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  "body": "\n## Definition\n\nLet $V$ be an [[oriented|913WEQERBE]] [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $E \\to V$ be an oriented real [[vector bundle|P4DZM975BV]] of rank $r$.\nLet $s$ be a smooth [[section|WWX7H5W1RP]] of $E$ whose zero set $X = s^{-1}(0)$ is compact.\nLet $U$ be an open neighbourhood of $X$ with compact closure in $V$.\nChoose a smooth section $s'$ of $E$ with these properties:\n\n1. $s'$ agrees with $s$ outside a compact subset of $U$.\n2. $s'$ is [[transverse|A2PDZ8GA9C]] to the zero section.\n3. No section $(1 - t) s + t s'$ with $t \\in [0, 1]$ has a zero on $\\overline{U} \\setminus U$.\n\nThen $Z = s'^{-1}(0) \\cap U$ is a compact oriented submanifold of dimension $d = n - r$.\nThe *virtual fundamental class* of $X$ in $U$ is the image of the [[fundamental class|ZEA0PEM3B2]] of $Z$ in [[singular homology|M0Q5CFJ7ZH]]:\n$$[X]^{\\mathrm{vir}} = [Z] \\in H_d(U; \\Q).$$\nThe number $d$ is the *virtual dimension*.\n\n## Remarks\n\nThe class does not depend on the choice of $s'$.\nTwo choices are joined by a generic homotopy whose zero set is an oriented cobordism between the two zero sets inside $U$.\nThe classes for smaller neighbourhoods map to the classes for larger ones.\nIf $s$ itself is transverse to the zero section, then $X$ is a compact oriented manifold of dimension $d$ and $[X]^{\\mathrm{vir}}$ is its fundamental class.\nFor a general Kuranishi structure the same construction is done chart by chart with multivalued perturbations, which is why rational coefficients are used.\n\nExample: let $V = S^2$, $E = T S^2$ and $s = 0$, so $X = S^2$ and $d = 0$.\nA generic vector field on $S^2$ has isolated zeros whose signed count is $2$, the value of the [[Euler class|BJX6QS0ANT]] of $T S^2$ on the fundamental class of $S^2$.\nSo $[X]^{\\mathrm{vir}} = 2 \\, [\\mathrm{pt}] \\in H_0(S^2; \\Q)$, although $X$ has dimension $2$.\n\n\nNon-example: for $V = \\R$, $E = \\R \\times \\R$ and $s = 0$, the zero set $X = \\R$ is not compact.\nThe perturbations $s' = \\varepsilon$ and $s' = \\varepsilon x$ have $0$ and $1$ zeros, so no well-defined class arises; this is why compactness is required.\n",
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