Graphics Reference
In-Depth Information
[
(
)
]
Æ
nk
+
()
j:
BR
,G
Vect
B
n
n
by
[
(
(
)
)
]
[
()
=
nk
+
j
f
f
*
g
R
.
n
One can show that j is a bijection whenever
B
is paracompact and k is large
compared with n.
The first four facts are easy to show. It is the relationship of the Grassmann
manifolds to the classification of vector bundles in Fact (5) and the consequences of
this that are nontrivial and the most significant for algebraic topology. One has a very
good understanding of the structure of these manifolds. Unfortunately, there is no
space to expand on this here and all we can do is refer the interested reader to [Stee51],
[Munk61], [Huse66], or [MilS74] for more information. However, Grassmann mani-
folds also play a role in algebraic geometry and we shall run into them again in
Chapter 10. We finish this section by showing that they are actually manifolds, justi-
fying the name. (Stiefel manifolds are also but we shall leave that as Exercise 8.14.1.)
The Grassmann manifold G
n
(
R
n+k
) is a compact nk-dimensional
8.14.1. Theorem.
C
•
manifold.
Sketch of proof.
The compactness follows from the fact that it is the continuous
image of the Stiefel manifold, which is compact because it is a closed subspace of a
compact space. This also shows second countability. One way to prove that G
n
(
R
n+k
)
is Hausdorff, is to show that one can define a continuous real-valued function on the
space that takes on different values at any two given points. Let
V
,
W
ΠG
n
(
R
n+k
). Pick
a point
p
in
R
n+k
that belongs to the linear subspace
V
but not to
W
. The function
(
)
Æ
nk
+
fG
n
:
R
R
defined by
()
=
(
)
f
X
dist
p, X
will do the job. It remains to show that the space is locally Euclidean.
Let
V
be an n-dimensional linear subspace of
R
n+k
and let B = (
v
1
,
v
2
,...,
v
n
) be a
basis for
V
. We can represent B by means of an n ¥ (n + k) matrix M
B
whose rows
are the vectors
v
i
. Assume that the first n columns of M
B
are linearly independent. It
is easy to show that out of all the matrices M
B
we get as B ranges over all bases of
V
there is a unique one M
V
of the form
=
(
)
,
MI N
(8.47)
V
V
where I is the n ¥ n identity matrix and N
V
is an n ¥ k matrix. Now let
nk
+
p
v
:
RV
Æ
