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mal weights ensuring the invariance are ’1 on the source and ’3 on the target.
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.


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


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 ))

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

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).
’’ ’

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

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
37. Examples and applications 319

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