General position of equivariant maps
General position of equivariant maps
复制标题
等变图的一般位置
DOI：
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发表时间：
1977
期刊：
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通讯作者：
E. Bierstone
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文献类型：
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作者：
E. Bierstone
.引入了关于紧李群作用等变的光滑映射的一般位置的自然通有概念。若G是紧李群，M，N是光滑G-流形，则相对于N的闭不变子流形P处于一般位置的光滑等变映射F：M ->> N在Whitney拓扑中是开稠密的. P在一般位置上的等变映射的逆像是Whitney分层的。由一般位置上的邻近等变映射构成的逆像是拓扑环境同位素的。在局部上下文中，设V，W是线性G-空间，F：V -» W是光滑等变映射。令Fx，...，Fk是V上光滑不变函数环上光滑等变映射模的齐次多项式生成元的有限集合。., hk使得F>> U °图h，其中图h是h（x）-（hx（x），.，hk（x）），且U（x，h）>> 2* xh·F，（x）. V X R* 的真实的仿射代数子簇（i/<<0）的同构类由V，W唯一确定（直到与仿射空间的乘积）。如果图h：V -» V X R* 在x e V处横向于（11 = 0）的最小惠特尼分层，则F被称为相对于06瓦特K处于一般位置。
. A natural generic notion of general position for smooth maps which are equivariant with respect to the action of a compact Lie group is introduced. If G is a compact Lie group, and M, N are smooth G-manifolds, then the set of smooth equivariant maps F: M -» N which are in general position with respect to a closed invariant submanifold P of N, is open and dense in the Whitney topology. The inverse image of P, by an equivariant map in general position, is Whitney stratified. The inverse images, by nearby equivariant maps in general position, are topologically ambient isotopic. In the local context, let V, W be linear G-spaces, and F: V -» W a smooth equivariant map. Let Fx,..., Fk be a finite set of homogeneous polynomial generators for the module of smooth equivariant maps, over the ring of smooth invariant functions on V. There are invariant functions //,,.. .,hk such that F » U ° graph h, where graph h is the graph of h(x) — (hx(x),..., hk(x)), and U(x, h) » 2*_ xh¡F,(x). The isomorphism class of the real affine algebraic subvariety (i/«0) of V X R* is uniquely determined (up to product with an affine space) by V, W. F is said to be in general position with respect to 06 WatOe K if graph h: V -» V X R* is transverse to the minimum Whitney stratification of (11 = 0), at x e V.