Kozlov, Dmitry N. Resonance category. (English) Zbl 1086.32023 Comment. Math. Helv. 80, No. 1, 197-223 (2005). 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.\(X=S^2\)). Reviewer: Marek Golasiński (Toruń) 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 Keywords:partition; resonance; space of polynomials; stratification; symmetric smash product Citations:Zbl 0208.24003; Zbl 0225.14005 PDFBibTeX XMLCite \textit{D. N. Kozlov}, Comment. Math. Helv. 80, No. 1, 197--223 (2005; Zbl 1086.32023) Full Text: DOI arXiv Link