Gómez-Tato, A.; Macías-Virgós, E.; Pereira-Sáez, M. J. Trace map, Cayley transform and LS category of Lie groups. (English) Zbl 1214.55006 Ann. Global Anal. Geom. 39, No. 3, 325-335 (2011). Let \(X\) be a topological space. A subset \(V \subset X\) is called categorical if \(V\) is contractible in \(X\). The Lusternik-Schnirelmann (LS)-category of \( X \) is the least positive integer \(n\) such that \(X\) can be covered by \( n+1 \) categorical open subsets. The LS-category is difficult to compute. Using the Cayley transform, the authors find a simpler method to compute the LS-category of Lie groups and homogeneous spaces. Explicit computations for \(U(n)\), \( U(2n)/Sp(n)\) and \(U(n) / O(n)\) are given. A relationship between the Cayley transform and the height function allows to give an explicit categorical covering of \(Sp(2)\). The obstacles to generalize these results to \(Sp(n)\) are related to left eigenvalues of symplectic matrices. Reviewer: Jean-Baptiste Gatsinzi (Gaborone) Cited in 3 ReviewsCited in 9 Documents MSC: 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 22E15 General properties and structure of real Lie groups Keywords:LS category; Cayley transform; Unitary and symplectic groups; Bott-Morse function, Left eigenvalues PDFBibTeX XMLCite \textit{A. Gómez-Tato} et al., Ann. Global Anal. Geom. 39, No. 3, 325--335 (2011; Zbl 1214.55006) Full Text: DOI arXiv References: [1] Baker A.: Right eigenvalues for quaternionic matrices: a topological approach. Linear Algebra Appl. 286(1–3), 303–309 (1999) · Zbl 0941.15013 [2] Brenner J.L.: Matrices of quaternions. Pacific. J. Math. 1, 329–335 (1951) · Zbl 0043.01402 [3] Cayley A.: Sur quelques propriétés des déterminants gauches. J. Reine Angew. Math. 32, 119–123 (1846) · ERAM 032.0912cj [4] Cornea, O., Lupton, G., Oprea, J., Tanré, D.: Lusternik–Schnirelmann category. 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