A generalization of Thom's transversality theorem

Abstract

We prove a generalization of Thom's transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\ra P$ to the preimage $(j^sf)^{-1}(A)$ of a submanifold $A\subseteq J^s(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction $g|_{(j^sf)^{-1}(A)}$ is also generic. We also present an example of $A$ where the theorem fails.

Full text

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arXiv:1001.2054v1 [math.DG] 13 Jan 2010
A GENERALIZATION OF THOM’S TRANSVERSALITY
THEOREM
LUK´AˇS VOKˇR´INEK
Abstract. We prove a generalization of Thom’s transversality theorem. It
gives conditions under which the jet map f∗|Y : Y ⊆ Jr(D,M) → Jr(D,N)
is generically (for f : M → N) transverse to a submanifold Z ⊆ Jr(D,N).
We apply this to study transversality properties of a restriction of a fixed map
g : M → P to the preimage (jsf)−1(A) of a submanifold A ⊆ Js(M,N)
in terms of transversality properties of the original map f. Our main result
is that for a reasonable class of submanifolds A and a generic map f the
restriction g|(jsf)−1(A)is also generic. We also present an example of A where
the theorem fails.
0. Introduction
We start by reminding that for smooth manifolds M and N the set C∞(M,N)
of smooth maps is endowed with two topologies called weak (compact-open) and
strong (Whitney) topology. They agree when M is compact. We say that a subset
of a topological space is residual if it contains a countable intersection of open dense
subsets. The Baire property for C∞(M,N) then guarantees that it is automatically
dense. This holds for both topologies but is almost exclusively used for the strong
one. Clearly every residual subset of C∞(M,N) for the strong topology is also
residual for the weak topology. The following is our main theorem in which we
denote by Jr
imm(D,M) the subspace of all jets of immersions.
Theorem A. Let D, M, N be manifolds, Y ⊆ Jr
submanifolds. Let us further assume that σY ⋔ σZ, where
σY = σ|Y : Y ⊆ Jr(D,M) → D
imm(D,M) and Z ⊆ Jr(D,N)
andσZ= σ|Z: Z ⊆ Jr(D,N) → D
are the restrictions of the source maps. For a smooth map f : M → N let f∗|Y
denote the map
Y??
??Jr
imm(D,M)
f∗
??Jr(D,N) .
Then the subset
X :=?f ∈ C∞(M,N)
??f∗|Y ⋔ Z?
is residual in C∞(M,N) with the strong topology, and open if Z is closed (as a
subset) and the target map τY : Y → M is proper.
We are interested in the theorem mainly because of the following two applica-
tions, the first of which is the classical theorem of Thom.
2000 Mathematics Subject Classification. 57R35, 57R45.
Key words and phrases. Transversality, residual, generic, restriction, fibrewise singularity.
Research was supported by the grant MSM 0021622409 of the Czech Ministry of Education
and by the grant 201/05/2117 of the Grant Agency of the Czech Republic.
1

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2LUK´AˇS VOKˇR´INEK
Theorem B (Thom’s Transversality Theorem). Let M, N be manifolds, Z ⊆
Jr(M,N) a submanifold. Then the subset
X := {f ∈ C∞(M,N) | jrf ⋔ Z}
is residual in C∞(M,N). It is moreover open provided that Z is closed (as a subset).
To explain the second application we need to introduce a bit of notation. Let us
consider the following diagram
f−1(A)
??
??
j
??
A??
??
g−1(B)??
??
??
M
f
??
g
??
N
B??
??P
of smooth manifolds and smooth maps between them where??
beddings. We assume that f ⋔ A and g ⋔ B for the two pullbacks to be defined
(which we emphasize by saying that they are transverse pullbacks) and also for
some technical reasons. Suppose that we fix g and allow ourselves to change f (but
only in such a way that f ⋔ A). It is not hard to describe when the composition
gj = g|f−1(A)is transverse to B. A more general condition on gj is whether it
satisfies some form of jet transversality. First we have to solve the problem of not
having the source of Jr(f−1(A),P) fixed.
??indicates em-
Construction. Let D be a d-dimensional manifold and
Diff(Rd,0) = invG0(Rd,Rd)0
the group1of germs at 0 of local diffeomorphisms Rd→ Rdfixing 0. Define a
principal Diff(Rd,0)-bundle
ChartsD= invG0(Rd,D)
ev0
− − − → D
of germs at 0 of local diffeomorphisms Rd→ D. If F is any manifold with a (smooth
in some sense) action of Diff(Rd,0) then we can construct an associated bundle
D[F] := ChartsD×Diff(Rd,0)F −→ D.
Any bundle of this form is “local”. Observe that this construction is functorial in
D on the category of d-dimensional manifolds and local diffeomorphisms. As an
example the bundle Jr(D,P) of r-jets of maps D → P is a local bundle as
D[Jr
0(Rd,P)] = ChartsD×Diff(Rd,0)Jr
0(Rd,P)∼= Jr(D,P)
where Jr
is provided by the map
0(Rd,P) is the subspace of Jr(Rd,P) of r-jets with source 0. The bijection
[u,α] ?→ α ◦ jr
u(0)(u−1).
Having a Diff(Rd,0)-invariant submanifold B ⊆ Jr
subbundle D[B] ⊆ Jr(D,P) for any d-dimensional manifold D. This allows us to
0(Rd,P) we get an associated
1If we wanted to give Diff(Rd,0) a topology we could do so by inducing the topology via the
map Diff(Rd,0) → J∞
invertible ∞-jets.
0(Rd,Rd)0. Or one could replace Diff(Rd,0) by its image - the subspace of

