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  "body": "\n## Statement\n\nLet $X$ be a topological space with $X = U \\cup V$, where $U$ and $V$ are open in $X$.\nAssume that $U$, $V$ and $U \\cap V$ are [[path-connected|0WB6DH499T]], and let $x_0 \\in U \\cap V$.\nWrite $\\pi_1(-)$ for $\\pi_1(-, x_0)$.\nLet $i_U \\colon \\pi_1(U \\cap V) \\to \\pi_1(U)$, $i_V \\colon \\pi_1(U \\cap V) \\to \\pi_1(V)$, $j_U \\colon \\pi_1(U) \\to \\pi_1(X)$ and $j_V \\colon \\pi_1(V) \\to \\pi_1(X)$ be maps induced by inclusions.\nThey are [[homomorphisms|0JYTG5K9Y4]] of [[fundamental groups|S1XK3HXHFX]].\nLet $\\Phi \\colon \\pi_1(U, x_0) * \\pi_1(V, x_0) \\to \\pi_1(X, x_0)$ be the homomorphism from the [[free product|TM7QS7P7X9]] that restricts to $j_U$ and $j_V$.\nThen $\\Phi$ is surjective, and its kernel is the [[normal subgroup|0CB0DX5BVV]] generated by the elements $i_U(\\omega)\\, i_V(\\omega)^{-1}$ for $\\omega \\in \\pi_1(U \\cap V, x_0)$.\nHence\n$$\\pi_1(X, x_0) \\cong \\pi_1(U, x_0) *_{\\pi_1(U \\cap V, x_0)} \\pi_1(V, x_0),$$\nthe free product with amalgamation.\n\n## Remarks\n\nSurjectivity comes from cutting a loop in $X$ into pieces that each lie in $U$ or in $V$, by a Lebesgue number argument.\nThe cut points are then joined to $x_0$ by paths inside $U \\cap V$.\nThe description of the kernel comes from cutting a path homotopy in $X$ into small squares in the same way.\nExample: for the figure eight $S^1 \\vee S^1$, let $U$ be the wedge minus a point of the first circle, and $V$ the wedge minus a point of the second circle.\nEach of them deformation retracts onto a circle, and $U \\cap V$ is contractible.\nSo $\\pi_1(S^1 \\vee S^1)$ is the free group on two generators.\nExample: for $n \\ge 2$, cover $S^n$ by the complements of the two poles; both sets are contractible and their intersection is path-connected, so $S^n$ is simply connected.\nThe hypothesis that $U \\cap V$ is path-connected is needed: for $S^1$ covered by two open arcs, the intersection has two components, both arcs are contractible, and yet $\\pi_1(S^1) \\cong \\Z$ is not trivial.\n\n## Explanations\n\n### L1 human\n\nThe theorem computes the loops of a space glued from two overlapping pieces.\nEach loop in the whole space is made of loops in the two pieces.\nThe only new relations come from loops that live in the overlap, since such a loop can be seen in either piece.\n\n### L2 human\n\nTogether with the [[fundamental group of the circle|6SQXPE0JPN]], the theorem computes the fundamental group of every connected CW complex from its $2$-skeleton.\nAttaching a $2$-cell adds one relation.\nThe result is a [[presentation|0JGEQXQ4AC]] with one generator for each $1$-cell outside a maximal tree and one relation for each $2$-cell.\nIt gives the fundamental groups of closed surfaces and of knot complements.\nA version with the [[fundamental groupoid|ZBHGXJF24E]] and a set of base points removes the condition that $U \\cap V$ is path-connected.\n",
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