Transversality and Lipschitz-Fredholm maps
We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theore...
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| author | Eftekharinasab, K. Eftekharinasab, K. |
| author_facet | Eftekharinasab, K. Eftekharinasab, K. |
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"author": "K. Eftekharinasab",
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| description | We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the generalized Sard’s theorem for this category of man-ifolds. We also provide an answer to the well known problem concerning the existence of a submanifold structure on the preimage of a transversal submanifold. |
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Збiрник праць
Iн-ту математики НАН України
2015, т.12, №6, 89-104
K. Eftekharinasab
Insitute of Mathematics of NAS of Ukraine
kaveheft@gmail.com
Transversality and Lipschitz-Fredholm
maps
We study transversality for Lipschitz-Fredholm maps in the context of
bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm
maps of a fixed index between Fréchet spaces has the transverse stability
property. We give a straightforward extension of the Smale transversality
theorem by using the generalized Sard’s theorem for this category of man-
ifolds. We also provide an answer to the well known problem concerning
the existence of a submanifold structure on the preimage of a transversal
submanifold.
Вивчається поняття трансверсальностi вiдображень Лiпшица-
Фредгольма у контекстi обмежених многовидiв Фреше. Доведено, що
множина всiх вiдображень Лiпшица-Фредгольма фiксованого iндексу
мiж просторами Фреше має властивiсть стiйкостi трансверсальних
перетинiв. Дано пряме узагальнення теореми Смейла про транс-
версальнiсть, для доведення якого використовується узагальнення
теореми Сарда на цю категорiю многовидiв. Також отримано вiд-
повiдь на вiдоме питання про iснування структури пiдмноговиду на
прообразi трансверсального пiдмноговиду.
1. Introduction
In [1] we proved a version of the classical Sard-Smale theorem
for a category of generalized Fréchet manifolds, bounded (or MCk)
Fréchet manifolds, introduced in [2]. Our approach to the theo-
rem’s generalization is based on the assumption that Fredholm
operators need to be globally Lipschitz. A reason for this interest
is that there exists an appropriate topology on L(E,F ), the space
of all linear globally Lipschitz maps between Fréchet spaces E and
© K. Eftekharinasab, 2015
90 K. Eftekharinasab
F , that leads to the openness of the set of linear isomorphisms in
L(E,F ), [1, Proposition 2.2]. This result in turn yields the open-
ness of the collection of Fredholm operators in L(E,F ), [1, Theo-
rem 3.2]. The other reason is that Lipschitzness is consistent with
the notion of differentiability, bounded (or MCk-) differentiabil-
ity, that we apply. If E,F are Fréchet spaces and if U is an open
subset of E, a map f : U → F is called bounded (or MC1-) dif-
ferentiable if it is Keller-differentiable, the directional derivative
d f(p) belongs to L(E,F ) for all p ∈ U , and the induced map
d f : U → L(E,F ) is continuous. Thus, we can naturally define
the index of a Fredholm map between manifolds.
We should point out that the mentioned results stems from the
essential fact that under a certain condition we can endow the
space L(E,F ) with a topological group structure. Also, the group
of automorphisms of a Fréchet space E, Aut(E), is open in L(E,E)
[3, Proposition 2.1]. But, in general, the group of automorphisms
of a Fréchet space does not admit a non-trivial topological group
structure. Thus, without some restrictions it would be impossible
to establish openness of sets of linear isomorphisms and Fredholm
operators. This is a major obstruction in developing the Fredholm
theory for Fréchet spaces.
A crucial step in the proof of an infinite dimensional version
of Sard’s theorem is that, roughly speaking, for a Fredholm map
f : M → N of manifolds, at each point p ∈ M , we may find local
charts (p ∈ U ⊆ M,φ) and (f(p) ∈ V ⊆ N,ψ) such that in the
charts f has a representation of the form f(u, v) = (u, η(u, v)),
where η : φ(U) → Rn is a smooth map. This is a consequence
of an inverse function theorem. One of the main significance of
the category of bounded Fréchet manifolds is the availability of
an inverse function theorem in the sense of Nash and Moser [2,
Theorem 4.7]. However, the bounded differentiability is strong
and in some cases the class of bounded maps can be quite small,
e.g. when the identity component of L(E,F ) contains only the
zero map [3, Remark 2.16].
Transversality and Lipschitz-Fredholm maps 91
We have argued that why we have utilized this particular cat-
egory of Fréchet manifolds. A salient example of these manifolds
is the space of all smooth sections of a fiber bundle over closed or
non-compact manifolds ( [2, Theorem 3.34]). On the other hand,
it turns out that these generalized manifolds can surpass the ge-
ometry of Fréchet manifolds. On these manifold we are able to
give a precise analytic meaning to some essential geometric objects
(such as connection maps, vector fields and integral curves), [4].
Therefore, we would expect their applications to problems in global
analysis.
The present work studies the differential topology of Lipschitz-
Fredholm maps in the bounded Fréchet setting. We show that
the set of Lipschitz-Fredholm operators of index l between Fréchet
spaces E and F is open in the space of linear globally Lipschitz
maps endowed with the fine topology (Proposition 3.5). We say
that a set of maps has the transverse stability property for the fine
topology if maps in a fine neighborhood of a given map have the
same transversality property i.e. if f : E → F is a map transversal
to a closed subspace F of F , then any map in a fine neighbor-
hood of f is transversal to F. We then prove that the set of all
Lipschitz-Fredholm maps of a fixed index between Fréchet spaces
has the transverse stability property (Theorem 3.6). We also study
transversality for Lipschitz-Fredholm maps between manifolds. We
give a straightforward generalization of the Smale transversality
theorem ( [5, Theorem 3.1]) by using our generalized Sard’s the-
orem (Theorem 3.9). Finally, we prove that if f : M → N is an
MCk Lipschitz-Fredholm map of manifolds which is transversal to
a finite dimensional submanifold A of N , then f−1(A) is a sub-
manifold (Theorem 4.2).
We stress that these results can not be proved without strong
restrictions. However, the basic concepts of infinite dimensional
differential topology such as submanifold and transversality can
be simply come over from the Banach setting.
92 K. Eftekharinasab
Our motivation for the present work, in the light of [4], lay in
the desire to develop transversality tools for the degree theory, in-
cluding the Leray-Schauder degree, for Lipschitz-Fredholm maps,
to derive applications to the study of solutions to systems of non-
linear partial differential or integral equations on spaces of smooth
sections which are not linear.
2. Preliminaries
We shall recall the required definitions from the category of
MCk manifolds briefly but in a self-contained way for the conve-
nience of the reader, which also give us the opportunity to establish
our notations for the rest of the paper. For more studies we refer
to [1, 2, 4].
Let (F, d) be a Fréchet space whose topology is defined by a
complete translational-invariant metric d. A metric with abso-
lutely convex balls will be called a standard metric. Every Fréchet
space admits a standard metric which defines its topology. We
shall always define the topology of Fréchet spaces with this type
of metrics.
Let (E, g) and (F, d) be Fréchet spaces and let Lg,d(E,F ) be
the set of all linear maps L : E → F such that
Lip(L)g,d := sup
x∈E\{0}
d(L(x), 0)
g(x, 0)
< ∞.
The transversal-invariant metric
Dg,d(L,H) = Lip(L−H)g,d (2.1)
on Lg,d(E,F ) turns it into an Abelian topological group ( [1, Re-
mark 2.1]). A map ϕ ∈ Lg,d(E,F ) is called Lipschitz-Fredholm
operator if its kernel has finite dimension and its image is closed
and has finite co-dimension. The index of ϕ, Indϕ, is defined by
Indϕ = dimkerϕ− codim Imgϕ. We denote by LF(E,F ) the set
of all Lipschitz-Fredholm operators, and by LFl(E,F ) the subset
of LF(E,F ) consisting of those operators of index l.
Transversality and Lipschitz-Fredholm maps 93
Proposition 2.1. [1, Proposition 2.2] The set of linear iso-
morphisms from E into F , Iso (E,F ), is open in Lg,d(E,F ) with
respect to the topology induced by the Metric (2.1).
Theorem 2.2. [1, Theorem 3.2] LF(E,F ) is open in Lg,d(E,F )
with respect to the topology defined by the metric (2.1). Further-
more, the function T → IndT is continuous on LF(E,F ), hence
constant on connected components of LF(E,F ).
A subset G of a Fréchet space F is called topologically comple-
mented (or it splits in F ), if F is homeomorphic to the topological
direct sum G ⊕ H, where H is a subspace of F . We call H a
topological complement of G in F .
Theorem 2.3. [2, Theorem 3.14] Let E be a Fréchet space. Then
(1) Every finite-dimensional subspace of E is closed.
(2) Every closed subspace G ⊂ E with
codim(G) = dim(E/G) < ∞
is topologically complemented in E.
