A note on some periodicity of Ad-cohomology groups of lens spaces
Let p be an odd prime. Let \(L^ n_ 0(q)\) be the 2n-skeleton of the standard lens space mod q, \(L^ n(q)=S^{2n+1}/({\mathbb{Z}}/q)\). The author obtains some periodicity of Ad-cohomology groups of \(L=L^ n_ 0(p^ r)\). The main theorems are as follows. Theorem 1.1. Let \(M=r+[(n-1)/(p- 1)]\). If i-j\(\equiv 0\) mod (p-1)p\({}^{M-1}\), there exist the isomorphisms: \(Ad^{2i}(L)\cong Ad^{2j}(L)\), \(Ad^{2i+1}(L)\cong Ad^{2j+1}(L)\). Theorem 1.2. Let \(N=r+\nu_ p(i)\). If \(\nu_ p(i)\equiv \nu_ p(j)\) and i-j\(\equiv 0\) mod (p-1)p\({}^{N-1}\), there exists the isomorphism: \(Ad^{2i}(L)\cong Ad^{2j}(L)\).
scientific article; zbMATH DE number 3990380