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Stratifications of inertia spaces of compact Lie group actions. (English) Zbl 1364.58002

For a smooth \(G\)-manifold \(M\), where \(G\) is a compact Lie group \(G\), the associated inertia space is studied as a differentiable space in the sense of K. Spallek [Math. Ann. 180, 269–296 (1969; Zbl 0169.52901)]. Moreover, an explicit Whitney stratification is constructed for the inertia space, turning it into a triangulable differential stratified space. If \(G\) acts locally free on \(M\), the inertia space recovers the inertia orbifold. In this case, the constructed stratification coincides with the orbit type stratification. Finally, the authors consider differential forms on the inertia space and obtain a de Rham type theorem for differential forms on the inertia space with respect to the stratification.
In more detail, the article under review contains the following content: After the introduction, a review of differentiable (stratified) spaces is provided (Section 2).
Then the inertia groupoid \(\Lambda\mathcal G\) and the inertia space \(\mathcal G\backslash\Lambda\mathcal G\) for a proper Lie groupoid \(\mathcal G\) are discussed in Section 3. Subsequently, this construction is specialised to the inertia space of the transformation groupoid \(G\ltimes M\) for the action of a compact Lie group \(G\). This inertia space, called the inertia space \(\Lambda(G\backslash M)\) of the \(G\)-space, is then studied. In this situation, additional structure on the inertia space is provided by slices for the \(G\)-action. The additional information is crucial to obtain the results in the sequel. Furthermore, slice arguments are used as important tools in the following proofs. The third section culminates in the description of the topology of \(\Lambda(G\backslash M)\) and the construction of a “canonical stratification” for the inertia space (Theorem 3.8).
Section 4 then deals with a finer stratification of the inertia space \(\Lambda(G\backslash M)\), the so called “orbit Cartan type stratification”. This stratification turns out to be a Whitney stratification (Theorem 4.1). Before a proof of Theorem 4.1 is given (occupying the whole of Section 4.4), the different stratifications for the inertia space are discussed extensively in an example section. In these examples, the authors discuss connections between stratifications which were either constructed in the paper under review or well known in the literature (e.g. the orbit stratification of the inertia orbifold for a locally free \(G\)-action).
Finally, in Section 5 the authors prove a de Rham theorem for the inertia space \(\Lambda(G\backslash M)\). To this end, a complex \(\Omega^\ast(\Lambda G\backslash M))\) of differential forms on the inertia space is constructed. Then Theorem 5.1 asserts that the cohomology of the complex \(\Omega^\ast(\Lambda G\backslash M))\) naturally coincides with the singular (or Čech) cohomology of \(\Lambda(G\backslash M)\). This Theorem is analogous to Sjamaar’s de Rham theorem for singular symplectic reduced spaces [R. Sjamaar, Pac. J. Math. 220, No. 1, 153–166 (2005; Zbl 1096.53051)].

MSC:

58A35 Stratified sets
57S15 Compact Lie groups of differentiable transformations
22C05 Compact groups
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
57R18 Topology and geometry of orbifolds
58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis
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