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  "body": "\n## Definition\n\nLet $M$ be a closed [[topological manifold|QQQ9B1DK4P]] of dimension $n$, that is, a [[compact|0TVJZYB6SH]] manifold without boundary, and let $x \\mapsto \\mu_x$ be an [[orientation|913WEQERBE]] of $M$.\nA *fundamental class* of $M$ is a class $[M] \\in H_n(M; \\Z)$ in [[singular homology|M0Q5CFJ7ZH]] with the following property.\nFor every $x \\in M$, the map $H_n(M) \\to H_n(M, M \\setminus \\set{x})$ induced by inclusion sends $[M]$ to $\\mu_x$.\n\n## Remarks\n\nFor a closed oriented $n$-manifold a fundamental class exists and is unique.\nIf $M$ is also connected, then $H_n(M; \\Z) \\cong \\Z$, generated by $[M]$.\nWith $\\Z/2$ coefficients every closed manifold, orientable or not, has a unique fundamental class in $H_n(M; \\Z/2)$.\nThe groups $H_n(M, M \\setminus \\set{x})$ are [[relative homology|4BVWH0BN28]] groups.\n\nExample: for $M = S^1$ with its counterclockwise orientation, the loop $\\sigma \\colon [0, 1] \\to S^1$, $\\sigma(t) = e^{2\\pi i t}$, is a singular $1$-cycle.\nIts class generates $H_1(S^1) \\cong \\Z$ and restricts to the counterclockwise local generator at each point, so $[S^1] = [\\sigma]$.\n\nNon-example: the compactness hypothesis is needed, since $H_n(\\R^n) = 0$ and so $\\R^n$ has no fundamental class.\nThe orientation hypothesis is needed with $\\Z$ coefficients, since $H_2(\\R P^2; \\Z) = 0$ for the [[real projective plane|1DDYB6VVSR]].\n",
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