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= {Z : Z(Rk )⊥A(Rk )}.

So ¦A : TA O(k, ∞) ’ VA O(k, ∞) turns out to be ¦A (Z) = AAt Z = AωA (Z),
where ωA (Z) := At Z = ’Z t A. Then ω ∈ „¦1 (O(k, ∞), o(k)) is an O(k)-equivariant
form which reproduces the generators of fundamental vector ¬elds of the principal
right action, so it is a principal O(k)-connection (37.19):

((rU )— ω)A (Z) = U t At ZU = Ad(U ’1 )ωA (Z),
ωA (AY ) = Y for Y ∈ o(k).

Step 3. The classifying process.
Let (p : P ’ M, O(k)) be a principal bundle with a principal connection form
ωP ∈ „¦1 (P, o(k)). We consider the obvious representation of O(k) on Rk and the
associated vector bundle E = P [Rk ] = P —O(k) Rk with its induced ¬ber Riemannian
metric gv and induced linear connection .
Now we choose a Riemannian metric gM on the base manifold M , which we pull
back to the horizontal bundle Hor E (with respect to ) in T E via the ¬berwise
isomorphism T p| Hor E : Hor E ’ T M . Then we use the vertical lift vlE :
E —M E ’ V E ‚ T E to heave the ¬ber metric gv to the vertical bundle. Finally,
we declare the horizontal and the vertical bundle to be orthogonal, and thus we
get a Riemannian metric gE := (T p| Hor E)— gM • (vprE )— gv on the total space

47.6
518 Chapter X. Further Applications 47.6

E. By the theorem of [Nash, 1956] (see also [G¨nther, 1989] and [Gromov, 1986]),
u
there is an isometric embedding (which can be chosen real analytic, if all data
are real analytic) i : (E, gE ) ’ RN into some high dimensional Euclidean space
which in turn is contained in R(N) . Let j := dv i(0E ) : E ’ R(N) be the vertical
d
derivative along the zero section of E which is given by dv i(0x )(ux ) = dt |0 i(tux ).
Then j : E ’ R(N) is a ¬ber linear smooth mapping which is isometric on each
¬ber.
Let us now identify the principal bundle P with the orthonormal frame bundle
O(Rk , (E, gv )) of its canonically associated Riemannian vector bundle. Then j— :
P u ’ j —¦ u ∈ O(k, ∞) de¬nes a smooth mapping which is O(k)-equivariant and
therefore ¬ts into the following pullback diagram

w O(k, ∞)
j—
P
p
u u
π

w G(k, ∞).
f
M

The factored smooth mapping f : M ’ G(k, ∞) is therefore classifying for the
bundle P , so that f — O(k, ∞) ∼ P .
=
In order to show that the canonical connection is pulled back to the given one we
consider again the associated Riemannian vector bundle E ’ M from above. Note
that T i(Hor E) is orthogonal to T i(V E) in R(N) , and we have to check that this
is still true for T j, see (37.26). This is a local question on M , so let E = U — Rn ,
then we have as in (29.9)

T (U — Rn ) = U — Rn — Rm — Rn (x, v; ξ, ω) ’
Ti
’ (i(x, v), d1 i(x, v).ξ + d2 i(x, v)ω) ∈ R(N) — R(N)

Tj
’’ (d2 i(x, 0)v, d1 d2 i(x, 0)(ξ, v) + d2 i(x, 0)ω),
Ti
V (U — Rn ) (x, v; 0, ω) ’ (i(x, v), d2 i(x, v)ω),

Tj
’’ (d2 i(x, 0)v, d2 i(x, 0)ω),
Hor (U — Rn ) (x, v; ξ, “x (ξ, v)) ’ (i(x, v), d1 i(x, v).ξ + d2 i(x, v)“x (ξ, v)),
Tj
’’ (d2 i(x, 0)v, d1 d2 i(x, 0)(ξ, v) + d2 i(x, 0)“x (ξ, v)),
0 = d2 i(x, v)ω, d1 i(x, v).ξ + d2 i(x, v)“x (ξ, v) for all ξ, ω.

