The ¬rst summand D0 = a a of D is an operator which is natural on the

functions with the speci¬ed weights on conformally ¬‚at manifolds and the sec-

ond summand is a correction for the general case. In view of this example, the

question is how far we can modify the natural operators (homogeneous in the

order and acting between bundles corresponding to irreducible representations of

CO(m, R)) found on the ¬‚at manifolds by adding some corrections. The answer

is rather nice: with some few exceptions this is always possible and the order

of the correction term is less by two (or more) than that of D0 . Moreover, the

correction involves only the Ricci curvature and its covariant derivatives. This

was deduced in [Eastwood, Rice, 87] in dimension four, and in [Baston, 90] for

dimensions greater than two (the complex representations are treated explicitely

and the authors assert that the real analogy is available with mild changes). In

particular, there are no corrections necessary for the ¬rst order operators, which

where completely classi¬ed by [Fegan, 76]. Nevertheless, the concrete formu-

las for the operators (¬rst of all for the curvature terms) are rarely available.

Another disadvantage of this approach is that we have no information on the

operators which vanish on the conformally ¬‚at manifolds, even we do not know

how far the extension of a given operator to the whole category is determined.

The description of all linear natural operators on the conformally ¬‚at mani-

folds is based on the general ideas as presented at the begining of this section.

This means we have to ¬nd the morphisms of g-modules W — ’ (Tn V )— , where

∞

g is the algebra of formal vector ¬elds on Rn with ¬‚ows consisting of conformal

morphisms. One can show that g = o(n + 1, 1), the pseudo-orthogonal algebra,

with grading g = g’1 • g0 • g1 = Rn • co(n, R) • Rn— . The lemmas 34.5 and 34.6

remain true and we see that (Tn V )— is the so called generalized Verma module

∞

corresponding to the representation of CO(n, R) on V . Each homomorphism

W — ’ (Tn V )— extends to a homomorphism of the generalized Verma modules

∞

(Tn W )— ’ (Tn V )— and so we have to classify all morphisms of generalized

∞ ∞

Verma modules. These were described in [Boe, Collingwood, 85a, 85b]. In par-

ticular, if we start with usual functions (i.e. with conformal weight zero), then

all conformally invariant operators which form a ˜connected pattern™ involving

the functions are drawn in 33.18. (The latter means that there are no more

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

Remarks 295

operators having one of the bundles indicated on the diagram as the source or

target.) A very interesting point is a general principal coming from the repre-

sentation theory (the so called Jantzen-Zuzkermann functors) which asserts that

once we have got such a ˜connected pattern™ all other ones are obtained by a

general procedure. Unfortunately this ˜translation procedure™ is not of a clear

geometric character and so we cannot get the formulas for the corresponding

operators in this way, cf. [Baston, 90]. The general theory mentioned above

implies that all the operators from the diagram in 33.18 admit the extension to

the whole category of conformal manifolds, except the longest arrow „¦0 ’ „¦m .

By the ˜translation procedure™, the same is ensured for all such patterns, but

the question whether there is an extension for the exceptional ˜long arrows™ is

not solved in general. Some of them do extend, but there are counter examples

of operators which do not admit any extension, see [Branson, 89], [Graham, to

appear].

Another more direct approach is used by [Branson, 85, 89] and others. They

write down a concrete general formula in terms of the Riemannian invariants

and they study the action of the conformal rescaling of the metric. Since it is

su¬cient to study the in¬nitesimal condition on the invariance with respect to

the rescaling of the metric, they are able to ¬nd series of conformally invariant

operators. But a classi¬cation is available for the ¬rst and second order operators

only.

Remarks

Proposition 30.4 was proved by [Kol´ˇ, Michor, 87]. Proposition 31.1 was

ar

deduced in [Kol´ˇ, 87a]. The natural transformations J r ’ J r were determined

ar

in [Kol´ˇ, Vosmansk´, 89]. The exchange map eΛ from 32.4 was introduced by

ar a

[Modugno, 89a].

The original proof of the Gilkey theorem on the uniqueness of the Pontryagin

forms, [Gilkey, 73], was much more combinatorial and had not used H. Weyl™s

theorem. Our approach is similar to [Atiyah, Bott, Patodi, 73], but we do

not need their polynomiality assumption. The Gilkey theorem was generalized

in several directions. For the case of Hermitian bundles and connections see

[Atiyah, Bott, Patodi, 73], for oriented Riemannian manifolds see [Stredder, 75],

the metrics with a general signature are treated in [Gilkey, 75]. The uniqueness of

the Levi-Civit` connection among the polynomial conformal natural connections

a

on Riemannian manifolds was deduced by [Epstein, 75]. The classi¬cation of the

¬rst order liftings of Riemannian metrics to the tangent bundles covers the results

due to [Kowalski, Sekizawa, 88], who used the so called method of di¬erential

equations in their much longer proof. Our methods originate in [Slov´k, 89] and

a

an unpublished paper by W. M. Mikulski.

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

296

CHAPTER VIII.

PRODUCT PRESERVING FUNCTORS

We ¬rst present the theory of those bundle functors which are determined by

local algebras in the sense of A. Weil, [Weil, 51]. Then we explain that the Weil

functors are closely related to arbitrary product preserving functors Mf ’ Mf .

In particular, every product preserving bundle functor on Mf is a Weil functor

and the natural transformations between two such functors are in bijection with

the homomorphisms of the local algebras in question.

In order to motivate the development in this chapter we will tell ¬rst a math-

ematical short story. For a smooth manifold M , one can prove that the space

of algebra homomorphisms Hom(C ∞ (M, R), R) equals M as follows. The ker-

nel of a homomorphism • : C ∞ (M, R) ’ R is an ideal of codimension 1 in

C ∞ (M, R). The zero sets Zf := f ’1 (0) for f ∈ ker • form a ¬lter of closed

sets, since Zf © Zg = Zf 2 +g2 , which contains a compact set Zf for a function

f which is unbounded on each non compact closed subset. Thus f ∈ker • Zf is

not empty, it contains at least one point x0 . But then for any f ∈ C ∞ (M, R)

the function f ’ •(f )1 belongs to the kernel of •, so vanishes on x0 and we have

f (x0 ) = •(f ).

An easy consequence is that Hom(C ∞ (M, R), C ∞ (N, R)) = C ∞ (N, M ). So

the category of algebras C ∞ (M, R) and their algebra homomorphisms is dual to

the category Mf of manifolds and smooth mappings.

But now let D be the algebra generated by 1 and µ with µ2 = 0 (sometimes

called the algebra of dual numbers or Study numbers, it is also the truncated

polynomial algebra of degree 1). Then it turns out that Hom(C ∞ (M, R), D) =

T M , the tangent bundle of M . For if • is a homomorphism C ∞ (M, R) ’ D,

then π —¦ • : C ∞ (M, R) ’ D ’ R equals evx for some x ∈ M and •(f ) ’

f (x).1 = X(f ).µ, where X is a derivation over x since • is a homomorphism.

