Actions of compact Lie groups and the equivariant Whitehead group (Q579672)

Let G be a compact Lie group and \(f: X\to Y\) a G homotopy equivalence between finite G CW-complexes. The generalized Whitehead torsion of such a map is an element in the group \(Wh_ G(X)\), generalizing the classical idea of Whitehead torsion. The author shows that this group has a direct sum decomposition: \[ Wh_ G(X)=\sum Wh(\pi_ 0(WH)^*_{\alpha}). \] The summation ranges over certain components of H fixed point sets, \(H\subset G\). Here \(WH=NH/H\) is the Weyl group of H in G, and the subscript \(\alpha\) indicates those elements mapping \(\alpha\) to itself. The superscript * indicates that we are considering the possibly noncompact Lie group of transformations of the universal covering space of \(\alpha\) which cover the action of \(WH_{\alpha}\) on \(\alpha\). Finally, \(\pi_ 0\) denotes the group of components. Such a decomposition had been known before [\textit{H. Hauschild}, Manuscr. Math. 26, 63-82 (1978; Zbl 0402.57031)], but the author's approach allows a componentwise computation of the torsion invariant. The second result of this paper shows that the equivariant subdivision map is a simple G homotopy equivalence.