F was defined to be 1-amenable (resp. strongly 1-amenable) if (1.1) is exact at WF (resp. at
WF and WM) for every multiquadratic extension M of F. It was shown in [ELW1,(3.2)] and
[ELW4,(3.4),(3.7)] that local fields, global fields, and fields of transcendence degree c 1 over
a real closed field are strongly 1-amenable. In fact, when these papers were written, there
were no examples known of fields which were not strongly 1-amenable. Meanwhile, in his
investigations of central simple algebras with involution, the third author has considered in …

Let F be a field of characteristic not 2, and suppose M is a multiquadratic extension of F. That
is, M/F is a finite abelian extension of exponent 2, so that M= F (@) for some finite subgroup
GEF/F', where $= F-(0). In studying Brauer groups and products of quaternion algebras, the
second author was led to consider the homology groups N,(M/F) of-a certain complex gM, F
associated to the extension M/F. This complex appears in [11,(3.1); 301 and in (1.1) below.
The purpose of the present work is to investigate the first homology group N,(M/F) and to …