So X is a tangent vector of M with foot point x. Similarly we may show that

Hom(C ∞ (M, R), D — D) = T T M .

Now let A be an arbitrary commutative real ¬nite dimensional algebra with

unit. Let W (A) be the subalgebra of A generated by the idempotent and nilpo-

tent elements of A. We will show in this chapter, that Hom(C ∞ (M, R), A) =

Hom(C ∞ (M, R), W (A)) is a manifold, functorial in M , and that in this way we

have de¬ned a product preserving functor Mf ’ Mf for any such algebra. A

will be called a Weil algebra if W (A) = A, since in [Weil, 51] this construc-

tion appeared for the ¬rst time. We are aware of the fact, that Weil algebras

denote completely di¬erent objects in the Chern-Weil construction of character-

istic classes. This will not cause troubles, and a serious group of mathematicians

has already adopted the name Weil algebra for our objects in synthetic di¬er-

ential geometry, so we decided to stick to this name. The functors constructed

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

35. Weil algebras and Weil functors 297

in this way will be called Weil functors, and we will also present a covariant

approach to them which mimics the construction of the bundles of velocities,

due to [Morimoto, 69], cf. [Kol´ˇ, 86].

ar

We will discuss thoroughly natural transformations between Weil functors and

study sections of them, a sort of generalized vector ¬elds. It turns out that the

addition of vector ¬elds generalizes to a group structure on the set of all sections,

which has a Lie algebra and an exponential mapping; it is in¬nite dimensional

but nilpotent.

Conversely under very mild conditions we will show, that up to some covering

phenomenon each product preserving functor is of this form, and that natural

transformations between them correspond to algebra homomorphisms. This has

been proved by [Kainz-Michor, 87] and independently by [Eck, 86] and [Luciano,

88].

Weil functors will play an important role in the rest of the book, and we will

frequently compare results for other functors with them. They can be much

further analyzed than other types of functors.

35. Weil algebras and Weil functors

35.1. A real commutative algebra A with unit 1 is called formally real if for any

a1 , . . . , an ∈ A the element 1 + a2 + · · · + a2 is invertible in A. Let E = {e ∈

n

1

2

A : e = e, e = 0} ‚ A be the set of all nonzero idempotent elements in A. It is

not empty since 1 ∈ E. An idempotent e ∈ E is said to be minimal if for any

e ∈ E we have ee = e or ee = 0.

Lemma. Let A be a real commutative algebra with unit which is formally real

and ¬nite dimensional as a real vector space.

Then there is a decomposition 1 = e1 + · · · + ek into all minimal idempotents.

Furthermore A = A1 • · · · • Ak , where Ai = ei A = R · ei • Ni , and Ni is a

nilpotent ideal.

Proof. First we remark that every system of nonzero idempotents e1 , . . . , er

satisfying ei ej = 0 for i = j is linearly independent over R. Indeed, if we multiply

a linear combination k1 e1 + · · · + kr er = 0 by ei we obtain ki = 0. Consider a

non minimal idempotent e = 0. Then there exists e ∈ E with e = ee =: e = 0. ¯

Then both e and e ’ e are nonzero idempotents and e(e ’ e) = 0. To deduce the

¯ ¯ ¯ ¯

required decomposition of 1 we proceed by recurrence. Assume that we have a

decomposition 1 = e1 + · · · + er into nonzero idempotents satisfying ei ej = 0

for i = j. If ei is not minimal, we decompose it as ei = ei + (ei ’ ei ) as above.

¯ ¯

The new decomposition of 1 into r + 1 idempotents is of the same type as the

original one. Since A is ¬nite dimensional this proceedure stabilizes. This yields

1 = e1 + · · · + ek with minimal idempotents. Multiplying this relation by a

minimal idempotent e, we ¬nd that e appears exactly once in the right hand

side. Then we may decompose A as A = A1 • · · · • Ak , where Ai := ei A.

Now each Ai has only one nonzero idempotent, namely ei , and it su¬ces to

investigate each Ai separately. To simplify the notation we suppose that A = Ai ,

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298 Chapter VIII. Product preserving functors

so that now 1 is the only nonzero idempotent of A. Let N := {n ∈ A : nk =

0 for some k} be the ideal of all nilpotent elements in A.

We claim that any x ∈ A \ N is invertible. If not then xA ‚ A is a proper

ideal, and since A is ¬nite dimensional the decreasing sequence

A ⊃ xA ⊃ x2 A ⊃ · · ·

of ideals must become stationary. If xk A = 0 then x ∈ N , thus there is a k such

that xk+ A = xk A = 0 for all > 0. Then x2k A = xk A and there is some y ∈ A

with xk = x2k y. So we have (xk y)2 = xk y = 0, and since 1 is the only nontrivial

idempotent of A we have xk y = 1. So xk’1 y is an inverse of x as required.

So the quotient algebra A/N is a ¬nite dimensional ¬eld, so A/N equals R

√

or C. If A/N = C, let x ∈ A be such that x + N = ’1 ∈ C = A/N . Then

1 + x2 + N = N = 0 in C, so 1 + x2 is nilpotent and A cannot be formally real.

Thus A/N = R and A = R · 1 • N as required.

35.2. De¬nition. A Weil algebra A is a real commutative algebra with unit

which is of the form A = R · 1 • N , where N is a ¬nite dimensional ideal of

nilpotent elements.

So by lemma 35.1 a formally real and ¬nite dimensional unital commutative

algebra is the direct sum of ¬nitely many Weil algebras.

35.3. Some algebraic preliminaries. Let A be a commutative algebra with

unit and let M be a module over A. The semidirect product A[M ] of A and M

or the idealisator of M is the algebra (A — M, +, ·), where (a1 , m1 ) · (a2 , m2 ) =

(a1 a2 , a1 m2 + a2 m1 ). Then M is a (nilpotent) ideal of A[M ].

Let M m—n = {(tij ) : tij ∈ M, 1 ¤ i ¤ m, 1 ¤ j ¤ n} be the space of all

(m — n)-matrices with entries in the module M . If S ∈ Ar—m and T ∈ M m—n

then the product of matrices ST ∈ M r—n is de¬ned by the usual formula.

For a matrix U = (uij ) ∈ An—n the determinant is given by the usual formula

n

det(U ) = σ∈Sn sign σ i=1 ui,σ(i) . It is n-linear and alternating in the columns

of U .

Lemma. If m = (mi ) ∈ M n—1 is a column vector of elements in the A-module

M and if U = (uij ) ∈ An—n is a matrix with U m = 0 ∈ M n—1 then we have

det(U )mi = 0 for each i.

