Spectral Sequences And Their Applications

Let (Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifold M of dimension n. For de Rham cohomology twisted by a fixed odd cocycle, we give a description of differentials in the spectral sequence{Erp,q,dr} derived from the filtration (?) ofΩ*(M), and converging to the twisted de Rham cohomology above. Without any hypothesis on the elements in Erp,q, the differentials in this spectral sequence are given in terms of cup products and in terms of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in the second chapter of this paper.Let p be an odd prime and A the mod p Steenrod algebra. In order to compute the stable homotopy groups of spheres with the classical Adams spec-tral sequence, we must compute the2-term of the Adams spectral sequence, ExtA*,*(Zp,Zp). In the third chapter, we prove that in the cohomology of A the product (?) and (?) are nontrivial for n≥5, and trivial for n=3,4, whereδs+4 is actually (?) described by X. Wang and Q. Zheng, p≥11,0≤s＜p-4 and t(s,n)= 2(p-1)[(s+2)+(s+4)p+(s+3)p2+(s+4)p3+pn].