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Let   $R$   be a reduced real affine algebra of (Krull) dimension   $d \ge 2$   such that either   $R$   has no real maximal ideals, or the intersection of all real maximal ideals in   $R$   has height at least one. In this article, we prove the following: (1) the   $d$  -th Euler class group   $\textrm{E}^{d}(R)$  , defined by Bhatwadekar–Raja Sridharan, is canonically isomorphic to the Levine–Weibel Chow group of zero cycles   $\textrm{CH}_{0}(\textrm{Spec}(R))$  ; (2) the universal Mennicke symbol   $\textrm{MS}_{d+1}(R)$   is canonically isomorphic to the universal weak Mennicke symbol   $\textrm{WMS}_{d+1}(R)$  ; and (3) additionally, if   $R$   is a regular domain, then the Whitehead group   $\textrm{SK}_{1}(R)$   is canonically isomorphic to   $\frac{\textrm{SL}_{d+1}(R)}{\textrm{E}_{d+1}(R)}$  . As an application, we investigate some Eisenbud–Evans-type theorems.

Zero Cycles, Mennicke Symbols, and K 1 -stability of Certain Real Affine Algebras