Proof. We may compute in the idealisator A[M ], or assume without loss of gen-

erality that all mi ∈ A. Let u—j denote the j-th column of U . Then uij mj = 0

for all i means that m1 u—1 = ’ j>1 mj u—j , thus

det(U )m1 = det(m1 u—1 , u—2 , . . . , u—n )

= det(’ mj u—j , u—2 , . . . , u—n ) = 0

j>1

Lemma. Let I be an ideal in an algebra A and let M be a ¬nitely generated

A-module. If IM = M then there is an element a ∈ I with (1 ’ a)M = 0.

n

Proof. Let M = i=1 Ami for generators mi ∈ M . Since IM = M we have

n

mi = j=1 tij mj for some T = (tij ) ∈ I n—n . This means (1n ’ T )m = 0 for

m = (mj ) ∈ M n—1 . By the ¬rst lemma we get det(1n ’ T )mj = 0 for all j. But

det(1n ’ T ) = 1 ’ a for some a ∈ I.

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35. Weil algebras and Weil functors 299

Lemma of Nakayama. Let (A, I) be a local algebra (i.e. an algebra with a

unique maximal ideal I) and let M be an A-module. Let N1 , N2 ‚ M be

submodules with N1 ¬nitely generated. If N1 ⊆ N2 +IN1 then we have N1 ⊆ N2 .

In particular IN1 = N1 implies N1 = 0.

Proof. Let IN1 = N1 . By the lemma above there is some a ∈ I with (1’a)N1 =

0. Since I is a maximal ideal (so A/I is a ¬eld), 1 ’ a is invertible. Thus

N1 = 0. If N1 ⊆ N2 + IN1 we have I((N1 + N2 )/N2 ) = (N1 + N2 )/N2 thus

(N1 + N2 )/N2 = 0 or N1 ⊆ N2 .

35.4. Lemma. Any ideal I of ¬nite codimension in the algebra of germs

∞

En := C0 (Rn , R) contains some power Mk of the maximal ideal Mn of germs

n

vanishing at 0.

Proof. Consider the chain of ideals En ⊇ I + Mn ⊇ I + M2 ⊇ · · · . Since I has

n

¬nite codimension we have I +Mk = I +Mk+1 for some k. So Mk ⊆ I +Mn Mk

n n n n

which implies Mk ⊆ I by the lemma of Nakayama 35.3 since Mk is ¬nitely

n n

generated by the monomials of order k in n variables.

35.5. Theorem. Let A be a unital real commutative algebra. Then the fol-

lowing assertions are equivalent.

(1) A is a Weil algebra.

∞

(2) A is a ¬nite dimensional quotient of an algebra of germs En = C0 (Rn , R)

for some n.

(3) A is a ¬nite dimensional quotient of an algebra R[X1 , . . . , Xn ] of poly-

nomials.

(4) A is a ¬nite dimensional quotient of an algebra R[[X1 , . . . , Xn ]] of formal

power series.

(5) A is a quotient of an algebra J0 (Rn , R) of jets.

k

Proof. Let A = R · 1 • N , where N is the maximal ideal of nilpotent ele-

ments, which is generated by ¬nitely many elements, say X1 , . . . , Xn . Since

R[X1 , . . . , Xn ] is the free real unital commutative algebra generated by these

elements, A is a quotient of this polynomial algebra. There is some k such that

xk+1 = 0 for all x ∈ N , so A is even a quotient of the jet algebra J0 (Rn , R).

k

Since the jet algebra is itself a quotient of the algebra of germs and the algebra

of formal power series, the same is true for A. That all these ¬nite dimensional

quotients are Weil algebras is clear, since they all are formally real and have

only one nonzero idempotent.

If A is a quotient of the jet algebra J0 (Rn , R), we say that the order of A is

r

at most r.

35.6. The width of a Weil algebra. Consider the square N 2 of the nilpotent

ideal N of a Weil algebra A. The dimension of the real vector space N/N 2 is

called the width of A.

Let M ‚ R[x1 , . . . , xn ] denote the ideal of all polynomials without con-

stant term and let I ‚ R[x1 , . . . , xn ] be an ideal of ¬nite codimension which

is contained in M2 . Then the width of the factor algebra A = R[x1 , . . . , xn ]/I

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300 Chapter VIII. Product preserving functors

is n. Indeed the nilpotent ideal of A is M/I and (M/I)2 = M2 /I, hence

(M/I)/(M/I)2 ∼ M/M2 is of dimension n.

=

35.7. Proposition. If M is a smooth manifold and I is an ideal of ¬nite

codimension in the algebra C ∞ (M, R), then C ∞ (M, R)/I is a direct sum of

¬nitely many Weil algebras.

If A is a ¬nite dimensional commutative real algebra with unit, then we have

Hom(C ∞ (M, R), A) = Hom(C ∞ (M, R), W (A)), where W (A) is the subalgebra

of A generated by all idempotent and nilpotent elements of A (the so-called Weil

part of A). In particular W (A) is formally real.

Proof. The algebra C ∞ (M, R) is formally real, so the ¬rst assertion follows from

lemma 35.1. If • : C ∞ (M, R) ’ A is an algebra homomorphism, then the kernel

of • is an ideal of ¬nite codimension in C ∞ (M, R), so the image of • is a direct

sum of Weil algebras and is thus generated by its idempotent and nilpotent

elements.

35.8. Lemma. Let M be a smooth manifold and let • : C ∞ (M, R) ’ A be an

algebra homomorphism into a Weil algebra A.

Then there is a point x ∈ M and some k ≥ 0 such that ker • contains the

ideal of all functions which vanish at x up to order k.

Proof. Since •(1) = 1 the kernel of • is a nontrivial ideal in C ∞ (M, R) of ¬nite

codimension.

If Λ is a closed subset of M we let C ∞ (Λ, R) denote the algebra of all real

valued functions on Λ which are restrictions of smooth functions on M . For a

smooth function f let Zf := f ’1 (0) be its zero set. For a subset S ‚ C ∞ (Λ, R)

we put ZS := {Zf : f ∈ S}.

Claim 1. Let I be an ideal of ¬nite codimension in C ∞ (Λ, R). Then ZI is a

¬nite subset of Λ and ZI = … if and only if I = C ∞ (Λ, R).

ZI is ¬nite since C ∞ (Λ, R)/I is ¬nite dimensional. Zf = … implies that f is

invertible. So if I = C ∞ (Λ, R) then {Zf : f ∈ I} is a ¬lter of nonempty closed

sets, since Zf © Zg = Zf 2 +g2 . Let h ∈ C ∞ (M, R) be a positive proper function,

i.e. inverse images under h of compact sets are compact. The square of the

geodesic distance with respect to a complete Riemannian metric on a connected

manifold M is such a function. Then we put f = h|Λ ∈ C ∞ (Λ, R). The sequence

f, f 2 , f 3 , . . . is linearly dependent mod I, since I has ¬nite codimension, so

n

g = i=1 »i f i ∈ I for some (»i ) = 0 in Rn . Then clearly Zg is compact. So this

¬lter of closed nonempty sets contains a compact set and has therefore nonempty

intersection ZI = f ∈I Zf .

