LIFTING ISOTOPIES FROM ORBIT SPACES
LIFTING ISOTOPIES FROM ORBIT SPACES
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DOI：
10.1016/0040-9383(75)90005-1
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发表时间：
1975-01-01
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影响因子：
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通讯作者：
BIERSTONE, E
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作者：
BIERSTONE, E
猜想（同位素提升特性）。如果h: I x M/G+ M/G是一个光滑的同位素，而f0是M覆盖ho= h I (O) x M/G的等变微分同构，则存在f覆盖h的f: I x M* M的光滑等变同位素，我们在所有轨道具有相同维数的情况下建立了这个猜想。注意，该猜想暗示轨道空间的光滑同位素保持轨道类型，而在Palais[9]的（连续）覆盖同伦定理中，假设轨道类型保持。光滑同位素的提升强化了Bredon[5]所考虑的一类光滑群作用的某些分类定理。虽然经典群的Bredon双轴作用有不同维度的轨道，但它们的轨道空间有足够简单的奇点，可以通过直接计算建立同位素的提升（§4）。紧李群的线性表示的轨道空间具有自然的半代数结构，是不变多项式代数的有限生成子集的像。这种半代数结构是轨道空间分析的局部模型。一般来说，半解析集具有类似于惠特尼的解析变量分层的自然分层[16]；在轨道空间的情况下，这种分层与按轨道类型分层是一致的（§2）。同位素提升问题等价于相应的无穷小问题：将轨道空间（43）上的“光滑向量场”提升为光滑等变向量场。当所有轨道具有相同的维数时，局部模型是有限群的线性表示的轨道空间。局部模型中的光滑向量场空间是通过实解析向量场在光滑函数环上生成的（07）。虽然我们通过复化线性表示及其轨道空间建立了实解析向量场的提升（66）（0%现实是必要的）。众所周知，在有限群的复表示的轨道空间上的全纯微分算子一般只能提升到具有亚纯系数的微分算子。重点是，在复杂情况下，轨道空间的解析分层比按轨道类型分层更粗糙。
Conjecture (isotopy lifting property). If h: I x M/G+ M/G is a smooth isotopy, and f0 an equivariant diffeomorphism of M covering ho= h I (O) x M/G, then there is a smooth equivariant isotopy f: I x M* M of f. covering h.We establish this conjecture in the case that all orbits have the same dimension. Notice the conjecture implies that a smooth isotopy of an orbit space preserves orbit type, whereas in the (continuous) covering homotopy theorem of Palais [9] the preservation of orbit type is assumed. The lifting of smooth isotopies strengthens certain classification theorems for smooth group actions of a type considered by Bredon [5]. Though Bredon’s biaxial actions of classical groups have orbits of different dimensions, their orbit spaces have simple enough singularities that the lifting of isotopies can be established by direct calculation (§ 4). The orbit space of a linear representation of a compact Lie group has a natural semi-algebraic structure as the image of a finite set of generators for the algebra of invariant polynomials. This semi-algebraic structure is a local model for analysis on orbit spaces. In general, a semi-analytic set has a natural stratification analogous to Whitney’s stratification of analytic varieites [l6]; in the case of orbit spaces this stratification coincides with the stratification by orbit type (§ 2). The isotopy lifting problem is equivalent to a corresponding infinitesimal problem: lifting “smooth vector fields” on the orbit space (43) to smooth equivariant vector fields. When all orbits have the same dimension, the local model is the orbit space of a linear representation of a finite group. The space of smooth vector fields in the local model is generated over the ring of smooth functions by real analytic vector fields (07). Though we establish the lifting of real analytic vector fields (66) by complexifying the linear representation and its orbit space (0% reality is essential. It is well known that holomorphic differential operators on the orbit space of a complex representation of a finite group in general lift only to differential operators with meromorphic coefficients. The point is that the analytic stratification of the orbit space in the complex case is coarser than the stratification by orbit type.