judith Ortiz Cofer, Puerto Rican author (d. 2016)
 * 1954  – Sid Meier, Canadian-American game designer and programmer, created the Civilization series
 * 1954  – Mike Pickering, English DJ and saxophonist
 * 1955 – Steve Jobs, American businessman, co-founded Apple Computer and Pixar (d. 2011)
 * 1955  – Eddie Johnson, American basketball player (d. 2020)
 * 1955  – Alain Prost, French race car driver
 * 1956 – Judith Butler, American philosopher, theorist, and author
 * 1956  – Eddie Murray, American baseball player and coach
 * 1956  – Paula Zahn, American journalist and producer
 * 1958 – Sammy Kershaw, American singer-songwriter and guitarist
 * 1958  – Mark Moses, American actor
 * 1959 – Beth Broderick, American actress and director
 * 1959  – Mike Whitney, Australian cricketer and television host
 * 1961 – Emilio Rivera, American actor
 * 1961  – Erna Solberg, Norwegian politician, 35th Prime Minister of Norway
 * 1962 – Kelly Craft, American businesswoman and diplomat
 * 1962  – Michelle Shocked, American singer-songwriter
 * 1963 – Sanjay Leela Bhansali, Indian filmmaker and composer
 * 1963  – Prince Carlo, Duke of Castro
 * 1963  – Mike Vernon, Canadian ice hockey player
 * 1964 – Russell Ingall, British-Australian race car driver and sportscaster
 * 1964  – Elizabeth Wilson, American politician
 * 1965 – Paul Gruber, American football player
 * 1965  – Jane Swift, American businesswoman and politician, Governor of Massachusetts
 * 1966 – Billy Zane, American actor and producer
 * 1967 – Brian Schmidt, Australian astrophysicist and academic, Nobel Prize laureate
 * 1968 – Mitch Hedberg, American comedian and actor (d. 2005)
 * 1969 – Kim Seung-woo, South Korean actor
 * 1970 – Jeff Garcia, American football player and coach
 * 1970  – Neil Sullivan, Scottish footballer and coach
 * 1970  – Jonathan Ward, American actor
 * 1971 – Pedro de la Rosa, Spanish race car driver
 * 1971  – Gillian Flynn, American author, screenwriter, and producer
 * 1971  – Brian Savage, Canadian ice hockey player
 * 1972 – Teodor Currentzis, Greek conductor and composer
 * 1972  – Manon Rhéaume, Canadian ice hockey player and coach
 * 1973 – Alexei Kovalev, Russian ice hockey player and pilot
 * 1974 – Mike Lowell, Puerto Rican baseball player
 * 1974  – Bonnie Somerville, American actress
 * 1975 – Ashley MacIsaac, Canadian singer-songwriter and fiddler
 * 1976 – Marco Campos, Brazilian race car driver (d. 1995)
 * 1976  – Zach Johnson, American golfer
 * 1976  – Bradley McGee, Australian cyclist and coach
 * 1977 – Jason Akermanis, Australian footballer and coach
 * 1977  – Bronson Arroyo, American baseball player
 * 1977  – Floyd Mayweather Jr., American boxer
 * 1980 – Shinsuke Nakamura, Japanese wrestler and mixed martial artist
 * 1981 – Jonas Andersson, Swedish ice hockey player
 * 1981  – Felipe Baloy, Panamanian footballer
 * 1981  – Lleyton Hewitt, Australian tennis player
 * 1981  – Mohammad Sami, Pakistani cricketer
 * 1981  – Bob Sanders, American football player
 * 1982 – Nick Blackburn, American baseball player
 * 1982  – Fala Chen, Chinese actress and singer
 * 1982  – Klára Koukalová, Czech tennis player
 * 1982  – Emanuel Villa, Argentine footballer
 * 1984 – Wilson Bethel, American actor
 * 1984  – Corey Graves, American wrestler and sportscaster
 * 1985 – Nakash Aziz, Indian playback singer and composer
 * 1986 – Wojtek Wolski, Polish-Canadian ice hockey player
 * 1987 – Kim Kyu-jong, South Korean singer, dancer, and actor
 * 1987  – Ashley Walker, American-Romanian basketball player
 * 1988 – Rodrigue Beaubois, French basketball player
 * 1988  – Alexander Koch, American actor
 * 1988  – Connie Ramsay, Scottish judoka
 * 1989 – Trace Cyrus, American singer-songwriter and guitarist
 * 1989  – Daniel Kaluuya, English actor
 * 1989  – Kosta Koufos, Greek-American basketball player
 * 1990 – Dwayne Allen, American football player
 * 1990  – Derek Wolfe, American football player
 * 1991 – Tim Erixon, American-Swedish ice hockey player
 * 1991  – Madison Hubbell, American ice dancer
 * 1991  – O'Shea Jackson Jr., American actor and rapper
 * 1991  – Semih Kaya, Turkish footballer
 * 1994 – Jessica Pegula, American tennis player
 * 1994  – Earl Sweatshirt, American rapper
 * 1996 – Royce Freeman, American football player

