A Witt Group Structure on Orbit Spaces of Unimodular Rows

Ravi A. Rao
Tata Institute of Fundamental Research,
Mumbai, India.

Abstract:

L.N. Vaserstein initiated  the algebraic study of a
group structure on orbit spaces of unimodular rows. Generalized Mennicke
$n$-symbols were studied by Fossum-Foxby-Iversen, and A. Suslin related them
with his completion of the ``factorial powered'' unimodular row
$(a_0, a_1, a_2^2, \cdots, a_{n-1}^{n-1})$. W. van der Kallen combined these
themes, with existing topological intuition, to get a universal weak Mennicke
symbols interepretation of the group structure on orbit spaces of unimodular
rows of size bigger than half the Krull dimension. We show that there is also
a Witt group structure interpretation, as was shown by L.N. Vaserstein in
dimension two. Our key lemma enriches the possibility of the orbit spaces
having interesting combinatorial properties.