(3) Every finite-dimensional subspace of E is topologically comple-
mented.
(4) A linear subspace G of E has a topological complement H if
and only if there exists a continuous projection Pr of E onto
H, see [6].
Let E,F be Fréchet spaces, U an open subset of E, and
P : U → F
a continuous map. Let CL(E,F ) be the space of all continuous
linear maps from E to F topologized by the compact-open topol-
ogy. We say P is differentiable at the point p ∈ U if the directional
derivative dP (p) exists in all directions h ∈ E. If P is differen-
tiable at all points p ∈ U , if dP (p) : U → CL(E,F ) is continuous
for all p ∈ U and if the induced map
P � : U × E → F, (u, h) �→ dP (u)h
94 K. Eftekharinasab
is continuous in the product topology, then we say that P is Keller-
differentiable. We define P (k+1) : U × Ek+1 → F in the obvious
inductive fashion.
If P is Keller-differentiable, dP (p) ∈ Lg,d(E,F ) for all p ∈ U ,
and the induced map dP (p) : U → Lg,d(E,F ) is continuous, then
P is called bounded differentiable. We say P is MC0 and write
P 0 = P if it is continuous. We say P is an MC1 and write P (1) =
P � if it is bounded differentiable. Let Lg,d(E,F )0 be the connected
component of Lg,d(E,F ) containing the zero map. If P is bounded
differentiable and if V ⊆ U is a connected open neighborhood
of x0 ∈ U , then P �(V ) is connected and hence contained in the
connected component P �(x0)+Lg,d(E,F )0 of P �(x0) in Lg,d(E,F ).
Thus, the map Qx0 : V → Lg,d(E,F )0 defined by
Qx0(y) = P �(y)− P �(x0)
is again a map between subsets of Fréchet spaces. This enables a
recursive definition: if P is MC1 and V can be chosen for each
x0 ∈ U such that Qx0 : V → Lg,d(E,F )0 is MCk−1, then P is
called an MCk-map. We make a piecewise definition of P (k) by
P (k) |V := (Qx0)
(k−1) for x0 and V as before. The map P is MC∞
(or smooth) if it is MCk for all k ∈ N0. We shall denote the
derivative of P at p by DP (p). Note that MCk-differentiability
implies the usual Ck-differentiability for maps of finite dimensional
manifolds.
Within this framework we can define MCk Fréchet manifolds,
MCk-maps of manifolds and tangent bundle over MCk manifolds
in obvious fashion way. We assume that manifolds are connected
and second countable.
Let f : M → N (k � 1) be an MCk-map of manifolds. We
denote by Txf : TxM → Tf(x)N the tangent map of f at x ∈ M
from the tangent space TxM to the tangent space Tf(x)N . We
say that f is an immersion (resp. submersion) provided Txf is
injective (resp. surjective) and the range Img(Txf) (resp. the
kernel ker(Txf)) splits in Tf(x)N (resp. TxM) for any x ∈ M .
An injective immersion f : M → N which gives an isomorphism
Transversality and Lipschitz-Fredholm maps 95
onto a submanifold of N is called an embedding. A point x ∈ M
is called a regular point if D f(x) : TxM −→ Tf(x)N is surjective.
The corresponding value f(x) is a regular value. Points and values
other than regular are called critical points and values, respectively.
Let M and N be MCk manifolds, k � 1. A Lipschitz-Fredholm
map is an MC1-map f : M → N such that for each x ∈ M the
derivative D f(x) : TxM −→ Tf(x)N is a Lipschitz-Fredholm oper-
ator. The index of f , denoted by Ind f , is defined to be the index
of D f(x) for some x. Since f is MCk and M is connected in the
light of Theorem 2.2 the definition does not depend on the choice
of x.
3. Transversality and openness
Let F1 be a linear closed subspace of a Fréchet space F that
splits in F . Given MCk manifold M modelled on F , a subset
M1 of M is a submanifold of M modelled on F1 provided there
is MCk-atlas {(Ui,φi)}i∈I on M that induces an atlas on M1, i.e.
for any i ∈ I there are open subsets Vi,Wi of F, F1 such that
φi(Ui) = Vi⊕Wi and φi(Ui∩M1) = Vi⊕{0} is open in F1. We say
that M1 is a submanifold of Banach type if F1 is a Banach space,
and a submanifold of finite type if F1 = Rn for some n ∈ N.
