with real coe¬cients, for example. If M is compact, H1 (M ) ∈ Mf and H1 be-

comes a product preserving functor from the category of all compact manifolds

into Mf without a Weil algebra.

For a connected manifold M the singular homology group H1 (M, Z) with

integer coe¬cients is a countable discrete set, since it is the abelization of the

fundamental group π1 (M ), which is a countable group for a separable connected

manifold. Then again by the K¨nneth theorem H1 ( ; Z) is a product preserv-

u

ing functor from the category of connected manifolds into Mf without a Weil

algebra.

More generally let K be a ¬nite CW -complex and let [K, M ] denote the

discrete set of all (free) homotopy classes of continuous mappings K ’ M ,

where M is a manifold. Algebraic topology tells us that this is a countable set.

Clearly [K, ] then de¬nes a product preserving functor without a Weil algebra.

Since we may take the product of such functors with other product preserving

functors we see, that the Weil algebra does not determine the functor at all. For

conditions which exclude such behaviour see theorem 36.8 below.

36.5. Convention. Let A = A1 • · · · • Ak be a formally real ¬nite dimensional

commutative algebra with its decomposition into Weil algebras. In this section

we will need the product preserving functor TA := TA1 — . . . — TAk : Mf ’

Mf which is given by TA (M ) := TA1 (M ) — . . . — TAk (M ). Then 35.13.1 for

TA has to be modi¬ed as follows: πA,M : TA M ’ M k is a ¬ber bundle. All

other conclusions of theorem 35.13 remain valid for this functor, since they are

preserved by the product, with exception of 35.13.3, which holds for connected

manifolds only now. Theorem 35.14 remains true, but the covariant description

(we will not use it in this section) 35.15 and 35.16 needs some modi¬cation.

36.6. Lemma. Let F : Mf ’ Mf be a product preserving functor. Then the

mapping

χF,M : F (M ) ’ Hom(C ∞ (M, R), Al(F )) = TAl(F ) M

χF,M (x)(f ) := F (f )(x),

is smooth and natural in F and M .

Proof. Naturality in F and M is obvious. To show that χ is smooth is more

di¬cult. To simplify the notation we let Al(F ) =: A = A1 • · · · • Ak be the

decomposition of the formally real algebra Al(F ) into Weil algebras.

Let h = (h1 , . . . , hn ) : M ’ Rn be a closed embedding into some high

dimensional Rn . By theorem 35.13.2 the mapping TA (h) : TA M ’ TA Rn is also

a closed embedding. By theorem 35.14, step 1 of the proof (and by reordering the

product), the mapping ·Rn ,A : Hom(C ∞ (Rn , R), A) ’ An is given by ·Rn ,A (•) =

(•(xi ))n , where (xi ) are the standard coordinate functions on Rn . We have

i=1

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312 Chapter VIII. Product preserving functors

F (Rn ) ∼ F (R)n ∼ An ∼ TA (Rn ). Now we consider the commuting diagram

= = =

F (M )

χF,M

u

wT

·M,A

∞

Hom(C (M, R), A) A (M )

(h— )—

u u

TA (h)

wT

·Rn ,A

∞ n n

F (Rn )

Hom(C (R , R), A) A (R )

For z ∈ F (M ) we have

(·Rn ,A —¦ (h— )— —¦ χF,M )(z) = ·Rn ,A (χF,M (z) —¦ h— )

= χF,M (z)(x1 —¦ h), . . . , χF,M (z)(xn —¦ h)

= χF,M (z)(h1 ), . . . , χF,M (z)(hn )

= F (h1 )(z), . . . , F (hn )(z) = F (h)(z).

This is smooth in z ∈ F (M ). Since ·M,A is a di¬eomorphism and TA (h) is a

closed embedding, χF,M is smooth as required.

