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  "body": "\n## Statement\n\nLet $M$, $N$ and $P$ be [[smooth manifolds|F45VQ47GPV]].\nA map between stratified sets is *stratum-preserving* if it maps each stratum into a stratum.\n\n1. (First isotopy lemma.) Let $X \\subseteq M$ be a closed subset with a [[Whitney stratification|3F84AMVJNV]], and let $f \\colon M \\to P$ be a smooth map.\nAssume that $f|_X \\colon X \\to P$ is [[proper|15XEBGNGHC]] and that $f|_S \\colon S \\to P$ is a [[submersion|GTMT5ZFV00]] for every stratum $S$ of $X$.\nThen $f|_X$ is a locally trivial fibration with stratum-preserving trivialisations: every $p \\in P$ has an open neighbourhood $U$ and a stratum-preserving [[homeomorphism|0QT00PB5G2]] $h \\colon f^{-1}(U) \\cap X \\to (f^{-1}(p) \\cap X) \\times U$ with $\\mathrm{pr}_U \\circ h = f$.\n2. (Second isotopy lemma.) Let $X \\subseteq M$ and $Y \\subseteq N$ be closed Whitney stratified subsets, and let $f \\colon M \\to N$ and $g \\colon N \\to P$ be smooth maps with $f(X) \\subseteq Y$.\nAssume that $f|_X$ and $g|_Y$ are proper, and that $f$ maps each stratum of $X$ submersively onto a stratum of $Y$.\nAssume that $f|_X$ satisfies Thom's condition $a_f$, and that $g|_S$ is a submersion for every stratum $S$ of $Y$.\nThen $f|_X$ is locally trivial over $P$, in this sense.\nEvery $p \\in P$ has an open neighbourhood $U$ and stratum-preserving homeomorphisms $X \\cap (g f)^{-1}(U) \\to (X \\cap (g f)^{-1}(p)) \\times U$ and $Y \\cap g^{-1}(U) \\to (Y \\cap g^{-1}(p)) \\times U$.\nThese homeomorphisms commute with the projections to $U$ and with $f$.\n\n## Remarks\n\nThe proof uses the control data of a [[Thom–Mather stratified space|P9R3777ZJF]], which Mather constructed on every Whitney stratified set.\nOne lifts the coordinate vector fields of $U$ to controlled vector fields on $X$, smooth on each stratum and compatible with the tubular projections and distance functions.\nControlled vector fields have continuous flows that preserve each stratum, and properness makes the flows defined for the needed times.\nIntegrating them gives the trivialisation.\nThe flows are only continuous across strata, so the trivialisations are homeomorphisms, not diffeomorphisms.\n\nExample: in $M = \\R^3$ let $X$ be the union of the cylinder $\\set{x^2 + y^2 = 1}$ and the $t$-axis $\\set{x = y = 0}$, with these two strata, and let $f(x, y, t) = t$.\nEach stratum is mapped submersively onto $\\R$, and $f|_X$ is proper since $X \\cap \\set{\\abs{t} \\le c}$ is compact.\nThe first lemma gives a homeomorphism of $X$ with $(\\text{circle} \\sqcup \\text{point}) \\times \\R$ that keeps the two strata apart.\n\nProperness is needed: the inclusion of $M = X = \\R \\setminus \\set{0}$ into $P = \\R$, with one stratum, is a submersion.\nBut its fibres are points over $t \\ne 0$ and empty over $0$, so it is not locally trivial near $0$.\n\n## Explanations\n\n### L1 human\n\nA stratified space is a space made of smooth pieces of different dimensions, such as a cone made of its tip and its smooth part.\nThe first lemma is a version of Ehresmann's theorem for such spaces.\nIf a proper map has no critical points on any piece, then its fibres all look the same, up to a homeomorphism that respects the pieces.\nThe second lemma does the same for a map between two stratified spaces that moves in a family.\n\n### L2 human\n\nFor smooth manifolds, Ehresmann's fibration theorem says that a proper submersion is a locally trivial fibration.\nThe first isotopy lemma extends this to Whitney stratified sets, at the price of getting only homeomorphisms.\nIt shows, for instance, that a Whitney stratified set is locally topologically trivial along each stratum.\nSo the local topology near a point stays the same as the point moves within a connected stratum.\nThe second lemma is a key tool in Mather's proof that topologically stable maps are dense.\nThom's condition $a_f$ is the hypothesis that makes the two families of trivialisations compatible.\n\n",
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