In the last equation we replace v and ω both by tv and apply ‚t |0 to get the required
result

0 = d2 i(x, tv)tv, d1 i(x, tv).ξ + d2 i(x, tv)“x (ξ, tv) for all ξ, ω.
0 = d2 i(x, 0)v, d2 d1 i(x, 0).(v, ξ) + d2 i(x, 0)“x (ξ, v) for all ξ, ω.



47.6
47.7 47. Manifolds for algebraic topology 519

47.7. The Lie group GL(∞; R). The canonical embeddings Rn ’ Rn+1 onto
the ¬rst n coordinates induce injections GL(n) ’ GL(n + 1). The inductive limit
is given by
GL(∞; R) = GL(∞) := ’ GL(n)
lim
’ n’∞

in the category of sets or groups. Since each GL(n) also injects into L(R(N) , R(N) )
we can visualize GL(∞) as the set of all N — N-matrices which are invertible and
di¬er from the identity in ¬nitely many entries only.
We also consider the Lie algebra gl(∞) of all N — N-matrices with only ¬nitely
many nonzero entries, which is isomorphic to R(N—N) , and we equip it with this
convenient vector space structure. Then

gl(∞) = lim gl(n)
’’
n’∞


in the category of real analytic mappings, since it is a regular inductive limit in the
category of bounded linear mappings.

Claim. gl(∞) = L(RN , R(N) ) as convenient vector spaces. Composition is a boun-
ded bilinear mapping on gl(∞). The transposition

A ’ At = A —¦ i : RN ’ (RN ) ’ (RN ) = R(N)

on the space L(RN , R(N) ) induces a bibounded linear isomorphism of gl(∞) which
resembles the usual transposition of matrices.
N
Proof. Let T ∈ L(RN , R(N) ). Then T ∈ L(RN , RN ) = R(N) and hence is a
matrix with ¬nitely many non zero entries in every line. Since T has values in R(N) ,
there are also only ¬nitely many non zero entries in each column, since T (ej ) ∈ R(N) .
Suppose that T is not in gl(∞). Then the matrix of T has in¬nitely many nonzero
entries, so there are Tjik = 0 for ik ∞ and jk ∞ and such that jk is the last
k


index with nonzero entry in the line ik . Now one can choose inductively an element
(xi ) ∈ RN with T (x) ∈ R(N) , a contradiction. For both spaces the evaluations
/
evi,j generate the convenient vector space structure by (5.18), so the convenient
structures coincide.
Another argument leading to this conclusion is the following: Since both spaces are
nuclear we have for the injective tensor product

L(RN , R(N) ) ∼ R(N) —µ R(N) .
ˆ
=

By the same reason the injective and the projective tensor product coincide. Since
both spaces are (DF), separately continuous bilinear functionals are jointly continu-
ous, so the latter space coincides with the bornological tensor product R(N) —β R(N) ,
˜
which commutes with direct sums, since it is a left adjoint functor, so ¬nally we
get R(N—N) .

47.7
520 Chapter X. Further Applications 47.8

Composition is bounded since it can be written as
comp
L(RN , R(N) ) — L(RN , R(N) ) ’ L(RN , R(N) ) — L(RN , RN ) ’ ’ L(RN , R(N) ).
’’
The assertion about transposition is obvious, using L(RN , (RN ) ) ∼ L2 (RN ; R).
=

