# zbMATH — the first resource for mathematics

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)
##### MSC:
 57T99 Homology and homotopy of topological groups and related structures
##### Keywords:
unitary group; symplectic group; reflection map
Full Text: