Arone, Greg The Weiss derivatives of BO(-) and BU(-). (English) Zbl 1006.55009 Topology 41, No. 3, 451-481 (2002). Let \(\mathbb{F}\) be \(\mathbb{R}\) or \(\mathbb{C}\). Let \({\mathcal U}\) be an infinite-dimensional vector space over \(\mathbb{F}\) with a positive-definite inner product. Let \({\mathcal G}\) be the category of finite-dimensional vector subspaces of \({\mathcal U}\) and product-preserving linear maps (i.e. the morphisms are linear isometric inclusions). M. Weiss provided [Trans. Am. Math. Soc. 347, No. 10, 3743-3796 (1995); erratum ibid. 350, No. 2, 851-855 (1998; Zbl 0866.55020)] a general framework for studying continuous functors from \({\mathcal G}\) to \(Spaces_*\) (the category of spaces with non-degenerate basepoint). One of the main constructions of Weiss associates a “Taylor tower” of sorts to a given continuous functor \(F:{\mathcal G}\to Spaces_*\). In this paper the author studies an “orthogonal tower” of the functors \(V\to BO(V)\) and \(V\to BU(V)\). He describes the Weiss derivatives of these functors and calculates the homology of the layers. Since the orthogonal tower studied here is related to the Goodwillie tower of the identity functor in various ways [T. G. Goodwillie, K-Theory 4, No. 1, 1-27 (1990; Zbl 0741.57021); 5, No. 4, 295-332 (1992; Zbl 0776.55008); Calculus III: the Taylor series of a homotopy functor, Preprint, 1996], the main results of this paper are Weiss-tower analogues of previous results of G. Arone and M. Mahowald [Invent. Math. 135, No. 3, 743-788 (1999; Zbl 0997.55016)] and G. Z. Arone and W. G. Dwyer [Partition complexes, Tits buildings and symmetric products, Proc. Lond. Math. Soc., III. Ser. 82, No. 1, 229-256 (2001; Zbl 1028.55008)]. From these results we mention:Theorem 1. There exists a \(U(p^k)\)-equivariant map \(U(p^k)_+ \Lambda_{N_k} TS_p (2k)\to L_{p^k}\), which is a mod \(p\) homology equivalence.Theorem 2. Let \(n \geq 1\). The \(n\)th derivative of the functor \(V\to B\operatorname{Aut}(V)\) is the spectrum \(\text{Map}_* (L_n,\sum^\infty S^{Ad_n})\), where \(L_n\) is the unreduced suspension of the geometric realization of the category of non-trivial direct-sum decompositions of \(\mathbb{F}^n\).Theorem 3. There exists an \(O(n-1)\)-equivariant equivalence \[ \text{Map}_* (L^\mathbb{R}_n,\sum^\infty S^{Ad^\mathbb{R}_n}) \simeq\text{Map}_*(S^1\wedge K_n,\sum^\infty S^0) \wedge_{\Sigma_n} O(n-1)_+. \] Similarly, in the complex case there exists a \(U(n-1)\)-equivariant weak equivalence \[ \text{Map}_*(L_n^\mathbb{R}, \sum^\infty S^{Ad^\mathbb{C}_n}) \simeq\text{Map}_*(S^1\wedge K_n,\sum^\infty S^n) \wedge_{\Sigma_n} U(n-1)_+. \] Reviewer: Ioan Pop (Iaşi) Cited in 5 ReviewsCited in 14 Documents MSC: 55P65 Homotopy functors in algebraic topology 55P99 Homotopy theory Keywords:Weiss-tower Citations:Zbl 0866.55020; Zbl 0741.57021; Zbl 0776.55008; Zbl 0997.55016; Zbl 1028.55008 PDFBibTeX XMLCite \textit{G. Arone}, Topology 41, No. 3, 451--481 (2002; Zbl 1006.55009) Full Text: DOI