Wall invariant for the space satisfying condition \((T^{**})\) (Q1818027)

The author defines the condition \((T^{**})\) and locally nilpotent spaces as the extensive concept of nilpotent spaces and he studies their properties. Furthermore, he studies the Wall invariant of spaces satisfying the condition \((T^{**})\). All spaces are arcwise connected CW complexes unless otherwise stated and the category is denoted by \(T\). He asserts the following Theorem 3.2. For \(X\) satisfying condition \((T^{**})\) with \(\pi_1(X)\) finite, if the action \(\pi_1(X)\times H_n(\widetilde X)\to H_n(\widetilde X)\) is nilpotent for all \(n\geq 0\), then \(X\in T_N\). Theorem 3.3. Let \(F\to E@>p>> B\) be a fibration with \(F\) a finitely dominated space. If \(B\) is a finite space satisfying condition \((T^{**})\), the action \(\pi_1(B)\times H_n(\widetilde B)\to H_n(\widetilde B)\) is nilpotent for all \(n\geq 0\) and \(\pi_1(B)(\neq 0)\) is finite then \(\omega(E)=0\). Theorem 3.4. Let \(F\to E@>p>>B\) be a fibration with \(F\) a finitely dominated space. For finite \(B(\in T_{LN})\) if \(\pi_1(B)\) is finite and \(\pi_1(B)\neq 0\) or \(\pi_1(B)\) is infinite with the maximal condition on a normal subgroup of \(\pi_1(B)\), then \(\omega(E)=0\). Theorem 3.6. Let \(F\to E@>p>>B\) be a fibration under the following conditions: \(B\) is a finitely dominated space and \(\pi_1(B)\) acts nilpotently on the homology of the fiber \(F\). If \(F\) is a finite complex, such that \(\pi_1(F)\) is nontrivial, \(E\) is a space satisfying condition \((T^{**})\), and the action \(\pi_1(E)\times H_n(\widetilde E)\to H_n(\widetilde E)\) is nilpotent where \(n\geq 0\), with \(\pi_1(E)(\neq 0)\) finite, then \(\omega(E)\in\text{Ker} p_*\), where \(p_*:K_0(\mathbb{Z}\pi_1(E))\to K_0(\mathbb{Z}\pi_1(B))\).