an:00950585 Zbl 0905.57025 Morisugi, K.; ??shima, H. Note on reflection maps and self maps of $$U(n),Sp(n)$$ and $$U(2n)/Sp(n)$$ EN J. Math. Kyoto Univ. 36, No. 1, 143-149 (1996). 00032936 1996
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57T99 unitary group; symplectic group; reflection map 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 [\textit{M. C. Crabb} and \textit{K. Knapp}, Proc. R. Soc. Edinb., Sect. A 107, 87-107 (1987; Zbl 0633.55014); \textit{I. M. James}, The topology of Stiefel manifolds (1976; Zbl 0337.55017); \textit{J. Mukai} and \textit{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. [\textit{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 $$\mathbb{F}= (\mathbb{C}, \mathbb{H})$$''. M.Mimura (Okayama) Zbl 0633.55014; Zbl 0337.55017; Zbl 0563.55004; Zbl 0757.55011