Pre-1600

 * 616 – Æthelberht of Kent
 * 951 – Liu Yun, Chinese governor (jiedushi)
 * 1018 – Borrell, bishop of Vic
 * 1114 – Thomas, archbishop of York
 * 1386 – Charles III of Naples (b. 1345)
 * 1496 – Eberhard I, Duke of Württemberg (b. 1445)
 * 1525 – Jacques de La Palice, French nobleman and military officer (b. 1470)
 * 1525  – Guillaume Gouffier, seigneur de Bonnivet, French soldier (b. c. 1488)
 * 1525  – Richard de la Pole, last Yorkist claimant to the English throne (b. 1480)
 * 1530 – Properzia de' Rossi, Italian Renaissance sculptor
 * 1563 – Francis, Duke of Guise (b. 1519)
 * 1580 – Henry FitzAlan, 19th Earl of Arundel, English nobleman (b. 1511)
 * 1588 – Johann Weyer, Dutch physician and occultist (b. 1515)

1601–1900

 * 1666 – Nicholas Lanier, English composer and painter (b. 1588)
 * 1685 – Charles Howard, 1st Earl of Carlisle, English general and politician, Lord Lieutenant of Cumberland (b. 1629)
 * 1704 – Marc-Antoine Charpentier, French composer (b. 1643)
 * 1714 – Edmund Andros, English courtier and politician, 4th Colonial Governor of New York (b. 1637)
 * 1721 – John Sheffield, 1st Duke of Buckingham and Normanby, English poet and politician, Lord President of the Council (b. 1648)
 * 1732 – Francis Charteris, Scottish soldier (b. 1675)
 * 1777 – Joseph I of Portugal (b. 1714)
 * 1785 – Carlo Buonaparte, Corsican lawyer and politician (b. 1746)
 * 1799 – Georg Christoph Lichtenberg, German physicist and academic (b. 1742)
 * 1810 – Henry Cavendish, French-English physicist and chemist (b. 1731)
 * 1812 – Étienne-Louis Malus, French physicist and mathematician (b. 1775)
 * 1815 – Robert Fulton, American engineer (b. 1765)
 * 1825 – Thomas Bowdler, English physician and philanthropist (b. 1754)
 * 1856 – Nikolai Lobachevsky, Russian mathematician and academic (b. 1792)
 * 1876 – Joseph Jenkins Roberts, American-Liberian politician, 1st President of Liberia (b. 1809)
 * 1879 – Shiranui Kōemon, Japanese sumo wrestler, the 11th Yokozuna (b. 1825)