Then the (convenient) a¬ne space
Id +gl(∞) = lim (Id +gl(n))
’’
n’∞

is closed under composition, which is real analytic on it. The determinant is a real
analytic function there, too.
Now obviously GL(∞) = {A ∈ Id +gl(∞) : det(A) = 0}, so GL(∞) is an open
subset in Id +gl(∞) and is thus a real analytic manifold, in fact, it is the inductive
limit of all the groups GL(n) = {A ∈ Id∞ +gl(n) : det(A) = 0} in the category of
real analytic manifolds.
We consider the Killing form on gl(∞), which is given by the trace
k(X, Y ) := tr(XY ) for X, Y ∈ gl(∞).
This is the right concept, since for each n and X, Y ∈ gl(n) ‚ gl(∞) we have
1
trR(N) (XY ) = trRn (XY ) = trgl(n) (ad(X) ad(Y )) + 2 trR(N) (X) trR(N) (Y ) ,
2n

but ad(X)ad(Y ) ∈ L(R(N) , R(N) ) is not of trace class. We have the following short
exact sequence of Lie algebras and Lie algebra homomorphisms
tr
0 ’ sl(∞) ’ gl(∞) ’ R ’ 0.

t
It splits, using t ’ n · IdRn for an arbitrary n, but gl(∞) has no nontrivial abelian
ideal a, since we would have a © gl(n) ‚ R · Idn for every n. So gl(∞) is only the
semidirect product of R with the ideal sl(∞) and not the direct product.

47.8. Theorem. GL(∞) is a real analytic regular Lie group modeled on R(N)
with Lie algebra gl(∞) and is the inductive limit of the Lie groups GL(n) in
the category of real analytic manifolds. The exponential mapping is well de¬ned,
real analytic, and a local real analytic di¬eomorphism onto a neighborhood of the
identity. The Campbell-Baker-Hausdor¬ formula gives a real analytic mapping
near 0 and expresses the multiplication on GL(∞) via exp. The determinant
det : GL(∞) ’ R \ 0 is a real analytic homomorphism. We have a real ana-
lytic left action GL(∞) — R(N) ’ R(N) , such that R(N) \ 0 is one orbit, but the
injection GL(∞) ’ L(R(N) , R(N) ) does not generate the topology.

Proof. Since the exponential mappings are compatible with the inductive limits
and are di¬eomorphisms on open balls with radius π in norms in which the Lie
brackets are submultiplicative, all these assertions follow from the inductive limit
property. One may use the double of the operator norms.
Regularity is proved as follows: A smooth curve X : R ’ gl(∞) factors locally in
R into some gl(n), and we may integrate this piece of the resulting right invariant
time dependent vector ¬eld on GL(n).


47.8
47.10 47. Manifolds for algebraic topology 521

47.9. Theorem. Let g be a Lie subalgebra of gl(∞). Then there is a smoothly
arcwise connected splitting regular Lie subgroup G of GL(∞) whose Lie algebra is
g. The exponential mapping of GL(∞) restricts to that of G, which is a local real
analytic di¬eomorphism near zero. The Campbell-Baker-Hausdor¬ formula gives a
real analytic mapping near 0 and has the usual properties, also on G.

Proof. Let gn := g©gl(n), a ¬nite dimensional Lie subalgebra of g. Then gn = g.
The convenient structure g = ’ n gn coincides with the structure inherited as a
lim

complemented subspace, since gl(∞) carries the ¬nest locally convex structure.
So for each n there is a connected Lie subgroup Gn ‚ GL(n) with Lie algebra gn .
Since gn ‚ gn+1 we have Gn ‚ Gn+1 , and we may consider G := n Gn ‚ GL(∞).
Each g ∈ G lies in some Gn and may be connected to Id via a smooth curve there,
which is also smooth curve in G, so G is smoothly arcwise connected.
All mappings exp |gn : gn ’ Gn are local real analytic di¬eomorphisms near 0, so
exp : g ’ G is also a local real analytic di¬eomorphism near zero onto an open
neighborhood of the identity in G. A similar argument applies to evol so that G is
regular. The rest is clear.

47.10. Examples. In the following we list some of the well known examples of
simple in¬nite dimensional Lie groups which ¬t into the picture treated in this
section. The reader can easily continue this list, especially by complex versions.

The Lie group SL(∞) is the inductive limit

SL(∞) = {A ∈ GL(∞) : det(A) = 1}
= lim SL(n) ‚ GL(∞),
’’
n’∞


the connected Lie subgroup with Lie algebra sl(∞) = {X ∈ gl(∞) : tr(X) = 0}.

