“…First, about Theorem 1.1 (i) (a), the inclusion P 0 k,n → P 0 k+1,n , which is called a stabilization map, was constructed by Guest, Kozlowski and Yamaguchi in [7,8]. Moreover, the induced homomorphism H * (P 0 k,n ; Z) → H * (P 0 k+1,n ; Z) was studied in [8]. Second, about Theorem 1.1 (i) (b) and (iii) for l = 0, Guest, Kozlowski and Yamaguchi [7] and independently Kallel [9] established a more precise result.…”