Let C(E,R+) be the set of all continuous functions from E into
R+, h ∈ Lg,d(E,F ) and ε ∈ C(E,R+). A map f ∈ Lg,d(E,F ) is
called a ε-approximation to h if d(f(x), h(x)) < ε(x) for all x ∈ E,
we write d(f, h) < ε for short. If we take the ε-approximation to h
to be a neighborhood of h in the set Lg,d(E,F ), then we obtain a
topology. This topology is called the fine topology and we denote
the resulting space by L0
fine(E,F ).
Let M and N be MCk manifolds modelled on Fréchet spaces
E and F , respectively. Let MCk(M,N), 1 � k � ∞, be the set
of MCk-maps from M into N . Two maps f, h ∈ MCk(M,N) are
said to be k-equivalent at x ∈ M if T k
x f = T k
xh, where T k is the
k-th tangent map. We define the k-jet of f at x, jkxf , to be the
equivalence class of f . Let dk be a fiber metric on the tangent
96 K. Eftekharinasab
space T k
xM that induces a Fréchet topology which is isomorphic
to E. We describe the fine topology of order k on MCk(M,N) as
follows. Let ϕ ∈ MCk(M,N) and Ω := {Vi}i∈I be a locally finite
cover of M . Let �i : Vi → R+ be continuous for all i ∈ I. Then,
the sets
Θ(ϕ, Vi, �i) := {φ ∈ MCk(M,N) | dk(jkxφ, jkxϕ) < �i(x), x ∈ Vi}
constitute a basis for fine open neighborhoods of ϕ. In this case
we say that φ in a fine neighborhood of ϕ is an MCk fine approx-
imation to ϕ.
Lemma 3.1. The fine topology is finer than the topology induced
by the Metric (2.1).
Proof. We must show that if N(f, δ) is a δ-neighborhood of f , then
we can find � > 0 such that if Dg,d(f, h) < �, then h ∈ N(f, δ).
Given a map h ∈ Lg,d(E,F ), let
� := min
�
1, inf
x∈E\{0}
δ(x)
g(h(x), 0)
�
.
Now suppose Dg,d(f, h) < �, then we can easily see that d(f, h) < δ
and hence h ∈ N(f, δ). �
Remark 3.2. We know that (Proposition 2.1) Iso(E,F ) is open in
Lg,d(E,F ) endowed with the topology induced by the metric (2.1).
By the preceding lemma the fine topology is finer than the metric
topology, thereby, Iso(E,F ) is open in L0
fine(E,F ).
Definition 3.3. Let f : E → F be a Lipschitz-Fredholm operator
of Fréchet spaces. We say that f is transversal to a closed subspace
F0 ⊆ F and write f � F0 if
(1) Img(f) + F0 = F , and
(2) either F0 splits in F or f−1(F0) splits in E.
The following result characterizes the transversality of Lipschitz-
Fredholm operators.
Transversality and Lipschitz-Fredholm maps 97
Proposition 3.4. Let ϕ ∈ LFl(E,F ). Suppose F0 ⊆ F is a closed
subspace such that Img(ϕ) + F0 = F . Then ϕ � F0 if and only if
there are closed subspace F1 ⊆ F and E0 ⊆ E with F = F0 ⊕ F1
and E = E0⊕ (E1 := ϕ−1(F1)) such that ϕ1 := ϕ|E1 ∈ Iso(E1, F1).
Proof. Assume that such a closed subspace F0 is given and ϕ � F0.
(Img(ϕ) ∩ F0) splits in F0 because
m = dim(F0/ Img(ϕ)) � dim(F/ Img(ϕ)) < ∞
and hence by Theorem 2.3(2) there exists a subspace F ⊆ F0 of
dimension m such that F0 = (Img(ϕ) ∩ F0)⊕ F. Since
Img(ϕ) ∩ F ⊆ Img(ϕ) ∩ F0,
it follows that Img(ϕ) ∩ F = {0}. Also,
Img(ϕ) + F = (Img(ϕ) + (Img(ϕ) ∩ F0)) + F
= Img(ϕ) + F0 = F.
Thus, Img(ϕ)⊕ F = F , therefore,
codim Img(ϕ) = m, dimker(ϕ) = l +m.
Moreover, there exists a closed subspace E ⊆ E such that
E = ker(ϕ)⊕ E.
