7.2. SINGULAR CHAINS
169
7.2
Singular Chains
•
The
standard Euclidean
p
simplex
in
R
p
is the convex set
Δ
p
⊂
R
p
gener
ated by an ordered (
p
+
1)tuple (
P
0
,
P
1
, . . . ,
P
p
) of points in
R
p
P
0
=
(0
, . . . ,
0)
,
P
i
=
(0
, . . . ,
0
,
1
,
0
, . . . ,
0)
,
i
=
1
, . . . ,
p
,
all of whose components are 0 except for the
i
th component, which is equal
to 1.
•
We use the following notation
Δ
p
=
(
P
0
,
P
1
, . . . ,
P
p
)
.
•
Let
M
be an
n
dimensional manifold. A
singular
p
simplex
in
M
is a
di
ﬀ
erentiable map
σ
p
:
Δ
p
→
M
.
•
By slightly abusing notation we denote the image
σ
p
(
Δ
p
) of
Δ
p
in
M
under
the map
σ
p
just by
σ
p
.
•
Let
α
be a
p
form on
M
and
σ
p
be a
p
simplex in
M
. We deﬁne the integral
of
α
over
σ
p
by
Z
σ
p
α
=
Z
Δ
p
σ
*
p
α .
•
The
k
th face
of a standard
p
simplex
Δ
p
=
(
P
0
,
P
1
, . . . ,
P
p
) (a face opposite
to to the vertex
P
k
) is the convex set in
R
p
generated by an ordered
p
tuple
of points in
R
p
Δ
(
k
)
p

1
=
(
P