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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

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