Claim 2. If I is an ideal of ¬nite codimension in C ∞ (M, R) and if a function

f ∈ C ∞ (M, R) vanishes near ZI , then f ∈ I.

Let ZI ‚ U1 ‚ U 1 ‚ U2 where U1 and U2 are open in M such that f |U2 = 0.

The restriction mapping C ∞ (M, R) ’ C ∞ (M \ U1 , R) is a surjective algebra

homomorphism, so the image I of I is again an ideal of ¬nite codimension in

C ∞ (M \ U1 , R). But clearly ZI = …, so by claim 1 we have I = C ∞ (M \ U1 , R).

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

35. Weil algebras and Weil functors 301

Thus there is some g ∈ I such that g|(M \ U1 ) = f |(M \ U1 ). Now choose

h ∈ C ∞ (M, R) such that h = 0 on U1 and h = 1 o¬ U2 . Then f = f h = gh ∈ I.

Claim 3. For the ideal ker • in C ∞ (M, R) the zero set ZI consists of one point

x only.

Since ker • is a nontrivial ideal of ¬nite codimension, Zker • is not empty and

¬nite by claim 1. For any function f ∈ C ∞ (M, R) which is 1 or 0 near the points

in Zker • the element •(f ) is an idempotent of the Weil algebra A. Since 1 is

the only nonzero idempotent of A, the zero set ZI consists of one point.

Now by claims 2 and 3 the ideal ker • contains the ideal of all functions which

∞

vanish near x. So • factors to the algebra Cx (M, R) of germs at x, compare

∞

35.5.(2). Now ker • ‚ Cx (M, R) is an ideal of ¬nite codimension, so by lemma

35.4 the result follows.

35.9. Corollary. The evaluation mapping ev : M ’ Hom(C ∞ (M, R), R),

given by ev(x)(f ) := f (x), is bijective.

This result is sometimes called the exercise of Milnor, see [Milnor-Stashe¬,

74, p. 11]. Another (similar) proof of it can be found in the mathematical short

story in the introduction to chapter VIII.

Proof. By lemma 35.8, for every • ∈ Hom(C ∞ (M, R), R) there is an x ∈ M

and a k ≥ 0 such that ker • contains the ideal of all functions vanishing at

x up to order k. Since the codimension of ker • is 1, we have ker • = {f ∈

C ∞ (M, R) : f (x) = 0}. Then for any f ∈ C ∞ (M, R) we have f ’ f (x)1 ∈ ker •,

so •(f ) = f (x).

35.10. Corollary. For two manifolds M1 and M2 the mapping

C ∞ (M1 , M2 ) ’ Hom(C ∞ (M2 , R), C ∞ (M1 , R))

f ’ (f — : g ’ g —¦ f )

is bijective.

Proof. Let x1 ∈ M1 and • ∈ Hom(C ∞ (M2 , R), C ∞ (M1 , R)). Then evx1 —¦ •

is in Hom(C ∞ (M2 , R), R), so by 35.9 there is a unique x2 ∈ M2 such that

evx1 —¦ • = evx2 . If we write x2 = f (x1 ), then f : M1 ’ M2 and •(g) = g —¦ f for

all g ∈ C ∞ (M2 , R). This also implies that f is smooth.

35.11. Chart description of Weil functors. Let A = R · 1 • N be a Weil

algebra. We want to associate to it a functor TA : Mf ’ Mf from the category

Mf of all ¬nite dimensional second countable manifolds into itself. We will give

several descriptions of this functor, and we begin with the most elementary and

basic construction, the idea of which goes back to [Weil, 53].

Step 1. If p(t) is a real polynomial, then for any a ∈ A the element p(a) ∈ A is

uniquely de¬ned; so we have a (polynomial) mapping TA (p) : A ’ A.

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302 Chapter VIII. Product preserving functors

Step 2. If f ∈ C ∞ (R, R) and »1 + n ∈ R · 1 • N = A, we consider the Taylor

∞ f (j) (») j

expansion j ∞ f (»)(t) = t of f at » and we put

j=0 j!

∞

f (j) (») j

TA (f )(»1 + n) := f (»)1 + n,

j!

j=1

which is ¬nite sum, since n is nilpotent. Then TA (f ) : A ’ A is smooth and we

get TA (f —¦ g) = TA (f ) —¦ TA (g) and TA (IdR ) = IdA .

Step 3. For f ∈ C ∞ (Rm , R) we want to de¬ne the value of TA (f ) at the vec-

tor (»1 1 + n1 , . . . , »m 1 + nm ) ∈ Am = A — . . . — A. Let again j ∞ f (»)(t) =

1± ± m

for t ∈ Rm . Then

±∈Nm ±! d f (»)t be the Taylor expansion of f at » ∈ R

we put

1±

d f (»)n±1 . . . n±m ,

TA (f )(»1 1 + n1 , . . . , »m 1 + nm ) := f (»)1 + m

1

±!

|±|≥1

which is again a ¬nite sum.

Step 4. For f ∈ C ∞ (Rm , Rk ) we apply the construction of step 3 to each com-

ponent fj : Rm ’ R of f to de¬ne TA (f ) : Am ’ Ak .

Since the Taylor expansion of a composition is the composition of the Taylor

expansions we have TA (f —¦ g) = TA (f ) —¦ TA (g) and TA (IdRm ) = IdAm .

If • : A ’ B is a homomorphism between two Weil algebras we have •k —¦

TA f = TB f —¦ •m for f ∈ C ∞ (Rm , Rk ).

Step 5. Let π = πA : A ’ A/N = R be the projection onto the quotient ¬eld

of the Weil algebra A. This is a surjective algebra homomorphism, so by step 4

the following diagram commutes for f ∈ C ∞ (Rm , Rk ):

wA

TA f

Am k

m

u u

k

πA π A

wR

f

m k

R

If U ‚ Rm is an open subset we put TA (U ) := (πA )’1 (U ) = U — N m , which is

m

an open subset in TA (Rm ) := Am . If f : U ’ V is a smooth mapping between

open subsets U and V of Rm and Rk , respectively, then the construction of steps

3 and 4, applied to the Taylor expansion of f at points in U , produces a smooth

mapping TA f : TA U ’ TA V , which ¬ts into the following commutative diagram:

wT

‘ TA f

U — Nm V — Nk

TA U AV

‘“ &

‘

u )&

pr1 π m pr1

u &

k

πA

A

wV

f

U

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

35. Weil algebras and Weil functors 303

We have TA (f —¦ g) = TA f —¦ TA g and TA (IdU ) = IdTA U , so TA is now a covariant

functor on the category of open subsets of Rm ™s and smooth mappings between

them.

