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.