The operator Φ := ϕ|E ∈ L(E, Img(ϕ)) is injective onto Img(ϕ),
hence, by virtue of open mapping theorem is a homeomorphism
and therefore Φ ∈ Iso(E, Img(ϕ)). Let
E0 := Φ−1(Img(ϕ) ∩ F0) ⊆ E,
then E0 = ϕ−1(Img(ϕ) ∩ F0) = ker(ϕ)⊕ E0.
As E0 is complemented in E0, there is a continuous projection
Pr1 of E0 onto E0 (see Theorem 2.3(4)). If E0 is complemented
in E, then there exists a continuous projection Pr2 of E onto E0.
Thus, Pr1 ◦ Pr2 is a continuous projection from E to E0 and its
restriction to E is a again continuous projection onto E0, thereby,
E0 is complemented in E. This means there is a closed subspace
E1 ⊆ E (which is also closed in E) such that E = E1 ⊕ E0.
98 K. Eftekharinasab
By the same argument we have, if F0 is complemented in F , then
(Img(ϕ) ∩ F0) is complemented in Img(ϕ) because (Img(ϕ) ∩ F0)
is complemented in F0. This means there is a closed subspace
F1 ⊆ Img(ϕ) (which is also closed in F ) such that
Img(ϕ) = F1 ⊕ (Img(ϕ) ∩ F0).
Therefore, we have
E = ker (ϕ)⊕ E0 ⊕ E1 = E0 ⊕ E1,
F = (Img(ϕ) ∩ F0)⊕ F⊕ F1 = F0 ⊕ F1
and ϕ1 = Φ|E1 ∈ Iso(E1, F1). Moreover, E1 = ϕ−1
1 (F1). The
converse is obvious. �
Proposition 3.5. LFl(E,F ) is open in L0
fine(E,F ).
Proof. Let ϕ ∈ LFl(E,F ). We show that there exists ε > 0
such that any φ ∈ Lg,d(E,F ) which is ε-approximation to ϕ is
a Lipschitz-Fredholm operator of index l.
First we prove for the case l = 0, then we show that the general
case can be reduced to the case l = 0. Let L : E → F (called a
corrector) be a linear globally Lipschitz map having finite dimen-
sional range such that K := L+ϕ is an isomorphism. Such a linear
map always exists. Indeed, L can be any linear globally Lipschitz
map from E into F such that
ker(L)⊕ ker(ϕ) = E, Img(L)⊕ Img(ϕ) = F.
Choose ε ∈ (0, 1/2Lip(K−1)) small enough and suppose that φ,L ∈
L(E,F ) are ε-approximation to ϕε-approximation to ϕ, and the
dimension of the image of L is finite. Then K = L+ φ satisfies
d(K(x),K(x)) < 1/Lip(K−1),
for all x ∈ E, thus K is an isomorphism (see Remark 3.2) and
hence φ ∈ LF(E,F ) and Ind(φ) = 0.
Now suppose l > 0, define the linear globally Lipschitz operators
ϕl,φl : E → F × Rl by ϕl(x) := (ϕ(x), 0) and φl(x) := (φ(x), 0).
Then ϕl is a Lipschitz-Fredholm operator of index 0. By the above
Transversality and Lipschitz-Fredholm maps 99
argument φl is a Lipschitz-Fredholm operator of index 0 and hence
φ is a Lipschitz-Fredholm operator of index l. Likewise, the case
l < 0 can be proved. �
Theorem 3.6. Let ϕ ∈ LFl(E,F ), and suppose that F0 ⊆ F
is closed and ϕ � F0. Then any φ ∈ Lg,d(E,F ) in some fine
neighborhood of ϕ is transversal to F0.
Proof. By Proposition 3.4 there exist closed subsets
E0 ⊆ E, F1 ⊆ F, E1 := ϕ−1(F1)
such that
F = F0 ⊕ F1, E = E0 ⊕ E1, ϕ1 := ϕ|E1 ∈ Iso(E1, F1).
There is a continuous function δ(x) such that every linear globally
Lipschitz map ψ : E1 → F1 which is δ-approximation to ϕ1 is an
isomorphism (see Remark 3.2). Let π : F → F1 be the projection
given by π(f0 + f1) = f1, and let κ = IdF − π. It is immediate
that π is linear and globally Lipschitz and Img(κ) = F0. Choose
ε ∈ (0, δ/Lip(π)) small enough, in view of Proposition 3.5, we
can assume that each φ ∈ L(E,F ) which is ε-approximation to ϕ
belongs to LFl(E,F ).