Step 6. In 1.14 we have proved that the separable connected smooth manifolds

are exactly the smooth retracts of open subsets in Rm ™s. If M is a smooth

manifold, let i : M ’ Rm be an embedding, let i(M ) ‚ U ‚ Rm be a tubular

neighborhood and let q : U ’ U be the projection of U with image i(M ). Then

q is smooth and q —¦ q = q. We de¬ne now TA (M ) to be the image of the smooth

retraction TA q : TA U ’ TA U , which by 1.13 is a smooth submanifold.

If f : M ’ M is a smooth mapping between manifolds, we de¬ne TA f :

TA M ’ TA M as

TA (i —¦f —¦q) TA q

TA M ‚ TA U ’ ’ ’ ’ TA U ’ ’ TA U ,

’ ’ ’’ ’’

which takes values in TA M .

It remains to show, that another choice of the data (i, U, q, Rm ) for the man-

ifold M leads to a di¬eomorphic submanifold TA M , and that TA f is uniquely

de¬ned up to conjugation with these di¬eomorphisms for M and M . Since this

is a purely formal manipulation with arrows we leave it to the reader and give

instead the following:

Step 6™. Direct construction of TA M for a manifold M using atlases.

Let M be a smooth manifold of dimension m, let (U± , u± ) be a smooth atlas

of M with chart changings u±β := u± —¦ u’1 : uβ (U±β ) ’ u± (U±β ). Then the

β

smooth mappings

wT

TA (u±β )

TA (uβ (U±β )) A (u± (U±β ))

m m

u u

πA πA

w u (U

u±β

uβ (U±β ) ±β )

±

form again a cocycle of chart changings and we may use them to glue the open

sets TA (u± (U± )) = u± (U± ) — N m ‚ TA (Rm ) = Am in order to obtain a smooth

manifold which we denote by TA M . By the diagram above we see that TA M

will be the total space of a ¬ber bundle T (πA , M ) = πA,M : TA M ’ M , since

the atlas (TA (U± ), TA (u± )) constructed just now is already a ¬ber bundle atlas.

Thus TA M is Hausdor¬, since two points xi can be separated in one chart if

they are in the same ¬ber, or they can be separated by inverse images under

πA,M of open sets in M separating their projections.

This construction does not depend on the choice of the atlas. For two atlases

have a common re¬nement and one may pass to this.

If f ∈ C ∞ (M, M ) for two manifolds M , M , we apply the functor TA to

the local representatives of f with respect to suitable atlases. This gives local

representatives which ¬t together to form a smooth mapping TA f : TA M ’

TA M . Clearly we again have TA (f —¦ g) = TA f —¦ TA g and TA (IdM ) = IdTA M , so

that TA : Mf ’ Mf is a covariant functor.

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304 Chapter VIII. Product preserving functors

35.12. Remark. If we apply the construction of 35.11, step 6™ to the algebra

A = 0, which we did not allow (1 = 0 ∈ A), then T0 M depends on the choice

of the atlas. If each chart is connected, then T0 M = π0 (M ), computing the

connected components of M . If each chart meets each connected component of

M , then T0 M is one point.

35.13. Theorem. Main properties of Weil functors. Let A = R · 1 • N

be a Weil algebra, where N is the maximal ideal of nilpotents. Then we have:

1. The construction of 35.11 de¬nes a covariant functor TA : Mf ’ Mf

such that (TA M, πA,M , M, N dim M ) is a smooth ¬ber bundle with standard ¬ber

N dim M . For any f ∈ C ∞ (M, M ) we have a commutative diagram

wT

TA f

TA M AM

πA,M πA,M

u u

wM.

f

M

So (TA , πA ) is a bundle functor on Mf , which gives a vector bundle on Mf if

and only if N is nilpotent of order 2.

2. The functor TA : Mf ’ Mf is multiplicative: it respects products.

It maps the following classes of mappings into itself: immersions, initial im-

mersions, embeddings, closed embeddings, submersions, surjective submersions,

¬ber bundle projections. It also respects transversal pullbacks, see 2.19. For

¬xed manifolds M and M the mapping TA : C ∞ (M, M ) ’ C ∞ (TA M, TA M ) is

smooth, i.e. it maps smoothly parametrized families into smoothly parametrized

families.

3. If (U± ) is an open cover of M then TA (U± ) is also an open cover of TA M .

4. Any algebra homomorphism • : A ’ B between Weil algebras induces

a natural transformation T (•, ) = T• : TA ’ TB . If • is injective, then

T (•, M ) : TA M ’ TB M is a closed embedding for each manifold M . If • is

surjective, then T (•, M ) is a ¬ber bundle projection for each M . So we may

view T as a co-covariant bifunctor from the category of Weil algebras times Mf

to Mf .

Proof. 1. The main assertion is clear from 35.11. The ¬ber bundle πA,M :

TA M ’ M is a vector bundle if and only if the transition functions TA (u±β ) are

¬ber linear N dim M ’ N dim M . So only the ¬rst derivatives of u±β should act on

N , so any product of two elements in N must be 0, thus N has to be nilpotent

of order 2.

2. The functor TA respects products in the category of open subsets of Rm ™s

by 35.11, step 4 and 5. All the other assertions follow by looking again at the

chart structure of TA M and by taking into account that f is part of TA f (as the

base mapping).

3. This is obvious from the chart structure.

4. We de¬ne T (•, Rm ) := •m : Am ’ B m . By 35.11, step 4, this restricts to

a natural transformation TA ’ TB on the category of open subsets of Rm ™s and

by gluing also on the category Mf . Obviously T is a co-covariant bifunctor on

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

35. Weil algebras and Weil functors 305

the indicated categories. Since πB —¦ • = πA (• respects the identity), we have

T (πB , M ) —¦ T (•, M ) = T (πA , M ), so T (•, M ) : TA M ’ TB M is ¬ber respecting

for each manifold M . In each ¬ber chart it is a linear mapping on the typical

¬ber NA M ’ NB M .

dim dim

So if • is injective, T (•, M ) is ¬berwise injective and linear in each canonical

¬ber chart, so it is a closed embedding.

If • is surjective, let N1 := ker • ⊆ NA , and let V ‚ NA be a linear com-

plement to N1 . Then for m = dim M and for the canonical charts we have the

commutative diagram:

w T uM

u

T (•, M )

TA M B

wT

T (•, U± )

TA (U± ) B (U± )

u u

TA (u± ) TB (u± )

w u (U ) — N

Id —(•|NA )m

m m

u± (U± ) — NA ± ± B

w u (U ) — 0 — N

Id —0 — Iso

u± (U± ) — N1 — V m

m m

± ± B

So T (•, M ) is a ¬ber bundle projection with standard ¬ber (ker •)m .