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A GENERALIZATION OF THOM’S TRANSVERSALITY THEOREM3
talk about jet transversality conditions on a map D → P without specifying what
D (and hence also Jr(D,P)) is.
Let A ⊆ Js(M,N) be a submanifold and jsf ⋔ A. Then f∗A := (jsf)−1(A) is
a submanifold of M and for any Diff(Rd,0)-invariant submanifold B ⊆ Jr
and the associated submanifold
f∗A[B] ⊆ Jr(f∗A,P)
0(Rd,P)
we might ask whether jr(g|f∗A) is transverse to f∗A[B]. To state the theorem we
make the following notation: for a map g : M → P we write g ⋔ B in place of
g∗⋔ B where again
g∗: Jr
This condition is satisfied e.g. by all submersions. Also consider the following map
κ : Jr+s(M,N) ×MJr
(jr+s
x
ϕ,jr
0,imm(Rd,M) −→ Jr
0(Rd,P).
0,imm(Rd,M) −→ Jr
0ψ) ?−→ jr
0(Rd,Js(M,N)),
0(js(ϕ) ◦ ψ).
(1)
Theorem C. Let
M
f
??
g
??
N
P
be a pair of smooth maps. Let A ⊆ Js(M,N) and B ⊆ Jr
submanifolds with d = dimM −codimA. Assuming that B is Diff(Rd,0)-invariant,
κ ⋔ Jr
??jsf ⋔ A, jr(g|f∗A) ⋔ f∗A[B]?
is residual in C∞(M,N) with the strong topology. If either r = 0 or s = 0 the
transversality condition κ ⋔ Jr
0(Rd,P) be smooth
0(Rd,A) and g ⋔ B the subset
X :=?f ∈ C∞(M,N)
0(Rd,A) is automatically satisfied.
Remark. The transversality condition κ ⋔ Jr
lustrate by an example in the next section. An interesting question arises whether
there is a reasonable sufficient condition on A for which the transversality is auto-
matic (an example of such is e.g. s = 0).
0(Rd,A) cannot be removed as we il-
Remark. It is also possible to state conditions under which X is open.
1. An example
We describe here a family of examples. For k = 1 (and partially for k = 2) Theo-
rem C can be applied whereas for k ≥ 3 the theorem fails due to κ ?⋔ Jr
will not be interested in a particular choice of B since the transversality condition
in question depends only on A. Let us start with the following diagram
0(Rd,A). We
Dk× M
f
??
??
R
Dk
where we think of f as a family of functions M → R parametrized by a disc Dk.
To fit into our situation we should replace Dkby Rkbut the boundary is not the
issue in our example. Let s = 1 and let A ⊆ J1(Dk× M,R) denote the subspace

