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  "body": "\n## Definition\n\nLet $k \\ge 2$ and $n \\ge 2$ be integers.\nLet $P^n(k) = S^{n-1} \\cup_k e^n$ be the space obtained by attaching an $n$-cell to $S^{n-1}$ along a map of degree $k$; it is a [[Moore space|34SN01NWPS]] $M(\\Z/k, n - 1)$.\nLet $X$ be a [[pointed space|T9DG8DDE75]].\nThe *$n$-th homotopy group of $X$ with coefficients in $\\Z/k$* is the set of pointed [[homotopy classes|F2F143KQ2M]] of pointed maps\n$$\\pi_n(X; \\Z/k) = [P^n(k), X]_*.$$\n\n## Remarks\n\nFor $n \\ge 3$ the space $P^n(k)$ is the suspension of $P^{n-1}(k)$, so $\\pi_n(X; \\Z/k)$ is a group.\nFor $n \\ge 4$ it is a double suspension, so the group is abelian.\nFor $n = 2$ the set $\\pi_2(X; \\Z/k)$ is in general only a pointed set.\n\nFor $n \\ge 3$ the cofibre sequence $S^{n-1} \\xrightarrow{k} S^{n-1} \\to P^n(k)$ gives a short exact sequence\n$$0 \\to \\pi_n(X) \\otimes \\Z/k \\to \\pi_n(X; \\Z/k) \\to \\set{x \\in \\pi_{n-1}(X) : kx = 0} \\to 0,$$\nwhere $\\pi_n(X)$ is the ordinary [[homotopy group|P976WYEFMB]].\n\nExample: for $X = S^n$ with $n \\ge 3$, $\\pi_n(S^n) \\otimes \\Z/k \\cong \\Z/k$ and $\\pi_{n-1}(S^n) = 0$, so $\\pi_n(S^n; \\Z/k) \\cong \\Z/k$.\n\nNon-example: maps from $S^n$ to $X$ do not give $\\pi_n(X; \\Z/k)$, since $S^n$ is not a Moore space $M(\\Z/k, n-1)$; its homology in degree $n - 1$ is $0$ for $n \\ge 2$, not $\\Z/k$.\n",
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