The Lie group SO(∞, R) is the inductive limit

SO(∞) = {A ∈ GL(∞) : Ax, Ay = x, y for all x, y ∈ R(N) and det(A) = 1}
= lim SO(n) ‚ GL(∞),
’’
n’∞


the connected Lie subgroup of GL(∞) with the Lie algebra o(∞) = {X ∈ gl(∞) :
X t = ’X} of skew elements.

The Lie group O(∞) is the inductive limit

O(∞) = {A ∈ GL(∞) : Ax, Ay = x, y for all x, y ∈ R(N) }
= lim O(n) ‚ GL(∞).
’’
n’∞


It has two connected components, that of the identity is SO(∞).

47.10
522 Chapter X. Further Applications 48.1

The Lie group Sp(∞, R) is the inductive limit

Sp(∞, R) = {A ∈ GL(∞) : At JA = J}
= lim Sp(2n, R) ‚ GL(∞), where
’’
n’∞
01
« 
¬ ’1 0 ·
01 · ∈ L(R(N) , R(N) ).
¬ ·
J =¬
¬
’1 0
·
 
..
.

It is the connected Lie subgroup of GL(∞) with the Lie algebra sp(∞, R) = {X ∈
gl(∞) : X t J + JX = 0} of symplectically skew elements.

47.11. Theorem. The following manifolds are real analytically di¬eomorphic to
the homogeneous spaces indicated:

L(Rk , R∞’k )
Idk
GL(k, ∞) ∼ GL(∞) ,
=
GL(∞ ’ k)
0
O(k, ∞) ∼ O(∞)/(Idk —O(∞ ’ k)),
=
G(k, ∞) ∼ O(∞)/(O(k) — O(∞ ’ k)).
=

The universal vector bundle (E(k, ∞), π, G(k, ∞), Rk ) is de¬ned as the associated
bundle

E(k, ∞) = O(k, ∞)[Rk ]
= {(Q, x) : x ∈ Q} ‚ G(k, ∞) — R(N) .

The tangent bundle of the Grassmannian is then given by

T G(k, ∞) = L(E(k, ∞), E(k, ∞)⊥ ).


Proof. This is a direct consequence of the chart construction of G(k, ∞).



48. Weak Symplectic Manifolds

48.1. Review. For a ¬nite dimensional symplectic manifold (M, σ) we have the
following exact sequence of Lie algebras, see also (45.7):

gradσ γ
0 ’ H 0 (M ) ’ C ∞ (M, R) ’ ’ X(M, σ) ’ H 1 (M ) ’ 0.
’’ ’

Here H — (M ) is the real De Rham cohomology of M , the space C ∞ (M, R) is
equipped with the Poisson bracket { , }, X(M, σ) consists of all vector ¬elds ξ
with Lξ σ = 0 (the locally Hamiltonian vector ¬elds), which is a Lie algebra for the

48.1
48.3 48. Weak symplectic manifolds 523

Lie bracket. Furthermore, gradσ f is the Hamiltonian vector ¬eld for f ∈ C ∞ (M, R)
given by i(gradσ f )σ = df and γ(ξ) = [iξ σ]. The spaces H 0 (M ) and H 1 (M ) are
equipped with the zero bracket.
Given a symplectic left action : G — M ’ M of a connected Lie group G on M ,
the ¬rst partial derivative of gives a mapping : g ’ X(M, σ) which sends each
element X of the Lie algebra g of G to the fundamental vector ¬eld. This is a Lie
algebra homomorphism.

wC w X(M, σ) w H (M )
gradσ
x
x
γ
i ∞
0 1
H (M ) (M, R)