36.7. The universal covering of a product preserving functor. Let

F : Mf ’ Mf be a product preserving functor. We will construct another

product preserving functor as follows. For any manifold M we choose a universal

˜

cover qM : M ’ M (over each connected component of M separately), and we let

˜

π1 (M ) denote the group of deck transformations of M ’ M , which is isomorphic

to the product of all fundamental groups of the connected components of M . It

˜

is easy to see that π1 (M ) acts strictly discontinuously on TA (M ), and by lemma

˜

36.6 therefore also on F (M ). So the orbit space

˜ ˜

F (M ) := F (M )/π1 (M )

˜˜ ˜

is a smooth manifold. For f : M1 ’ M2 we choose any smooth lift f : M1 ’ M2 ,

˜

which is unique up to composition with elements of π1 (Mi ). Then F f factors

as follows:

w

˜

F (f )

˜ ˜

F (M1 ) F (M2 )

u u

w

˜

F (f )

˜ ˜

F (M ) F (M2 ).

˜

The resulting smooth mapping F (f ) does not depend on the choice of the lift

˜ ˜

f . So we get a functor F : Mf ’ Mf and a natural transformation q = qF :

˜ ˜

F ’ F , induced by F (qM ) : F (M ) ’ F (M ), which is a covering mapping. This

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

36. Product preserving functors 313

˜

functor F is again product preserving, because we may choose (M1 — M2 )∼ =

˜ ˜

M1 — M2 and π1 (M1 — M2 ) = π1 (M1 ) — π1 (M2 ), thus

˜

F (M1 — M2 ) = F ((M1 — M2 )∼ )/π1 (M1 — M2 ) =

˜ ˜ ˜ ˜

= F (M1 )/π1 (M1 ) — F (M2 )/π1 (M2 ) = F (M1 ) — F (M2 ).

˜

Note ¬nally that TA = TA if A is sum of at least two Weil algebras. As an exam-

ple consider A = R • R, then TA (M ) = M — M , but TA (S 1 ) = R2 /Z(2π, 2π) ∼

˜ =

1

S — R.

36.8. Theorem. Let F be a product preserving functor.

(1) If M is connected, then there exists a unique smooth mapping ψF,M :

TAl(F ) (M ) ’ F (M ) which is natural in F and M and satis¬es χF,M —¦

ψF,M = qTAl(F ),M :

w F (M )

h

ψF,M

TAl(F ) (M )

hqj 9

h

B9

9

χ F,M

TAl(F ) (M ).

(2) If F maps embeddings to injective mappings, then χF,M : F (M ) ’

TAl(F ) (M ) is injective for all manifolds M , and it is a di¬eomorphism for

connected M .

(3) If M is connected and ψF,M is surjective, then χF,M and ψF,M are cov-

ering mappings.

Remarks. Condition (2) singles out the functors of the form TA among all

product preserving functors. Condition (3) singles the coverings of the TA ™s. A

product preserving functor satisfying condition (3) will be called weakly local .

Proof. We let Al(F ) =: A = A1 • · · · • Ak be the decomposition of the formally

real algebra Al(F ) into Weil algebras. We start with a

Sublemma. If M is connected then χF,M is surjective and near each • ∈

Hom(C ∞ (M, R), A) = TA (M ) there is a smooth local section of χF,M .

Let • = •1 + · · · + •k for •i ∈ Hom(C ∞ (M, R), Ai ). Then by lemma 35.8 for

each i there is exactly one point xi ∈ M such that •i (f ) depends only on a ¬nite

jet of f at xi . Since M is connected there is a smoothly contractible open set

U in M containing all xi . Let g : Rm ’ M be a di¬eomorphism onto U . Then

(g — )— : Hom(C ∞ (Rm , R), A) ’ Hom(C ∞ (M, R), A) is an embedding of an open

neighborhood of •, so there is • ∈ Hom(C ∞ (Rm , R), A) depending smoothly on

¯

——

• such that (g ) (•) = •. Now we consider the mapping

¯

F (g)

·m

Hom(C ∞ (Rm , R), A) ’R ’ TA (Rm ) ∼ F (Rm ) ’ ’

’’ ’’

=

F (g) χM

’ ’ F (M ) ’ ’ Hom(C ∞ (M, R), A).

’’ ’

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314 Chapter VIII. Product preserving functors

We have (χM —¦ F (g) —¦ ·Rm )(•) = ((g — )— —¦ χRm —¦ ·Rm )(•) = (g — )— (•) = •,

¯ ¯ ¯

since it follows from lemma 36.6 that χRm —¦ ·Rm = Id. So the mapping sU :=

F (g) —¦ ·Rm —¦ (g —— )’1 : TA U ’ F (M ) is a smooth local section of χM de¬ned

near •. We may also write sU = F (iU ) —¦ (χF,U )’1 : TA U ’ F (M ), since for

contractible U the mapping χF,U is clearly a di¬eomorphism. So the sublemma

is proved.

