A group structure on certain orbit sets of unimodular rows

Some remarks on symplectic injective stability

“…In the papers [12], [13] W. van der Kallen shows that the orbit space of unimodular rows modulo the elementary action Um n (A)/E n (A) has an abelian group structure if 2 ≤ d ≤ 2n − 4, where d is the stable dimension of A. In fact he defines the notion of a weak Mennicke n-symbol in [13] and shows that the orbit space is bijective to the universal weak Mennicke n-symbol WMS n (A), under the above conditions.…”

A stably elementary homotopy

“…For this we recall the excision theorem of W. van der Kallen in [3], Theorem 3.21. First we recall the relative groups, the excision ring, etc., as defined in [3], §2.1.…”

A nice group structure on the orbit space of unimodular rows-II

“…Using this we deduce in Theorem 3.6 that if I is an ideal in a ring R, and the orbit spaces MSE n (R), and MSE n (R, I) have the usual group structures (see [25,26]), then the group structure on MSE n (R, I) is nice (i.e. is Mennicke-like) if it is nice for the Excision ring MSE n (R ⊕ I).…”

“…the ideal I. In [25] W. van der Kallen defined a group structure for the orbits of unimodular rows of length d + 1, where d was the dimension of the base ring, and studied the Excision property for orbit spaces MSE n (R, I) of unimodular rows of length n ≥ 3 modulo elementary action. (See Theorem 2.6).…”