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Note on reflection maps and self maps of \(U(n),Sp(n)\) and \(U(2n)/Sp(n)\). (English) Zbl 0905.57025
From the authors’ introduction: “Let \(U(n)\) and \(Sp(n)\) be the \(n\)-th unitary and symplectic group, respectively. We denote the complex numbers by \(\mathbb{C}\), and the quaternions by \(\mathbb{H}\). Let \(\mathbb{F}\) be \(\mathbb{C}, \mathbb{H}\) or \((\mathbb{C}, \mathbb{H})\). In order to describe uniformly for three cases, we write \[ G_n(\mathbb{F}) =\begin{cases} U(n) \quad & \text{if }\mathbb{F} =\mathbb{C} \\ Sp(n) \quad &\text{if }\mathbb{F} =\mathbb{H} \\ U(2n)/Sp(n) \quad & \text{if }\mathbb{F}= (\mathbb{C}, \mathbb{H}). \end{cases} \] When \(\mathbb{F}\) is \(\mathbb{C}\) or \(\mathbb{H}\), we denote by \(P(\mathbb{F}^n)\) and \(Q_n(\mathbb{F})\) the projective space and the quasi-projective space, respectively. We write \(Q_n (\mathbb{C}, \mathbb{H})= \Sigma P(\mathbb{H}^n)_+\), the suspension of the union of \(P(\mathbb{H}^n)\) and a point space. Recall from [M. C. Crabb and K. Knapp, Proc. R. Soc. Edinb., Sect. A 107, 87-107 (1987; Zbl 0633.55014); I. M. James, The topology of Stiefel manifolds (1976; Zbl 0337.55017); J. Mukai and S. Oka, Mem. Fac. Sci., Kyushu Univ., Ser. A 38, 277-284 (1984; Zbl 0563.55004)] that there is a map, called the reflection map, \(r:Q_n (\mathbb{F}) \to G_n (\mathbb{F})\) which induces an epimorphism on cohomology. Our result is
Theorem. For any integer \(k\), there exist maps \(c_k: Q_n(\mathbb{F})\to Q_n (\mathbb{F})\) and \(m_k: G_n (\mathbb{F})\to G_n (\mathbb{F})\) such that (1) the following diagram commutes \[ \begin{tikzcd} Q_n(\mathbb{F}) \ar[r,"r"]\ar[d,"c_k"'] & G_n (\mathbb{F})\ar[d,"m_k"]\\Q_n(\mathbb{F})\ar[r,"r"'] & G_n (\mathbb{F})\quad ; \end{tikzcd} \] (2) \(c_k\) induces the homomorphism of \(k\)-multiple on the integral cohomology; (3) \(m_k\) induces the homomorphism of \(k\)-multiple on the ring basis of the integral cohomology which will be given in Lemmas 2.1 and 4.1.
When \(\mathbb{F}\) is \(\mathbb{C}\) or \(\mathbb{H}\), setting \(m_k\) to be the \(k\)-times multiplication map, the Theorem may be well-known for experts. Since \(G_n(\mathbb{C}, \mathbb{H})\) is not an \(H\)-space for \(n\geq 2\) (cf. [Y. Hemmi, J. Pure Appl. Algebra 75, No. 3, 277-296 (1991; Zbl 0757.55011)]), the existence of the map \(m_k\) is not obvious when \(\mathbb{F}= (\mathbb{C}, \mathbb{H})\)”.
Reviewer: M.Mimura (Okayama)
57T99 Homology and homotopy of topological groups and related structures
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