(1) Now we start with the construction of ψF,M . We note ¬rst that it su¬ces

to construct ψF,M for simply connected M because then we may induce it for

not simply connected M using the following diagram and naturality.

w

ψF,M

˜

˜ ˜ ˜

TA (M ) TA M F (M )

u u

w F (M ).

ψF,M

TA (M )

Furthermore it su¬ces to construct ψF,M for high dimensional M since then we

w

have ψF,M —R

TA (M — R) F (M — R)

u u

w F (M ) — F (R).

ψF,M — IdF (R)

TA (M ) — F (R)

So we may assume that M is connected, simply connected and of high dimension.

For any contractible subset U of M we consider the local section sU of χF,M

constructed in the sublemma and we just put ψF,M (•) := sU (•) for • ∈ TA U ‚

TA M . We have to show that ψF,M is well de¬ned. So we consider contractible

U and U in M with • ∈ TA (U © U ). If π(•) = (x1 , . . . , xk ) ∈ M k as in

the sublemma, this means that x1 , . . . , xk ∈ U © U . We claim that there are

contractible open subsets V , V , and W of M such that x1 , . . . , xk ∈ V © V ©

W and that V ‚ U © W and V ‚ U © W . Then by the naturality of χ

we have sU (•) = sV (•) = sW (•) = sV (•) = sU (•) as required. For the

existence of these sets we choose an embedding H : R2 ’ M such that c(t) =

H(t, sin t) ∈ U , c (t) = H(t, ’ sin t) ∈ U and H(2πj, 0) = xj for j = 1, . . . , k.

This embedding exists by the following argument. We connect the points by

a smooth curve in U and a smooth curve in U , then we choose a homotopy

between these two curves ¬xing the xj ™s, and we approximate the homotopy by

an embedding, using transversality, again ¬xing the xj ™s. For this approximation

we need dim M ≥ 5, see [Hirsch, 76, chapter 3]. Then V , V , and W are just

small tubular neighborhoods of c, c , and H.

(2) Since a manifold M has at most countably many connected components,

there is an embedding I : M ’ Rn for some n. Then from

v w

F (i)

F (Rn )

F (M )

∼ χF,Rn

χF,M

u u

=

wT n

TA (M ) A (R ),

TA (i)

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36. Product preserving functors 315

lemma 36.6, and the assumption it follows that χF,M is injective. If M is fur-

thermore connected then the sublemma implies furthermore that χF,M is a dif-

feomorphism.

(3) Since χ—¦ψ = q, and since q is a covering map and ψ is surjective, it follows

that both χ and ψ are covering maps.

In the example F = TR•R considered at the end of 36.7 we get that ψF,S 1 :

˜

F (S 1 ) = R2 /Z(2π, 2π) ’ F (S 1 ) = S 1 — S 1 = R2 /(Z(2π, 0) — Z(0, 2π)) is the

covering mapping induced from the injection Z(2π, 2π) ’ Z(2π, 0) — Z(0, 2π).

36.9. Now we will determine all weakly local product preserving functors F on

the category conMf of all connected manifolds with Al(F ) equal to some given

formally real ¬nite dimensional algebra A with k Weil components. Let F be

such a functor.

For a connected manifold M we de¬ne C(M ) by the following transversal

pullback:

w

C(M ) F (M )

u u

wT

0

k

TRk (M ) M A M,

where 0 is the natural transformation induced by the inclusion of the subalgebra

Rk generated by all idempotents into A.

Now we consider the following diagram: In it every square is a pullback, and

each vertical mapping is a covering mapping, if F is weakly local, by theorem

w

36.8.

0

˜ ˜

Mk TA M

u

u

wT

˜k

M /π1 (M ) A (M )

u u

ψ

w F (M )

C(M )

u u

χ

wT

k

M A (M ).

