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  "body": "\n## Statement\n\nLet $n \\ne 2$ be a non-negative integer.\nLet $F(z, t)$ be a holomorphic function on a neighbourhood of $(0, 0)$ in $\\C^{n+1} \\times \\C$, and write $f_t(z) = F(z, t)$.\nAssume that, for all $t$ in a neighbourhood of $0$, $f_t(0) = 0$ and $f_t$ has an [[isolated singularity|VTGYEES3VS]] at $0$.\nAssume also that the [[Milnor number|1ZE51P7GRQ]] $\\mu(f_t)$ does not depend on $t$.\nThen, for all $t$ small enough, there is a [[homeomorphism|0QT00PB5G2]] $\\phi$ between neighbourhoods of $0$ in $\\C^{n+1}$ with $\\phi(0) = 0$ and $\\phi(f_0^{-1}(0)) = f_t^{-1}(0)$.\nIn words, a $\\mu$-constant family of isolated hypersurface singularities has constant topological type.\n\n## Remarks\n\nLê Dũng Tráng and C. P. Ramanujam proved the theorem in 1976.\nThe proof shows that the Milnor fibrations of $f_0$ and $f_t$ are related by an h-cobordism, and then applies the h-cobordism theorem of Smale.\nThat theorem needs the cobordism to have dimension at least $6$, which is why the case $n = 2$, surfaces in $\\C^3$, is excluded; this case is still open.\nThe converse holds: the Milnor number is a topological invariant of the hypersurface germ, so a family of constant topological type has constant $\\mu$.\n\nExample: $f_t(x, y) = x^4 + y^4 + t x^2 y^2$ for $t$ near $0$.\nFor $t \\ne \\pm 2$ the quartic form $f_t$ is a product of four distinct linear factors, so $f_t$ has an isolated singularity with $\\mu(f_t) = 9$.\nThe theorem says that all these germs are topologically the same as $x^4 + y^4$, four lines through the origin.\n\n## Explanations\n\n### L1 human\n\nDeform a singular hypersurface by changing a parameter, and keep track of the Milnor number, which is an integer computed by algebra.\nIf this number never jumps, the theorem says that the shape of the singularity, as seen by topology, does not change at all.\nSo a single integer controls the topology in families, except for surfaces in $\\C^3$, where the method breaks down and the question is still open.\n\n### L2 human\n\nThe theorem is a central result of equisingularity theory, which asks when a family of singularities should count as having the same singularity.\nFor plane curves, $n = 1$, constant $\\mu$ is equivalent to constant topological type, and to constant Puiseux characteristics.\nLater work showed that a $\\mu$-constant family is in fact topologically trivial as a family when $n \\ne 2$.\nThe Milnor number is upper semicontinuous in families, so it can only jump up at special parameter values.\n",
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