Algebraic Topology · Problem 2.1.15

Homology — Allen Hatcher

Mathematics — Topology · exact-sequences-homology

Problem

For an exact sequence

A \to B \to C \to D \to E

show that

C = 0

iff the map

A \to B

is surjective and

D \to E

is injective. Hence for a pair of spaces

(X, A)

, the inclusion

A \hookrightarrow X

induces isomorphisms on all homology groups iff

H_n(X, A) = 0

for all

n