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Rational orthogonal calculus. (English) Zbl 1386.55012
Orthogonal calculus, developed by M. Weiss [Trans. Am. Math. Soc. 347, No. 10, 3743–3796 (1995; Zbl 0866.55020)], studies homotopy properties of continuous functors from the category of finite-dimensional real inner product spaces and isometries to the category of based spaces. It constructs a Taylor tower for such functors, where the \(n\)th layer of the tower is determined by a spectrum with \(O(n)\)-action. In this paper, the author uses model categories to construct a rational version of orthogonal calculus. Thus, given a continuous functor \(F\) as above, he constructs a tower of approximations of \(F\) that depends only on the (objectwise) rational homology type of \(F\). The \(n\)th layer is given by a rational spectrum with \(O(n)\)-action. It follows from the work of J.P.C. Greenlees and B. Shipley [Bull. Lond. Math. Soc. 46, No. 1, 133–142 (2014; Zbl 1294.55002)] that these layers are classified by torsion \(H^\ast(\text{B}\mathrm{SO}(n);{\mathbb{Q}})[O(n)/\mathrm{SO}(n)]\)-modules.
55P42 Stable homotopy theory, spectra
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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