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7.2. SINGULAR CHAINS
173
•
Let
σ
p
:
Δ
p
→
M
be a singular
p
simplex in
M
. Then its boundary
∂σ
p
is
the integer (
p

1)chain in
M
deﬁned by
∂σ
p
=
σ
p
*
(
∂
Δ
p
)
.
•
In more detail
∂σ
p
=
σ
p
*
(
∂
Δ
p
)
=
p
X
k
=
0
(

1)
k
σ
p
*
(
Δ
(
k
)
p

1
)
=
p
X
k
=
0
(

1)
k
(
σ
p
◦
f
k
)
.
Recall that
σ
p
*
(
Δ
(
k
)
p

1
)
=
(
σ
p
◦
f
k
) is the
k
th face of the singular
p
simplex
σ
p
.
•
That is,
the boundary of the image of
Δ
p
is the image of the boundary of
Δ
p
.
•
The boundary of any singular
p
chain with coe
ﬃ
cients in
G
is deﬁned by,
for any
g
k
∈
G
,
σ
k
p
∈
S
p
(
M
),
∂
r
X
k
=
1
g
k
σ
k
p
=
r
X
k
=
1
g
k
∂σ
k
p
.
•
This leads to the
boundary homomorphism
∂
:
C
p
(
M
;
G
)
→
C
p

1
(
M
;
G
)
.
•
Let
F
:
M
→
V
be a map,
σ
p
be a singular
p
simplex in
M
and
F
*
σ
p
be the
induced singular
p
simplex in
V
. Then
∂
(
F
*
σ
p
)
=
∂
(
F
◦
σ
p
)
=
(
F
◦
σ
p
)
*
(
∂
Δ
p
)
=
(
F
*
◦
σ
p
*
)(
∂
Δ
p
)
=
F
*
[
σ
p
*
(
∂
Δ
p
)]
=
F
*
(
∂σ
p
)
•
More generally, let
c
p
=
r
X
k
=
1
g
k
σ
k
p
be a
p
chain on
M
and
F
*
c
p
be the induced
p
chain on
V
.
di
ﬀ
geom.tex; April 12, 2006; 17:59; p. 171
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CHAPTER 7. HOMOLOGY THEORY
•
Then
∂
(
F
*
c
p
)
=
F
*
(
∂
c
p
)
.
•
Therefore,
∂
◦
F
*
=
F
*
◦
∂ ,
in other words,
the boundary of an image is the image of the boundary
.
•
Thus, we obtain a
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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