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4LUK´AˇS VOKˇR´INEK
of all those jets which have zero derivative in the direction of M. Clearly A has
codimension dimM and thus d = k.
For a function ϕ : Dk× M → R we have j1
(x,y)ϕ ∈ A iff the composition
TyM??
??TxDk× TyM
dϕ??R
is zero. We express this by saying that the map
d|TM: J1(Dk× M,R) → T∗M
describes A as the preimage of 0. We will now show how to describe Jr
f∗A in fact) in a similar way. Compose the defining equation with an immersion
ψ : Rk→ Dk× M as in (1) and differentiate to get
0(Rk,A) (or
TyM ⊗ Rk??
incl⊗ψ∗
??S2(TxDk× TyM)
d2ϕ??R .
Differentiating further and putting all the maps together we get a single map
Jr+1(Dk× M,R) ×MJr
0,imm(Rk,Dk× M)
χ
− − → T∗M ⊗ (R ⊕ Rk⊕ ··· ⊕ SrRk)∗
describing f∗A. Moreover χ is a submersion at f∗A iff κ ⋔ Jr
the T∗M ⊗(SiRk)∗-coordinate is a sum of various restrictions of the derivatives of
ϕ with only a single term involving the highest derivative di+1ϕ, namely
0(Rk,A). Fixing jr
0ψ
TyM ⊗ SiRk??
incl⊗Siψ∗
??Si+1(TxDk× TyM)
di+1ϕ??R .
This implies that if the image of the derivative of ψ∗ at 0 intersects TM in a
subspace of dimension at most 1, the highest order term is a surjective linear map
and quite easily the whole map χ is a submersion (even for a fixed ψ). The opposite
implication also holds as considerations at pairs with jr+1ϕ = 0 show.
For k = 1 we have shown that κ ⋔ Jr
0(R,A) and therefore for a generic map f
the fibrewise singularity set Σf = f∗A is a 1-dimensional submanifold for which
the projection Σf → D1is generic. This means that only regular points and
folds appear. The regular points of the projection are exactly the fibrewise Morse
singularities of f while folds correspond to the fibrewise A3-singularities (those of
the form x3
m).
For k ≥ 2 it is not the case that κ ⋔ Jr
where the image of the tangential map ψ∗at 0 intersects the tangent space of the
fibre M in a subspace of dimension at least 2.
Nevertheless for k = 2 the conclusion of Theorem C still holds. This is be-
cause the condition κ ⋔ Jr
ψ(Rd) ⊆ f∗A and for a generic map f such ψ cannot be tangent to the fibre M
(if that was the case then the rank of the fibrewise Hessian of f at x would drop
by 2 and this does not happen generically). Therefore again Σf is a 2-dimensinal
submanifold of D2× M partitioned into three parts: the cusps of the projection
Σf → D2(where f attains an isolated fibrewise A4-singularity); the folds form
a 1-dimensional submanifold (where the fibrewise A3-singularities occur); and the
fibrewise Morse singularities.
For k = 3 the conclusion of Theorem C fails. This is due to the fact that for a
generic map between 3-manifolds the rank drops at most by 1 whereas for a generic
3-parameter family of functions M → R the fibrewise Hessian can drop rank by 2.
1± x2
2± ··· ± x2
0(Rk,A), namely at pairs (jr+s
x
ϕ,jr
0ψ)
0(Rk,A) is only required at those (jr+s
x
ϕ,jr
0ψ) for which

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A GENERALIZATION OF THOM’S TRANSVERSALITY THEOREM5
2. Proofs
First we will show how to translate transversality conditions for a restriction of
a fixed map g : M → P to the preimage f−1(A) of a submanifold along a map
f : M → N in terms of transversality conditions on f itself. At the end we prove
that such properties are generic in the sense that maps satisfying them form a
residual subset of C∞(M,N). First version does not involve any jet conditions.
Lemma 1. Let
f−1(A)
??
??
j
??
A??
??
g−1(B)??
i
??
??
M
f
??
g
??
N
B??
??P
be a diagram where we assume that f ⋔ A and g ⋔ B. Then the following conditions
are equivalent:
(i) gj ⋔ B,
(ii) fi ⋔ A,
(iii) f−1(A) ⋔ g−1(B).
Proof. Since (iii) is symmetric it is enough to show the equivalence of (i) and (iii).
But (i) is equivalent to the map
g∗: Txf−1(A) −→ Tg(x)P/Tg(x)B
induced by the derivative of g being surjective for every x ∈ f−1(A) ∩ g−1(B).
Because of the assumption g ⋔ B, we have a commutative diagram
Txf−1(A)
??
??
Tg(x)P/Tg(x)B
???
?
?
?
?
?
?
?
?
?
?
?
TxM/Txg−1(B)
∼
=
and so (i) is equivalent to (iii).
?
Now we will generalize the lemma to jet transversality conditions. Let us recall
that for a Diff(Rd,0)-invariant submanifold B ⊆ Jr
associated subbundle D[B] ⊆ Jr(D,P) for any d-dimensional manifold D.
0(Rd,P) we have constructed an
Lemma 2. For h : D → P the following conditions are equivalent
(i) h∗: Jr
(ii) jr(h) : D → Jr(D,P) is transverse to D[B].
0,imm(Rd,D) → Jr
0(Rd,P) is transverse to B,
Proof. Taking associated bundles (i) is clearly equivalent to the transversality of
h∗: Jr
imm(D,D) → Jr(D,P)