Graphics Reference
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of singular q-chains
of
X
, denoted by C
s
(
X
), by
s
()
=
{
Æ
()
=
}
C
X
f
:
S
Z
f T
0 for all but a finite number of T
Œ
S
q
.
q
The group operation “+” on C
s
(
X
) is the obvious one, namely, if f, g ΠC
s
(
X
) and
T ΠS
q
, then
(
)( )
=
()
+
(
.
fgT fT gT
+
By identifying T with the map f
T
:S
q
Æ
Z
, where
()
=
()
=
fT
f
1
0
,
,
T
T
¢
if T
¢
π
T,
T
we see ( just like in the case of the chain groups for a simplicial complex) that C
s
(
X
)
can be thought of as the set of all finite linear combinations n
1
T
1
+ n
2
T
2
+ ...+ n
k
T
k
,
n
i
Œ
Z
, of singular simplices T
i
of
X
.
q
Definition.
Given a singular q-simplex T :D
Æ
X
, define the
ith face
of T,
d
i
q
-
1
T: D
Æ
X
by
(
)
(
d
i
)
=
(
)
Tt t
,
,...,
t
Tt
,...,
t
, , ,...,
0
t
t
.
01
q
-
1
0
i
-
1
i
q
-
1
The
boundary map
∂
s
s
()
Æ
s
()
:
C
X
C
X
-1
is the homomorphism defined by the condition that
q
()
=
()
(
Â
i
)
s
i
∂
T
1
d
T
i
=
0
q
for each singular q-simplex T :D
Æ
X
.
One shows like before that ∂
q-1
°
∂
s
= 0, so that one can again define homology groups.
The
qth singular homology group
, denoted by H
s
(
X
), is defined by
Definition.
s
ker
∂
s
()
=
H
.
s
im
∂
q
+
1
Definition.
Given a continuous map f : X Æ Y, define a homomorphism
