Bak groups and equivariant surgery (Q1118867)

Let X, Y be compact, 1-connected oriented G-manifolds of codimension \(n\geq 5\), G a finite group. Let f: \(X\to Y\) be a G-normal map which is a boundary and singularity equivalence, and let \(X_ s\) denote the singular set of X. If G is of odd order, the ``gap hypothesis'' \[ (GH)\quad 2 \dim X_ s<\dim X \] suffices to obtain a well-defined surgery obstruction \(\sigma\) (f) in \(L^ h_ n({\mathbb{Z}}G;w)\). If G has even order, however, one needs the ``stronger gap hypothesis'' \[ (SGH)\quad \dim X_ s<[(\dim X-1)/2], \] as has already been observed by Lueck and Madsen. In the present paper the author uses an intermediate condition to define a surgery obstruction with values in a quotient of \(L^ h_ n({\mathbb{Z}}G;w)\), which in fact is a Witt-group of hermitian forms with respect to a certain form parameter in the sense of \textit{A. Bak} [K- theory of forms (1981; Zbl 0465.10013)]. This yields surgery obstructions also in the critical case \(\dim X_ s=[(\dim X-1)/2].\)