Algebraic Topology · Problem 2.2.38

Homology — Allen Hatcher

Mathematics — Topology · exact-sequences-homology

Problem

Show that a commutative diagram

\begin{array}{ccccccccc} \cdots & \to & C_{n+1} & & & B_n & \to & C_n & \\ & & & \searrow & A_n & \swarrow & & & \searrow\\ \cdots & \to & E_{n+1} & & & D_n & \to & E_n & \end{array}

with the two sequences across the top and bottom exact, gives rise to an exact sequence

\cdots \to E_{n+1} \to B_n \to C_n \oplus D_n \to E_n \to B_{n-1} \to \cdots

where the maps are obtained from those in the previous diagram in the obvious way, except that

B_n \to C_n \oplus D_n

has a minus sign in one coordinate.