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Let $G$ be a finite group acting smoothly on a smooth manifold $X$. Then the fixed loci $X^H$, for $H \subset G$ are smooth manifolds, and hence forming fixed-locus wise vector spaces of smooth differential $n$-forms (for any $n \in \mathbb{N}$) yields a functor
$\array{ G Orbits &\longrightarrow& VectorSpaces \\ G/H &\mapsto& \Omega^n_{dR}\big( X^H \big) }$Question: In which generality are these functors injective objects, and what’s the proof?
I can show this to be so for the case that $G$ is: of order 4 or cyclic of prime order.
From the proof of these special cases it is pretty clear how the general case will work, and I suppose I can prove any number of further special cases by a case-by-case analysis; but it remains unclear to me how to formulate the fully general proof.
A first inkling of how to approach this issue may be gleaned from the text offered as proof to Prop. 4.3 in Triantafillou 82 (where it’s PL dR forms instead of smooth forms, but the combinatorial part of the argument is the same), but I don’t see how these hints are more than that.
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