˜

Thus F (M ) = TA (M )/G, where G is the group of deck transformations of

˜

the covering C(M ) ’ M k , a subgroup of π1 (M )k containing π1 (M ) (with its

˜ ˜

diagonal action on M k ). Here g = (g1 , . . . , gk ) ∈ π1 (M )k acts on TA (M ) =

˜ ˜

TA1 (M ) — . . . — TAk (M ) via TA1 (g1 ) — . . . — TAk (gk ). So we have proved

36.10. Theorem. A weakly local product preserving functor F on the cat-

egory conMf of all connected manifolds is uniquely determined by specifying

a formally real ¬nite dimensional algebra A = Al(F ) and a product preserving

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316 Chapter VIII. Product preserving functors

k

functor G : conMf ’ Groups satisfying π1 ⊆ G ⊆ π1 , where π1 is the funda-

k

mental group functor, sitting as diagonal in π1 , and where k is the number of

Weil components of A.

The statement of this theorem is not completely rigorous, since π1 depends

on the choice of a base point.

36.11. Corollary. On the category of simply connected manifolds a weakly

local product preserving functor is completely determined by its algebra A =

Al(F ) and coincides with TA .

If the algebra Al(F ) = A of a weakly local functor F is a Weil algebra (the

unit is the only idempotent), then F = TA on the category conMf of connected

manifolds. In particular F is a bundle functor and is local in the sense of 18.3.(i).

36.12. Proof of theorem 36.1. Using the assumptions we may conclude that

πF,M : F (M ) ’ M is a ¬ber bundle for each M ∈ Mf , using 20.3, 20.7, and

20.8. Moreover for an embedding iU : U ’ M of an open subset F (iU ) : F (U ) ’

’1

F (M ) is the embedding onto F (M )|U = πF,M (U ). Let A = Al(F ). Then A can

have only one idempotent, for even the bundle functor pr1 : M — M ’ M is not

local. So A is a Weil algebra.

By corollary 36.11 we have F = TA on connected manifolds. Since F is local,

it is fully determined by its values on smoothly contractible manifolds, i.e. all

Rm ™s.

36.13. Lemma. For product preserving functors F1 and F2 on Mf we have

Al(F2 —¦ F1 ) = Al(F1 ) — Al(F2 ) naturally in F1 and F2 .

Proof. Let B be a real basis for Al(F1 ). Then

R · b) ∼

Al(F2 —¦ F1 ) = F2 (F1 (R)) = F2 ( F2 (R) · b,

=

b∈B b∈B

so the formula holds for the underlying vector spaces. Now we express the

multiplication F1 (m) : Al(F1 ) — Al(F1 ) ’ Al(F1 ) in terms of the basis: bi bj =

k

k cij bk , and we use

F2 (F1 (m)) = (F1 (m)— )— : Hom(C ∞ (Al(F1 ) — Al(F1 ), R), Al(F2 )) ’

’ Hom(C ∞ (Al(F1 ), R), Al(F2 ))

to see that the formula holds also for the multiplication.

Remark. We chose the order Al(F1 ) — Al(F2 ) so that the elements of Al(F2 )

stand on the right hand side. This coincides with the usual convention for writing

an atlas for the second tangent bundle and will be essential for the formalism

developed in section 37 below.

36.14. Product preserving functors on not connected manifolds. Let

F be a product preserving functor Mf ’ Mf . For simplicity™s sake we assume

that F maps embeddings to injective mappings, so that on connected manifolds

it coincides with TA where A = Al(F ). For a general manifold we have TA (M ) ∼

=

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

36. Product preserving functors 317

Hom(C ∞ (M, R), A), but this is not the unique extension of F |conMf to Mf ,

as the following example shows: Consider Pk (M ) = M — . . . — M (k times),

given by the product of Weil algebras Rk . Now let Pk (M ) = ± Pk (M± ) be the

c

disjoint union of all Pk (M± ) where M± runs though all connected components

c

of M . Then Pk is a di¬erent extension of Pk |conMf to Mf .

Let us assume now that A = Al(F ) is a direct sum on k Weil algebras,

A = A1 • · · · • Ak and let π : TA ’ Pk be the natural transformation induced

by the projection on the subalgebra Rk generated by all idempotents. Then also

F c (M ) = π ’1 (Pk (M )) ‚ TA (M ) is an extension of F |conMf to Mf which

c

di¬ers from TA . Clearly we have F c (M ) = ± F (M± ) where the disjoint union