Now we show that each such φ is transversal to F0. Let
Φ := (π ◦ ϕ)|E1 ∈ L(E1, F1).
Then d(Φ,ϕ1) � Lip(π)ε < δ and so Φ ∈ Iso(E1, F1) (see Re-
mark 3.2). Thus, we only need to prove F = Img(φ) + F0. Let
f ∈ F = F0 ⊕ F1 so f = f0 + f1, where fi ∈ Fi(i = 0, 1).
We have Φ−1(f1) = e1 ∈ E1 ⊆ E, x = φ(e1) ∈ Img(φ), and
y = f0 − κ(x) ∈ F0. Whence,
x+ y = π(x) + κ(x) + f0 − κ(x) = f0 + Φ(e1) = f0 + f1 = f,
therefore F = Img(φ) + F0. �
Now we prove the transversality theorem for MCk-Lipschitz-
Fredholm maps. It is indeed a consequence of the Sard’s theorem
for these maps, [1, Theorem 4.3]. A careful reading of the proof of
100 K. Eftekharinasab
the theorem shows that the minor assumption of endowing man-
ifolds with compatible metrics is superfluous and the theorem re-
mains valid for manifolds without compatible metrics. Thus, the
statement of the theorem is as follows:
Theorem 3.7 (Sard’s Theorem). Let M and N be MCk man-
ifolds, k � 1. If f : M → N is an MCk-Lipschitz-Fredholm map
with k > max{Ind f, 0}. Then, the set of regular values of f is
residual in N .
Definition 3.8. Let f : M → N be a Lipschitz-Fredholm map
and let ı : A �→ N be an MC1 embedding of a finite dimensional
manifold A. We say that f is transversal to ı and write f � ı if
D f(x)(TxM) + D ı(y)(TyA) = Tf(x)N , whenever f(x) = ı(y). It
is also said that the submanifold A := ı(A) is transversal to f .
The following theorem is the analogous of the Smale transver-
sality, [5, Theorem 3.1]. Its proof is just a slight modification of
the argument of Smale.
Theorem 3.9. Let M and N be MCk manifolds modelled on
spaces (F, d) and (E, g), respectively. Let f : M → N be an MCk-
Lipschitz-Fredholm map and let ı : A �→ N be an MC1-embedding
of a finite dimension manifold A with k > max{Ind f +dimA, 0}.
Then there exists an MC1 fine approximation g of ı such that g is
embedding and f � g. Furthermore, Suppose S is a closed subset
of A and f � ı(S), then g can be chosen so that ı = g on S.
Proof. Since manifolds are second countable we only need to work
in local coordinates. Assume that y ∈ A and n = dim ı(A). Since
ı(A) is an embedded submanifold of finite type of N , we may find
an open neighborhood U ⊂ Rn about y, a chart about ı(y) and a
splitting E = Rn⊕E1 such that ı(y) = ı(y, 0) in the neighborhoods.
Let π2 : E → E1 be the projection onto E1. Let V ⊂ U be a
neighborhood of y, and h a smooth real valued function which is 1
on V and 0 outside of U . Since π1 ◦f is locally Fredholm-Lipschitz
it follows by Sard’s Theorem (Theorem 3.7) that there is a regular
Transversality and Lipschitz-Fredholm maps 101
value z for π1 ◦ f which is close to 0. Now define
g(y) = h(y)(z, y) + [1− h(y)]ı(y).
It is immediate that f � g on V , and for z sufficiently close to 0,
g is MC1 fine approximation to ı. The second statement follows
by our definition of g. �
4. Transversal submanifolds
We will need the following inverse function theorem.
Theorem 4.1. [2, Theorem 4.7], Inverse Function Theorem for
MCk-maps. Let (E, g) be a Fréchet space with standard metric g.
Let U ⊂ E be open, x0 ∈ U and f : U ⊂ E → E an MCk-map,
k ≥ 1. If f �(x0) ∈ Aut (E), then there exists an open neighborhood
V ⊆ U of x0 such that f(V ) is open in E and f |V : V → f(V ) is
an MCk- diffeomorphism.
To avoid some technical complications we consider only mani-
folds without boundary in the sequel.
Theorem 4.2. Let M and N be MCk manifolds modelled on
spaces (F, d) and (E, g), respectively. Suppose that f : M → N
is an MCk-Lipschitz-Fredholm map of index l. Let A be a sub-
manifold of N with dimension m and let ı : A �→ N be the in-
clusion. If f is transversal to A, then f−1(A) is a submani-
fold of M of dimension l + m. For all x ∈ f−1(A) we have
Tx(f
−1(A)) = (Txf)
−1(Tf(x)A).