35.14. Theorem. Algebraic description of Weil functors. There are

bijective mappings ·M,A : Hom(C ∞ (M, R), A) ’ TA (M ) for all smooth man-

ifolds M and all Weil algebras A, which are natural in M and A. Via · the

set Hom(C ∞ (M, R), A) becomes a smooth manifold and Hom(C ∞ ( , R), A) is

a global expression for the functor TA .

Proof. Step 1. Let (xi ) be coordinate functions on Rn . By lemma 35.8 for

• ∈ Hom(C ∞ (Rn , R), A) there is a point x(•) = (x1 (•), . . . , xn (•)) ∈ Rn such

that ker • contains the ideal of all f ∈ C ∞ (Rn , R) vanishing at x(•) up to some

order k, so that •(xi ) = xi (•) · 1 + •(xi ’ xi (•)), the latter summand being

nilpotent in A of order ¤ k. Applying • to the Taylor expansion of f at x(•)

up to order k with remainder gives

‚ |±| f

(x(•)) •(x1 ’ x1 (•))±1 . . . •(xn ’ xn (•))±n

1

•(f ) = ±! ±

‚x

|±|¤k

= TA (f )(•(x1 ), . . . , •(xn )).

So • is uniquely determined by the elements •(xi ) in A and the mapping

·Rn ,A : Hom(C ∞ (Rn , R), A) ’ An ,

·(•) := (•(x1 ), . . . , •(xn ))

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306 Chapter VIII. Product preserving functors

is injective. Furthermore for g = (g 1 , . . . , g m ) ∈ C ∞ (Rn , Rm ) and coordinate

functions (y 1 , . . . , y m ) on Rm we have

(·Rm ,A —¦ (g — )— )(•) = (•(y 1 —¦ g), . . . , •(y m —¦ g))

= (•(g 1 ), . . . , •(g m ))

= TA (g 1 )(•(x1 ), . . . , •(xn )), . . . , TA (g m )(•(x1 ), . . . , •(xn )) ,

so ·Rn ,A is natural in Rn . It is also bijective since any (a1 , . . . , an ) ∈ An

de¬nes a homomorphism • : C ∞ (Rn , R) ’ A by the prescription •(f ) :=

TA f (a1 , . . . , an ).

Step 2. Let i : U ’ Rn be the embedding of an open subset. Then the image of

the mapping

(i— )— ·Rn ,A

Hom(C ∞ (U, R), A) ’ ’ Hom(C ∞ (Rn , R), A) ’ ’ ’ An

’’ ’’

’1

is the set πA,Rn (U ) = TA (U ) ‚ An , and (i— )— is injective.

To see this let • ∈ Hom(C ∞ (U, R), A). By lemma 35.8 ker • contains the

ideal of all f vanishing up to some order k at a point x(•) ∈ U ⊆ Rn , and since

•(xi ) = xi (•) · 1 + •(xi ’ xi (•)) we have

πA,Rn (·Rn ,A (• —¦ i— )) = πA (•(x1 ), . . . , •(xn )) = x(•) ∈ U.

n

As in step 1 we see that the mapping

’1

(a1 , . . . , an ) ’ (C ∞ (U, R) f ’ TA (f )(a1 , . . . , an ))

πA,Rn (U )

is the inverse to ·Rn ,A —¦ (i— )— .

Step 3. The two functors Hom(C ∞ ( , R), A) and TA : Mf ’ Set coincide

on all open subsets of Rn ™s, so they have to coincide on all manifolds, since

smooth manifolds are exactly the retracts of open subsets of Rn ™s by 1.14.1.

Alternatively one may check that the gluing process described in 35.11, step

6, works also for the functor Hom(C ∞ ( , R), A) and gives a unique manifold

structure on it which is compatible to TA M .

35.15. Covariant description of Weil functors. Let A be a Weil algebra,

which by 35.5.(2) can be viewed as En /I, a ¬nite dimensional quotient of the

∞

algebra En = C0 (Rn , R) of germs at 0 of smooth functions on Rn .

De¬nition. Let M be a manifold. Two mappings f, g : Rn ’ M with f (0) =

∞

g(0) = x are said to be I-equivalent, if for all germs h ∈ Cx (M, R) we have

h —¦ f ’ h —¦ g ∈ I.

The equivalence class of a mapping f : Rn ’ M will be denoted by jA (f )

and will be called the A-velocity at 0 of f . Let us denote by JA (M ) the set of

all A-velocities on M .

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

35. Weil algebras and Weil functors 307

There is a natural way to extend JA to a functor Mf ’ Set. For every

smooth mapping f : M ’ N between manifolds we put JA (f )(jA (g)) := jA (f —¦g)

for g ∈ C ∞ (Rn , M ).

Now one can repeat the development of the theory of (n, r)-velocities for the

more general space JA (M ) instead of J0 (Rn , M ) and show that JA (M ) is a

k

smooth ¬ber bundle over M , associated to a higher order frame bundle. This

development is very similar to the computations done in 35.11 and we will in

fact reduce the whole situation to 35.11 and 35.14 by the following

35.16. Lemma. There is a canonical equivalence

JA (M ) ’ Hom(C ∞ (M, R), A),

jA (f ) ’ (C ∞ (M, R) g ’ jA (g —¦ f ) ∈ A),

which is natural in A and M and a di¬eomorphism, so the functor JA : Mf ’

FM is equivalent to TA .

Proof. We just have to note that JA (R) = En /I = A.

Let us state explicitly that a trivial consequence of this lemma is that the Weil

functor determined by the Weil algebra En /Mk+1 = J0 (Rn , R) is the functor

k

n

r

Tn of (n, r)-velocities from 12.8.

35.17. Theorem. Let A and B be Weil algebras. Then we have:

(1) We get the algebra A back from the Weil functor TA by TA (R) = A

with addition +A = TA (+R ), multiplication mA = TA (mR ) and scalar

multiplication mt = TA (mt ) : A ’ A.

(2) The natural transformations TA ’ TB correspond exactly to the algebra

homomorphisms A ’ B

Proof. (1) This is obvious. (2) For a natural transformation • : TA ’ TB its

value •R : TA (R) = A ’ TB (R) = B is an algebra homomorphisms. The inverse

of this mapping is already described in theorem 35.13.4.

35.18. The basic facts from the theory of Weil functors are completed by the

following assertion, which will be proved in more general context in 36.13.

Proposition. Given two Weil algebras A and B, the composed functor TA —¦ TB

is a Weil functor generated by the tensor product A — B.

Corollary. (See also 37.3.) There is a canonical natural equivalence TA —¦ TB ∼

=

∼ B — A.

