Date: Sun, 14 May 2000 16:51:07 +0800 (HKT)
From: Siu Lok Shun <lssiu@hkusua.hku.hk>
Subject: algebraic topology and the real closed field

Let R be a real closed field and R[i] is its algebraic closure. It is
known that all properties of the real field and the complex field also
holds for R and R[i] respectively except the completeness. Accordingly I
guess all result of algebraic topology can also be transfered to these two
fields, for example,
        (i) \pi_n(S^n)\cong Z and (ii) \pi_3(GL_n(H))\cong Z,
where H is the quatertnion field.
The results (i) and (ii) are used by Wood (1985) to prove the existence of
singular eigenvalue of quaternion matrices. If my guess is true, then his
result can be generalized to any quaternion algebra over real closed
field. But is it true?

Your sincerely,

Siu, Lok-Shun.