Proof. If f−1(A) = ∅ the theorem is clearly valid so let us assume
that f−1(A) �= ∅. Let (ψ, U) be a chart at f(x0) ∈ A in N with
the submanifold property for A. Let U1, U2 be open subsets of
E,Rm such that ψ(U) = U1 ⊕ U2,ψ(U ∩ A) = U1 ⊕ {0}, and
ψ(f(x0)) = (0, 0). Let (V,ϕ) be a chart at x0 in M such that
ϕ(x0) = 0, ϕ : V → ϕ(V ) ⊂ F and f(V ) ⊂ U . Let
f := ψ ◦ f ◦ ϕ−1 : ϕ(V ) → ψ(U)
102 K. Eftekharinasab
be the local representative of f . Then f(0) = (0, 0) and by hy-
pothesis f is a Lipschitz-Fredholm map, in particular, D f(0) ∈
LFl(F,E). The tangent map Tf(x0)ı : Tf(x0)A → Tf(x0)N is in-
jective with closed split image. Hence Tf(x0)A can be identified
with a closed split subspace of Tf(x0)N . Thus D f(x0) is transver-
sal to Tf(x0)A. Therefore, keeping in the mind the definition of
the differential in terms of tangent maps, D f(0) is transversal
to Tψ(Tf(x0)A) = U1 ⊕ {0} =: E1. Then, by virtue of Proposi-
tion 3.4 there are closed subsets F1 ⊂ F , E0 ⊂ E such that F =
F1⊕(F0 := D f(0)(E0)), E = E1⊕E0, Δ := D f(0) |F0∈ Iso(F0, E0)
and Δ1 := D f(0) |F1∈ Iso(F1, E1). Moreover, dimF0 = m+ l.
Consider the projection π : F → F1 given by
π(f0 + f1) = f1.
Since F1 and F0 are closed and complementary it follows that
obviously the map κ = IdF − π is the unique projection with
Img (κ) = F0 and ker(κ) = F1. Let π1 : E → E0 be the projection
given by π1(e0 + e1) = e0. Then, Π := Δ−1 ◦ π1 ◦ D f(0) is a pro-
jection with Img (Π) = F0 and F1 ⊆ ker (Π). Since F = F0 ⊕ F1,
it follows that F1 = ker (Π) and therefore Π = κ.
Now define the map H : ϕ(V ) → F of class MCk by
H(x) = π(x) +Δ−1 ◦ π1 ◦ f(x).
We obtain that H(0) = 0 and
DH(0) = π +Δ−1 ◦ π1 ◦D f(0) = π + κ = IdF .
If we choose V small enough, then by Theorem 4.1 H is an MCk-
diffeomorphism onto an open neighborhood U ⊆ ψ(U) of ψ(f(x0) =
(0, 0). Let Φ = H ◦ ϕ−1, then (Φ, F0) is a chart for x0 on V with
the submanifold property. Because we have
x ∈ f−1(A) ⇔ ψ(f(x)) ∈ U1 ⊕ {0} ⇔ f(ϕ(x)) ∈ U1 ⊕ {0}
⇔ H(ϕ(x)) ∈ F0.
Transversality and Lipschitz-Fredholm maps 103
Let p ∈ A, γ : R → M a smooth curve sending zero to p, and j1pγ
the 1-jet of γ at p.
j1pγ ∈ TpA ⇔ j1ϕ(p)(ϕ ◦ γ) = Tϕ(j1pγ) ∈ ϕ(V )× F, ϕ ◦ γ ⊂ ϕ(V )
⇔ T f(j1ϕ(p)(ϕ ◦ γ)) ∈ ψ(U)× E
⇔ f(ϕ ◦ γ) = ψ(f ◦ γ) ⊂ ψ(U)
⇔ d
dt
ψ(f ◦ γ) |t=0= v =
d
dt
ψ([ψ−1(ψ(f(x)) + tv)]),
ψ(f ◦ γ) ⊂ ψ(U)
⇔ j1f(p)(f ◦ γ) = j1p [ψ
−1(ψ(f(p)) + tv)] ∈ Tf(p)A
⇔ j1pγ ∈ (Tpf)
−1(Tf(p)A)
This proves the second assertion. �
If manifolds have nonempty boundary we just need to modify
the proof by extending the considered maps.
