In this paper, a new family of homotopy elements in the stable homotopy groups of spheres represented by h1hnhm γ？s in the Adams spectral sequence is detected, where n-2≥m≥5 and 3≤s 〈p.
In 1981, Cohen constructed an infinite family of homotopy elements ζk∈ π＊（S） represented by h0bk ∈ ExtA3,2（p-1）（pk＋1＋1）（z/p,Z/p） in the Adams spectral sequence, where p 〉 2 and k ≥ 1. In this paper, we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζn-1β2γs＋3 is nontrivial in the stable homotopy groups of spheres πt（s,n）-s-8（S）, where p ≥7, n 〉 3, 0≤s 〈p-5 andt（s,n） =2（p-1）[pn＋（s＋3）p2＋（s＋4）p＋（s＋3）]＋s.
Let （Ω＊ （M）, d） be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex （Ω＊ （M）, d ＋ H∧） and its associated twisted de Rham cohomology H＊ （M, H）. The authors show that there exists a spectral sequence {Ep/r.q, dr } derived from the filtration Fp（Ω＊ （M）） = （¤i〉p Ωi（M） of Ω＊ （M）, which converges to the twisted de Rham cohomology H＊（M, H）. It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.