1901–present

 * 1910 – Osman Hamdi Bey, Turkish archaeologist and painter (b. 1842)
 * 1914 – Joshua Chamberlain, American general and politician, 32nd Governor of Maine (b. 1828)
 * 1925 – Hjalmar Branting, Swedish journalist and politician, 16th Prime Minister of Sweden, Nobel Prize laureate (b. 1860)
 * 1927 – Edward Marshall Hall, English lawyer and politician (b. 1858)
 * 1929 – André Messager, French pianist, composer, and conductor (b. 1853)
 * 1930 – Hermann von Ihering, German-Brazilian zoologist (b. 1850)
 * 1953 – Robert La Follette Jr., American politician, senator of Wisconsin (b. 1895)
 * 1953  – Gerd von Rundstedt, German field marshal (b. 1875)
 * 1967 – Mir Osman Ali Khan, Last Nizam of Hyderabad State (b. 1886)
 * 1970 – Conrad Nagel, American actor (b. 1897)
 * 1974 – Margaret Leech, American historian and author (b. 1895)
 * 1975 – Hans Bellmer, German artist (b. 1902)
 * 1975  – Nikolai Bulganin, Russian marshal and politician, 6th Premier of the Soviet Union (b. 1895)
 * 1978 – Alma Thomas, American painter and educator (b.1891)
 * 1982 – Virginia Bruce, American actress (b. 1910)
 * 1986 – Rukmini Devi Arundale, Indian Bharatnatyam dancer (b. 1904)
 * 1986  – Tommy Douglas, Scottish-Canadian minister and politician, 7th Premier of Saskatchewan (b. 1904)
 * 1990 – Tony Conigliaro, American baseball player (b. 1945)
 * 1990  – Malcolm Forbes, American sergeant and publisher (b. 1917)
 * 1990  – Sandro Pertini, Italian journalist and politician, 7th President of Italy (b. 1896)
 * 1990  – Johnnie Ray, American singer-songwriter and pianist (b. 1927)
 * 1991 – John Daly, American journalist and game show host (b. 1914)
 * 1991  – George Gobel, American actor (b. 1919)
 * 1991  – Webb Pierce, American singer-songwriter and guitarist (b. 1921)
 * 1993 – Danny Gallivan, Canadian sportscaster (b. 1917)
 * 1993  – Bobby Moore, English footballer and manager (b. 1941)
 * 1994 – Jean Sablon, French singer and actor (b. 1906)
 * 1994  – Dinah Shore, American actress and singer (b. 1916)
 * 1998 – Antonio Prohías, Cuban-American cartoonist (b. 1921)
 * 1998  – Henny Youngman, English-American comedian and violinist (b. 1906)
 * 1999 – Andre Dubus, American short story writer, essayist, and memoirist (b. 1936)
 * 2001 – Theodore Marier, American composer and educator, founded the Boston Archdiocesan Choir School (b. 1912)
 * 2001  – Claude Shannon, American mathematician, cryptographer, and engineer (b. 1916)
 * 2002 – Leo Ornstein, Ukrainian-American pianist and composer (b. 1893)
 * 2004 – John Randolph, American actor (b. 1915)
 * 2005 – Coşkun Kırca, Turkish diplomat, journalist and politician (b. 1927)
 * 2006 – Octavia E. Butler, American author and educator (b. 1947)
 * 2006  – Don Knotts, American actor and comedian (b. 1924)
 * 2006  – John Martin, Canadian broadcaster, co-founded MuchMusic (b. 1947)
 * 2006  – Dennis Weaver, American actor, director, and producer (b. 1924)
 * 2007 – Bruce Bennett, American shot putter and actor (b. 1906)
 * 2007  – Damien Nash, American football player (b. 1982)
 * 2008 – Larry Norman, American singer-songwriter and producer (b. 1947)
 * 2010 – Dawn Brancheau, senior animal trainer at SeaWorld (b. 1969)
 * 2011 – Anant Pai, Indian author and illustrator (b. 1929)
 * 2012 – Agnes Allen, American baseball player and therapist (b. 1930)
 * 2012  – Oliver Wrong, English nephrologist and academic (b. 1925)
 * 2013 – Virgil Johnson, American singer (b. 1935)
 * 2013  – Con Martin, Irish footballer and manager (b. 1923)
 * 2014 – Franny Beecher, American guitarist (b. 1921)
 * 2014  – Alexis Hunter, New Zealand-English painter and photographer (b. 1948)
 * 2014  – Carlos Páez Vilaró, Uruguayan painter and sculptor (b. 1923)
 * 2014  – Harold Ramis, American actor, director, producer, and screenwriter (b. 1944)
 * 2015 – Mefodiy, Ukrainian metropolitan (b. 1949)
 * 2015  – Rakhat Aliyev, Kazakh politician and diplomat (b. 1962)
 * 2016 – Peter Kenilorea, Solomon Islands politician, 1st Prime Minister of the Solomon Islands (b. 1943)
 * 2016  – Nabil Maleh, Syrian director, producer, and screenwriter (b. 1936)
 * 2016  – George C. Nichopoulos, American soldier and physician (b. 1927)
 * 2018 – Sridevi, Indian actress (b. 1963)
 * 2018  – Haukur Hilmarsson, Icelandic political activist and internationalist volunteer fighter (b. 1986)
 * 2020 – Katherine Johnson, American physicist and mathematician (b. 1918)
 * 2021 – Ronald Pickup, English actor (b. 1940)
 * 2023 – Edith Roger, Norwegian dancer and choreographer (b. 1922)

Holidays and observances

 * Christian feast day:
 * Blessed Ascensión Nicol y Goñi
 * Lindel Tsen and Paul Sasaki (Anglican Church of Canada)
 * Modest (bishop of Trier)
 * Sergius of Cappadocia
 * February 24 (Eastern Orthodox liturgics)
 * Dragobete (Romania)
 * Engineer's Day (Iran)
 * Flag Day in Mexico
 * Independence Day, celebrates the independence of Estonia from the Russian Empire in 1918; the Soviet period is considered to have been an illegal annexation.
 * National Artist Day (Thailand)
 * Sweden Finns' Day (Sweden)

title: February 24