Corollary 4.3. Let f : M → N be an MCk-Lipschitz-Fredholm
map of index l. If y is a regular value of f , then the level set
f−1(y) is a submanifold of dimension l and its tangent space at x
is kerTxf .
Proof. The set {y} is transversal to f so the result follows from
the theorem. �
Corollary 4.4. Let f : M × N → O be a smooth Lipschitz-
Fredholm map of manifolds, we write fx := f(·, x), and let A be a
closed finite dimension submanifold of O. Assume that f � A and
for all (m,n) ∈ f−1
n (A) the composition
(TmM
D fn(m)−−−−−→ Tfn(m)O
Q−→ Tfn(m)O/TnS)
is Lipschitz-Fredholm. Then there is a residual set of n in O for
which the map fn : M → O is transversal to A.
Proof. By hypothesis the kernel of Q ◦ D f(x) is complemented
for all x ∈ f−1(A). By the preceding theorem B := f−1(A) is a
Fréchet submanifold. The map f |B is smooth Lipschitz-Fredholm,
104 K. Eftekharinasab
therefore by Sard’s theorem there is a residual set of regular values
of it in O. If n ∈ N is a regular value of f |B, then fn is transversal
to A. �
References
[1] Eftekharinasab K. Sard’s theorem for mappings between
Fréchet manifolds // Ukrainian Math. J. — 2011. — 62, 12. —
P. 1896–1905.
[2] Müller O. A metric approach to Fréchet geometry // Journal
of Geometry and Physics . — 2008. — 58, 11. — P. 1477–1500.
[3] Glöckner H. Implicit functions from topological vector spaces
in the presence of metric estimates // arXiv:math/6612673. —
2006.
[4] Eftekharinasab K. Geometry of bounded manifolds // Rocky
Mountain J. Math. — to appear.
[5] Smale S. An infinite dimensional version of Sard’s theorem //
Amer. J. Math. — 1965. — 87. — P. 861–866.
[6] Köthe Gottfried. Topological vector spaces. I. Translated from
the German by D. J. H. Garling. Die Grundlehren der mathe-
matischen Wissenschaften, Band 159. — N. Y.: Springer-Verlag,
1969. — P. xv+456.
|
| id | oai:trim.imath.kiev.ua:article-133 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:02:48Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/84/f8b25bf070e9ded652f092c10975fa84.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1332018-01-23T11:58:55Z Transversality and Lipschitz-Fredholm maps Eftekharinasab, K. Eftekharinasab, K. We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the generalized Sard’s theorem for this category of man-ifolds. We also provide an answer to the well known problem concerning the existence of a submanifold structure on the preimage of a transversal submanifold. Вивчається поняття трансверсальностi вiдображень Лiпшица-Фредгольма у контекстi обмежених многовидiв Фреше. Доведено, що множина всiх вiдображень Лiпшица-Фредгольма фiксованого iндексу мiж просторами Фреше має властивiсть стiйкостi трансверсальних перетинiв. Дано пряме узагальнення теореми Смейла про транс-версальнiсть, для доведення якого використовується узагальнення теореми Сарда на цю категорiю многовидiв. Також отримано вiдповiдь на вiдоме питання про iснування структури пiдмноговиду на прообразi трансверсального пiдмноговиду. Інститут математики НАН України 2015-12-15 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/133 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 6 (2015): Topology of maps on low-dimensional manifolds; 89-104 Сборник Трудов Института математики НАН Украины; Том 12 № 6 (2015): Топологiя вiдображень маловимiрних многовидiв; 89-104 Збірник Праць Інституту математики НАН України; Том 12 № 6 (2015): Топологiя вiдображень маловимiрних многовидiв; 89-104 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/133/241 Авторське право (c) 2015 K. Eftekharinasab |
| spellingShingle | Eftekharinasab, K. Eftekharinasab, K. Transversality and Lipschitz-Fredholm maps |
| title | Transversality and Lipschitz-Fredholm maps |
| title_full | Transversality and Lipschitz-Fredholm maps |
| title_fullStr | Transversality and Lipschitz-Fredholm maps |
| title_full_unstemmed | Transversality and Lipschitz-Fredholm maps |
| title_short | Transversality and Lipschitz-Fredholm maps |
| title_sort | transversality and lipschitz-fredholm maps |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/133 |
| work_keys_str_mv | AT eftekharinasabk transversalityandlipschitzfredholmmaps AT eftekharinasabk transversalityandlipschitzfredholmmaps |