TB —¦ TA generated by the exchange algebra isomorphism A — B =

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

308 Chapter VIII. Product preserving functors

36. Product preserving functors

36.1. A covariant functor F : Mf ’ Mf is said to be product preserving, if

the diagram

F (pr1 ) F (pr2 )

F (M1 ) ← ’ ’ F (M1 — M2 ) ’ ’ ’ F (M2 )

’’ ’’

is always a product diagram. Then F (point) = point, by the following argument:

u (pr ) F (point u— point) F (pr )T (point)

∼ wF

F

£

∼

1 2

RRR

F (point)

f

= =

RR R R

f f

1 2

point

Each of f1 , f , and f2 determines each other uniquely, thus there is only one

mapping f1 : point ’ F (point), so the space F (point) is single pointed.

The basic purpose of this section is to prove the following

Theorem. Let F be a product preserving functor together with a natural trans-

formation πF : F ’ Id such that (F, πF ) satis¬es the locality condition 18.3.(i).

Then F = TA for some Weil algebra A.

This will be a special case of much more general results below. The ¬nal proof

will be given in 36.12. We will ¬rst extract uniquely a sum of Weil algebras from

a product preserving functor, then we will reconstruct the functor from this

algebra under mild conditions.

36.2. We denote the addition and the multiplication on the reals by +, m :

R2 ’ R, and for » ∈ R we let m» : R ’ R be the scalar multiplication by » and

we also consider the mapping » : point ’ R onto the value ».

Theorem. Let F : Mf ’ Mf be a product preserving functor. Then either

F (R) is a point or F (R) is a ¬nite dimensional real commutative and formally real

algebra with operations F (+), F (m), scalar multiplication F (m» ), zero F (0),

and unit F (1), which is called Al(F ). If • : F1 ’ F2 is a natural transformation

between two such functors, then Al(•) := •R : Al(F1 ) ’ Al(F2 ) is an algebra

homomorphism.

Proof. Since F is product preserving, we have F (point) = point. All the laws

for a commutative ring with unit can be formulated by commutative diagrams

of mappings between products of the ring and the point. We do this for the ring

R and apply the product preserving functor F to all these diagrams, so we get

the laws for the commutative ring F (R) with unit F (1) with the exception of

F (0) = F (1) which we will check later for the case F (R) = point. Addition F (+)

and multiplication F (m) are morphisms in Mf , thus smooth and continuous.

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

36. Product preserving functors 309

For » ∈ R the mapping F (m» ) : F (R) ’ F (R) equals multiplication with the

element F (») ∈ F (R), since the following diagram commutes:

e ee ee e

eeeeg )

F (R)

e

F (m

u

»

∼

=

w F (R) — F (R) Aw F (R)

Id —F (»)

9

99 9

F (R) — point

u 9 F (m)

∼

=

w F (R — R)

F (Id —»)

F (R — point)

We may investigate now the di¬erence between F (R) = point and F (R) = point.

In the latter case for » = 0 we have F (») = F (0) since multiplication by F (»)

equals F (m» ) which is a di¬eomorphism for » = 0 and factors over a one pointed

space for » = 0. So for F (R) = point which we assume from now on, the group

homomorphism » ’ F (») from R into F (R) is actually injective.

In order to show that the scalar multiplication » ’ F (m» ) induces a contin-

uous mapping R — F (R) ’ F (R) it su¬ces to show that R ’ F (R), » ’ F (»),

is continuous.

(F (R), F (+), F (m’1 ), F (0)) is a commutative Lie group and is second count-

able as a manifold since F (R) ∈ Mf . We consider the exponential mapping

exp : L ’ F (R) from the Lie algebra L into this group. Then exp(L) is

an open subgroup of F (R), the connected component of the identity. Since

{F (») : » ∈ R} is a subgroup of F (R), if F (») ∈ exp(L) for all » = 0, then

/

F (R)/ exp(L) is a discrete uncountable subgroup, so F (R) has uncountably many

connected components, in contradiction to F (R) ∈ Mf . So there is »0 = 0 in

R and v0 = 0 in L such that F (»0 ) = exp(v0 ). For each v ∈ L and r ∈ N,

hence r ∈ Q, we have F (mr ) exp(v) = exp(rv). Now we claim that for any

sequence »n ’ » in R we have F (»n ) ’ F (») in F (R). If not then there is a

sequence »n ’ » in R such that F (»n ) ∈ F (R) \ U for some neighborhood U of

F (») in F (R), and by considering a suitable subsequence we may also assume

2

that 2n (»n+1 ’ ») is bounded. By lemma 36.3 below there is a C ∞ -function

»0

f : R ’ R with f ( 2n ) = »n and f (0) = ». Then we have

F (»n ) = F (f )F (m2’n )F (»0 ) = F (f )F (m2’n ) exp(v0 ) =

= F (f ) exp(2’n v0 ) ’ F (f ) exp(0) = F (f (0)) = F (»),

contrary to the assumption that F (»n ) ∈ U for all n. So » ’ F (») is a contin-

/

uous mapping R ’ F (R), and F (R) with its manifold topology is a real ¬nite

dimensional commutative algebra, which we will denote by Al(F ) from now on.

The evaluation mapping evIdR : Hom(C ∞ (R, R), Al(F )) ’ Al(F ) is bijective

since it has the right inverse x ’ (C ∞ (R, R) f ’ F (f )x ). But by 35.7 the

evaluation map has values in the Weil part W (Al(F )) of Al(F ), so the algebra

Al(F ) is generated by its idempotent and nilpotent elements and has to be

formally real, a direct sum of Weil algebras by 35.1.

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

310 Chapter VIII. Product preserving functors

Remark. In the case of product preserving bundle functors the smoothness of

» ’ F (») is a special case of the regularity proved in 20.7. In fact one may also

conclude that F (R) is a smooth algebra by the results from [Montgomery-Zippin,

55], cited in 5.10.

36.3. Lemma. [Kriegl, 82] Let »n ’ » in R, let tn ∈ R, tn > 0, tn ’ 0 strictly

monotone, such that

»n ’ »n+1

,n ∈ N

(tn ’ tn+1 )k

is bounded for all k. Then there is a C ∞ -function f : R ’ R with f (tn ) = »n

and f (0) = » such that f is ¬‚at at each tn .

Proof. Let • ∈ C ∞ (R, R), • = 0 near 0, • = 1 near 1, and 0 ¤ • ¤ 1 elsewhere.

Then we put

for t ¤ 0,

±

»

t ’ tn+1

(»n ’ »n+1 ) + »n+1 for tn+1 ¤ t ¤ tn ,

f (t) = •

tn ’ tn+1

for t1 ¤ t,

»1

and one may check by estimating the left and right derivatives at all tn that f

is smooth.

36.4. Product preserving functors without Weil algebras. Let F :

Mf ’ Mf be a functor with preserves products and assume that it has

the property that F (R) = point. Then clearly F (Rn ) = F (R)n = point and

F (M ) = point for each smoothly contractible manifold M . Moreover we have:

Lemma. Let f0 , f1 : M ’ N be homotopic smooth mappings, let F be as

above. Then F (f0 ) = F (f1 ) : F (M ) ’ F (N ).