χx

g

A linear lift χ : g ’ C ∞ (M, R) of with gradσ —¦χ = exists if and only if γ —¦ = 0
in H 1 (M ). This lift χ may be changed to a Lie algebra homomorphism if and only
if the 2-cocycle χ : g — g ’ H 0 (M ), given by (i —¦ χ)(X, Y ) = {χ(X), χ(Y )} ’
¯ ¯
χ([X, Y ]), vanishes in H 2 (g, H 0 (M )), for if χ = δ± then χ ’ i —¦ ± is a Lie algebra
¯
homomorphism.
If χ : g ’ C ∞ (M, R) is a Lie algebra homomorphism, we may associate the moment
mapping µ : M ’ g = L(g, R) to it, which is given by µ(x)(X) = χ(X)(x) for
x ∈ M and X ∈ g. It is G-equivariant for a suitably chosen (in general a¬ne)
action of G on g . See [Weinstein, 1977] or [Libermann, Marle, 1987] for all this.

48.2. We now want to carry over to in¬nite dimensional manifolds the procedure
of (48.1). First we need the appropriate notions in in¬nite dimensions. So let M
be a manifold, which in general is in¬nite dimensional.
A 2-form σ ∈ „¦2 (M ) is called a weak symplectic structure on M if it is closed
(dσ = 0) and if its associated vector bundle homomorphism σ ∨ : T M ’ T — M is
injective.
A 2-form σ ∈ „¦2 (M ) is called a strong symplectic structure on M if it is closed (dσ =
0) and if its associated vector bundle homomorphism σ ∨ : T M ’ T — M is invertible
with smooth inverse. In this case, the vector bundle T M has re¬‚exive ¬bers Tx M :
Let i : Tx M ’ (Tx M ) be the canonical mapping onto the bidual. Skew symmetry
of σ is equivalent to the fact that the transposed (σ ∨ )t = (σ ∨ )— —¦ i : Tx M ’ (Tx M )
satis¬es (σ ∨ )t = ’σ ∨ . Thus, i = ’((σ ∨ )’1 )— —¦ σ ∨ is an isomorphism.

48.3. Every cotangent bundle T — M , viewed as a manifold, carries a canonical
weak symplectic structure σM ∈ „¦2 (T — M ), which is de¬ned as follows (see (43.9)
for the ¬nite dimensional version). Let πM : T — M ’ M be the projection. Then


the Liouville form θM ∈ „¦1 (T — M ) is given by θM (X) = πT — M (X), T (πM )(X)—

for X ∈ T (T — M ), where denotes the duality pairing T — M —M T M ’ R.
,
Then the symplectic structure on T — M is given by σM = ’dθM , which of course
in a local chart looks like σE ((v, v ), (w, w )) = w , v E ’ v , w E . The associated
mapping σ ∨ : T(0,0) (E — E ) = E — E ’ E — E is given by (v, v ) ’ (’v , iE (v)),

48.3
524 Chapter X. Further Applications 48.6

where iE : E ’ E is the embedding into the bidual. So the canonical symplectic
structure on T — M is strong if and only if all model spaces of the manifold M are
re¬‚exive.

48.4. Let M be a weak symplectic manifold. The ¬rst thing to note is that the
hamiltonian mapping gradσ : C ∞ (M, R) ’ X(M, σ) does not make sense in general,
since σ ∨ : T M ’ T — M is not invertible. Namely, gradσ f = (σ ∨ )’1 —¦ df is de¬ned
only for those f ∈ C ∞ (M, R) with df (x) in the image of σ ∨ for all x ∈ M . A
similar di¬culty arises for the de¬nition of the Poisson bracket on C ∞ (M, R).
σ
De¬nition. For a weak symplectic manifold (M, σ) let Tx M denote the real linear
∨ —
σ
subspace Tx M = σx (Tx M ) ‚ Tx M = L(Tx M, R), and let us call it the smooth
cotangent space with respect to the symplectic structure σ of M at x in view of
the embedding of test functions into distributions. These vector spaces ¬t together
to form a subbundle of T — M which is isomorphic to the tangent bundle T M via
σ ∨ : T M ’ T σ M ⊆ T — M . It is in general not a splitting subbundle.

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