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Resonance category. (English) Zbl 1086.32023

Given a pointed space \(X\), write \(X^{(n)}=\underbrace{X\wedge\cdots\wedge X}_n/S_n\) for the \(n\)-fold symmetric smash product of \(X\) with the canonical stratification, where \(S_n\) is the symmetric group.
The author defines a new category \(\mathcal{R}\), called a resonance category to view that stratification of \(X^{(n)}\) as a a certain functor, called a resonance functor from \(\mathcal{R}\) to the category Top\(^\ast\) of pointed spaces. To illustrate this abstract framework, the author chooses the spaces of real (resp., complex) polynomials to study the Arnold problem [V. I.Arnold, Trans.Moscow Math. Soc. 21(1970), 30–52 (1971), translation from Tr. Mosk. Mat. O.-va 21, 27–46 (1970; Zbl 0208.24003 )] of computing the algebraic invariants of these strata, for \(X=S^1\) (resp.\(X=S^2\)).

MSC:

32S20 Global theory of complex singularities; cohomological properties
18B30 Categories of topological spaces and continuous mappings (MSC2010)
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
58K15 Topological properties of mappings on manifolds
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