Spectral Sequences

Abstract

In Chapter 10 we saw that if k_* is an ordinary homology theory with coefficients G and X is a CW-complex, then we can often compute k*(X)by the following prescription: the groups k_q(X^q/X^q-1)and the boundary operators Δ: k_q(X^q/X^q-1)→ k_q-1(X^q-1/X ^q-2)of the triple (X^q,X^q-1,X^q-2) together form a chain complex {k_q(X^q/X^q-1),Δ}, and the homology of this chain complex turns out to be k_*(X). This is true essentially because k_n(X^q/X^q-1)= 0 if n ≠ q, from which it follows that k _n (X^q) = 0 if n > q and k_n(X^q) ≅ k_n(X)if n < q, among other things.