Proof. A continuous homotopy h : M —[0, 1] ’ N between f0 and f1 may ¬rst be

reparameterized in such a way that h(x, t) = f0 (x) for t < µ and h(x, t) = f1 (x)

for 1 ’ µ < t, for some µ > 0. Then we may approximate h by a smooth

mapping without changing the endpoints f0 and f1 . So ¬nally we may assume

that there is a smooth h : M — R ’ N such that h —¦ insi = fi for i = 0, 1 where

inst : M ’ M — R is given by inst (x) = (x, t). Since

u w F (R)

F (pr1 ) F (pr2 )

F (M — R)

F (M )

F (M ) — point point

is a product diagram we see that F (pr1 ) = IdF (M ) . Since pr1 —¦ inst = IdM we

get also F (inst ) = IdF (M ) and thus F (f0 ) = F (h) —¦ F (ins0 ) = F (h) —¦ F (ins1 ) =

F (f1 ).

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

36. Product preserving functors 311

Examples. For a manifold M let M = M± be the disjoint union of its con-

˜

nected components and put H1 (M ) := ± H1 (M± ; R), using singular homology

˜ ˜

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

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

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

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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 ) ∼

=

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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

runs again over all connected components of M .

Proposition. Any product preserving functor F : Mf ’ Mf which maps

embeddings to injective mappings is of the form F = Gc — . . . — Gc for product

n

1

preserving functors Gi which also map embeddings to injective mappings.

Proof. Let again Al(F ) = A = A1 • · · · • Ak be the decomposition into Weil

algebras. We conclude from 36.8.2 that χF,M : F (M ) ’ TA (M ) is injective for

each manifold M . We have to show that the set {1, . . . , k} can be divided into

equivalence classes I1 , . . . , In such that F (M ) ⊆ TA (M ) is the inverse image

under π : TA (M ) ’ Pk (M ) of the union of all N1 — . . . — Nk where the Ni run

through all connected components of M in such a way that i, j ∈ Ir for some r

implies that Ni = Nj . Then each Ir gives rise to Gc = T c .

r i∈Ir Ai

To ¬nd the equivalence classes we consider X = {1, . . . , k} as a discrete man-

ifold and consider F (X) ⊆ TA (X) = X k . Choose an element i = (i1 , . . . , ik ) ∈

F (X) with maximal number of distinct members. The classes Ir will then be

the non-empty sets of the form {s : is = j} for 1 ¤ j ¤ k. Let n be the number

of di¬erent classes.

Now let D be a discrete manifold. Then the claim says that

F (D) = {(d1 , . . . , dk ) ∈ Dk : s, t ∈ Ir implies ds = dt for all r}.

Suppose not, then there exist d = (d1 , . . . , dk ) ∈ F (D) and r, s, t with s, t ∈ Ir

and ds = dt . So among the pairs (i1 , d1 ), . . . , (ik , dk ) there are at least n + 1

distinct ones. Let f : X — D ’ X be any function mapping those pairs to

1, . . . , n + 1. Then F (f )(i, d) = (f (i1 , d1 ), . . . , f (ik , dk )) ∈ F (X) has at least

n + 1 distinct members, contradicting the maximality of n. This proves the

claim for D and also F (Rm — D) = Am — F (D) is of the right form since the

connected components of Rm — D correspond to the points of D.

Now let M be any manifold, let p : M ’ π0 (M ) be the projection of M

onto the (discrete) set of its connected components. For a ∈ F (M ) the value

F (p)(a) ∈ F (π0 (M )) just classi¬es the connected component of Pk (M ) over

which a lies, and this component of Pk (M ) must be of the right form. Let

x1 , . . . , xk ∈ M such s, t ∈ Ir implies that xs and xt are in the same connected

component Mr , say, for all r. The proof will be ¬nished if we can show that the

¬ber π ’1 (x1 , . . . , xk ) ‚ TA (M ) is contained in F (M ) ‚ TA (M ). Let m = dim M

(or the maximum of dim Mi for 1 ¤ i ¤ n if M is not a pure manifold) and let

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

318 Chapter VIII. Product preserving functors

N = Rm — {1, . . . , n}. We choose y1 , . . . , yk ∈ N and a smooth mapping g :

N ’ M with g(yi ) = xi which is a di¬eomorphism onto an open neighborhood

of the xi (a submersion for non pure M ). Then clearly TA (g)(π ’1 (y1 , . . . , yk )) =

π ’1 (x1 , . . . , xk ), and from the last step of the proof we know that F (N ) contains

π ’1 (y1 , . . . , yk ). So the result follows.

By theorem 36.10 we know the minimal data to reconstruct the action of F

on connected manifolds. For a not connected manifold M we ¬rst consider the

surjective mapping M ’ π0 (M ) onto the space of connected components of M .

Since π0 (M ) ∈ Mf , the functor F acts on this discrete set. Since F is weakly

local and maps points to points, F (π0 (M )) is again discrete. This gives us a

product preserving functor F0 on the category of countable discrete sets.

If conversely we are given a product preserving functor F0 on the category of

countable discrete sets, a formally real ¬nite dimensional algebra A consisting

of k Weil parts, and a product preserving functor G : conMf ’ groups with

k

π1 ⊆ G ⊆ π1 , then clearly one can construct a unique product preserving weakly

local functor F : Mf ’ Mf ¬tting these data.

37. Examples and applications

37.1. The tangent bundle functor. The tangent mappings of the algebra

structural mappings of R are given by

T R = R2 ,

T (+)(a, a )(b, b ) = (a + b, a + b ),

T (m)(a, a )(b, b ) = (ab, ab + a b),

T (m» )(a, a ) = (»a, »a ).

So the Weil algebra T R = Al(T ) =: D is the algebra generated by 1 and δ with

δ 2 = 0. It is sometimes called the algebra of dual numbers or also of Study

numbers. It is also the truncated polynomial algebra of order 1 on R. We will

write (a + a δ)(b + b δ) = ab + (ab + a b)δ for the multiplication in T R.

By 35.17 we can now determine all natural transformations over the category

Mf between the following functors.

(1) The natural transformations T ’ T consist of all ¬ber scalar multipli-

cations m» for » ∈ R, which act on T R by m» (1) = 1 and m» (δ) = ».δ.

(2) The projection π : T ’ IdMf is the only natural transformation.

37.2. Lemma. Let F : Mf ’ Mf be a multiplicative functor, which is also

a natural vector bundle over IdMf in the sense of 6.14, then F (M ) = V — T M

for a ¬nite dimensional vector space V with ¬berwise tensor product. Moreover

for the space of natural transformations between two such functors we have

N at(V — T, W — T ) = L(V, W ).

Proof. A natural vector bundle is local, so by theorem 36.1 it coincides with

TA , where A is its Weil algebra. But by theorem 35.13.(1) TA is a natural

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37. Examples and applications 319