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Supplement: The Long Exact
Homology Sequence and
Applications
S1. Chain Complexes
In the supplement, we will develop some of the building blocks for algebraic topology.
As we go along, we will make brief comments [in brackets] indicating the connection
between the algebraic machinery and the topological setting, but for best results here,
please consult a text or attend lectures on algebraic topology.
S1.1 Defnitions and Comments
A
chain complex
(or simply a
complex
)
C
∗
is a family of
R
modules
C
n
,
n
∈
Z
, along
with
R
homomorphisms
d
n
:
C
n
→
C
n
−
1
called
diferentials
, satisfying
d
n
d
n
+1
= 0 for
all
n
. A chain complex with only Fnitely many
C
n
’s is allowed; it can always be extended
with the aid of zero modules and zero maps. [In topology,
C
n
is the abelian group of
n

chains
, that is, all formal linear combinations with integer coeﬃcients of
n
simplices in a
topological space
X
. The map
d
n
is the
boundary operator
, which assigns to an
n
simplex
an
n
−
1chain that represents the oriented boundary of the simplex.]
The kernel of
d
n
is written
Z
n
(
C
∗
) or just
Z
n
; elements of
Z
n
are called
cycles
in
dimension
n
. The image of
d
n
+1
is written
B
n
(
C
∗
) or just
B
n
; elements of
B
n
are called
boundaries
in dimension
n
. Since the composition of two successive di±erentials is 0, it
follows that
B
n
⊆
Z
n
. The quotient
Z
n
/B
n
is written
H
n
(
C
∗
) or just
H
n
; it is called the
n
th
homology module
(or
homology group
if the underlying ring
R
is
Z
).
[The key idea of algebraic topology is the association of an algebraic object, the col
lection of homology groups
H
n
(
X
), to a topological space
X
. If two spaces
X
and
Y
are homeomorphic, in fact if they merely have the same homotopy type, then
H
n
(
X
)
and
H
n
(
Y
) are isomorphic for all
n
. Thus the homology groups can be used to distin
guish between topological spaces; if the homology groups di±er, the spaces cannot be
homeomorphic.]
Note that any exact sequence is a complex, since the composition of successive maps
is 0.
1
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S1.2 Defnition
A
chain map
f
:
C
∗
→
D
∗
from a chain complex
C
∗
to a chain complex
D
∗
is a collection
of module homomorphisms
f
n
:
C
n
→
D
n
, such that for all
n
, the following diagram is
commutative.
C
n
f
n
/
d
n
²
D
n
d
n
C
n
−
1
f
n
−
1
/
D
n
−
1
We use the same symbol
d
n
to refer to the diFerentials in
C
∗
and
D
∗
.
[If
f
:
X
→
Y
is a continuous map of topological spaces and
σ
is a singular
n
simplex
in
X
, then
f
#
(
σ
)=
f
◦
σ
is a singular
n
simplex in
Y
, and
f
#
extends to a homomorphism
of
n
chains. If we assemble the
f
#
’s for
n
=0
,
1
,...
, the result is a chain map.]
S1.3 Proposition
A chain map
f
takes cycles to cycles and boundaries to boundaries. Consequently, the
map
z
n
+
B
n
(
C
∗
)
→
f
n
(
z
n
)+
B
n
(
D
∗
) is a wellde±ned homomorphism from
H
n
(
C
∗
) to
H
n
(
D
∗
). It is denoted by
H
n
(
f
).
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 Spring '11
 sdd
 Algebra, Algebraic Topology

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