Linearity of homotopy representations. II (Q1325182)

[For Part I see Osaka J. Math. 29, No. 3, 595-606 (1992; Zbl 0778.57022).] Let \(G\) be a finite group. A finite \(G\)-CW complex \(X\) is called a homotopy representation of \(G\) if, for any subgroup \(H\) of \(G\), the \(H\)- fixed point set \(X^ H\) is homotopy equivalent to a \((\dim X^ H)\)- dimensional sphere or it is empty. Furthermore, if \(X\) is \(G\)-homotopy equivalent to a finite \(G\)-CW complex, it is called finite, if \(X\) is \(G\)-homotopy equivalent to a linear \(G\)-sphere, it is called linear, and if \(X* S(V)\) is linear for some linear \(G\)-sphere \(S(V)\), it is called stably linear. One defines the dimension function \(\text{Dim} X\) of \(X\) from the set of subgroups of \(G\) to the integers by setting \(\text{Dim} X(H) = \dim X^ H + 1\). If \(X^ H\) is empty, one sets \(\dim X^ H = - 1\). One calls \(\text{Dim} X\) linear if \(\text{Dim} X = \text{Dim} S(V)\) for some representation \(V\) of \(G\). The author proves the following results: Theorem A. All homotopy representations of \(G\) with linear dimension functions are linear if and only if \(G\) is isomorphic to \(C_{p^ m}\) \((m \geq 0)\), \(C_ 6\), \(D_{2p^ m}\) \((m \geq 1)\), \(D_{12}\), \(A_ 4\), or \(S_ 4\) \((p\) prime), where \(C_ n\) denotes the cyclic group of order \(n\), \(D_{2n}\) denotes the dihedral group of order \(2n\), and \(S_ n\) (resp. \(A_ n)\) denotes the symmetric group (resp. alternating group) on \(n\) letters. Corollary B. All homotopy representations of \(G\) are linear if and only if \(G\) is the cyclic \(p\) group \(C_{p^ m}\) or the dihedral 2 group \(D_{2^ m}\). Theorem C. Let \(G\) be an abelian group. Then all linear homotopy representations of \(G\) with linear dimension functions are linear if and only if \(G\) is an abelian \(p\)-group or \((C_ 2)^ m \times (C_ 3)^ n\) \((m,n \geq 1)\).