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Hatcher-Algebraic-Topology
· Problem 2.1.16
Homology
Mathematics — Graph Theory · relative-homology
Problem
(a) Show that
H
0
(
X
,
A
)
=
0
H_0(X, A) = 0
H
0
(
X
,
A
)
=
0
iff
A
A
A
meets each path-component of
X
X
X
. (b) Show that
H
1
(
X
,
A
)
=
0
H_1(X, A) = 0
H
1
(
X
,
A
)
=
0
iff
H
1
(
A
)
→
H
1
(
X
)
H_1(A) \to H_1(X)
H
1
(
A
)
→
H
1
(
X
)
is surjective and each path-component of
X
X
X
contains at most one path-component of
A
A
A
.
Related theory
Relative Homology And Excision
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