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˜ ˜
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

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

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