Some applications of transversality for infinite dimensional manifolds

We present some transversality results for a category of Fr\'{e}chet manifolds, the so-called $MC^k$-Fr\'{e}chet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.


Bounded Fréchet manifolds
In this section, we shall briefly recall the basics of MC k -Fréchet manifolds for the convenience of readers, which also allows us to establish our notations for the rest of the paper. For more studies, we refer to [1,2].
Throughout the paper we assume that E, F are Fréchet spaces and CLpE, F q is the space of all continuous linear mappings from E to F topologized by the compact-open topology. If T is a topological space by U Ď T we mean U is open in T .
Let ϕ : U Ď E Ñ F be a continuous map. If the directional (Gâteaux) derivatives D ϕpxqh " lim tÑ0 ϕpx`thq´ϕpxq t exist for all x P U and all h P E, and the induced map D ϕpxq : U Ñ CLpE, F q is continuous for all x P U, then we say that ϕ is a Keller's differentiable map of class C 1 c . The higher directional derivatives and C k c -mappings, k ě 2, are defined in the obvious inductive fashion. To define bounded or MC k -differentiability, we endow a Fréchet space F with a translation invariant metric ̺ defining its topology, and then introduce the metric concepts which strongly depend on the choice of ̺. We consider only metrics of the following form ̺px, yq " sup nPN 1 2 n x´y F,n 1` x´y F,n , where ¨ F,n is a collection of seminorms generating the topology of F . Let σ be a metric that defines the topology of a Fréchet space E. Let L σ,̺ pE, F q be the set of all linear mappings L : E Ñ F which are (globally) Lipschitz continuous as mappings between metric spaces E and F , that is where LippLq is the (minimal) Lipschitz constant of L.
An MC k -Fréchet manifold is a Hausdorff second countable topological space modeled on a Fréchet space with an atlas of coordinate charts such that the coordinate transition functions are all MC k -mappings. We define MC k -mappings between Fréchet manifolds as usual. Henceforth, we assume that M and N are connected MC k -Fréchet manifolds modeled on Fréchet spaces pF, ̺q and pE, σq, respectively.
A mapping ϕ P L σ,̺ pE, F q is called Lipschitz-Fredholm operator if its kernel has finite dimension and its image has finite co-dimension. The index of ϕ is defined by Ind ϕ " dim ker ϕ´codim Img ϕ.
We denote by LFpE, F q the set of all Lipschitz-Fredholm operators, and by LF l pE, F q the subset of LFpE, F q consisting of those operators of index l.
An MC k -Lipschitz-Fredholm mapping ϕ : M Ñ N, k ě 1, is a mapping such that for each x P M the derivative D ϕpxq : T x M ÝÑ T f pxq N is a Lipschitz-Fredholm operator. The index of ϕ, denoted by Ind ϕ, is defined to be the index of D ϕpxq for some x which does not depend on the choice of x, see [1,Definition 3.2 ].
Let ϕ : M Ñ N pk ě 1q be an MC k -mapping. We denote by T x ϕ : T x M Ñ T ϕpxq N the tangent map of f at x P M from the tangent space T x M to the tangent space T ϕpxq N. We say that ϕ is an immersion (resp. submersion) provided T x ϕ is injective (resp. surjective) and the range ImgpT x ϕq (resp. the kernel kerpT x ϕq) splits in T ϕpxq N (resp. T x M) for any x P M.
An injective immersion f : M Ñ N which gives an isomorphism onto a submanifold of N is called an embedding.
The corresponding value f pxq is a regular value. Points and values other than regular are called critical points and values, respectively.
Let ϕ : M Ñ N be an MC k -mapping, k ě 1. We say that ϕ is transversal to a submanifold S Ď N and write ϕ&S if either ϕ´1pSq " H, or if for each x P ϕ´1pSq (1) pT x ϕqpT x Mq`T ϕpxq S " T ϕpxq N, and (2) pT x ϕq´1pT ϕpxq Sq splits in T x M. In terms of charts, ϕ&S when x P ϕ´1pSq there exist charts pφ, Uq around x and pψ, Vq around ϕpxq such that Then the composite mapping is an MC k -submersion, where Pr V 2 is the projection onto V 2 .

Transversality theorems
We generalize [2,Theorem 4.2] and [2, Corollary 4.1] for not necessarily Lipschitz-Fredholm mappings and finite dimensional submanifolds. We shall need the following version of the inverse function theorem for MC k -mappings.
Proposition 2.1. Let ϕ : M Ñ N be an MC k -mapping, S Ă N an MC k -submanifold and x P ϕ´1pSq. Then ϕ&S if and only if there are charts pU, φq around x with φpxq " 0 E and pV, ψq around y " ϕpxq in S with ψpyq " 0 F such that the following hold: (1) There are subspaces E 1 and E 2 of E, and F 1 and F 2 of F such that E " E 1 ' E 2 and F " F 1 ' F 2 . Moreover, ψpS X Vq " F 1 and where 0 E P E i Ď E i and 0 F P F i Ď F i , i " 1, 2.
(2) In the charts the local representative of ϕ has the form where ϕ : E 1`E2 Ñ F 1 is an MC k -mapping,φ is an MC k -isomorphism of E 2 onto F 2 and Pr E 2 : E Ñ E 2 is the projection.
Proof. Sufficiency: Let pU, φq and pV, ψq be charts that satisfy the assumptions we will prove ϕ&S. In the charts, by using the identifications T x M » E, T y » F , the tangent map T x ϕ : Let Pr F 2 : F Ñ F 2 be the projection onto F 2 . Since ϕ 1 φψ p0 E q " pϕq 1`φ˝P r E 2 and pϕq 1 p0q : E Ñ F 1 , it follows that for all e P E, e " e 1`e2 P E 1 ' E 2 it is a surjective mapping of E onto F 2 . Moreover, we have Necessity: Suppose ϕ&S. Since S is an MC k -submanifold of N and y " ϕpxq P S, there is a chart pW, Ûq around y having the submanifold property for S in N: By the inverse mapping theorem 2.1, τ is a local MC k -diffeomorphism. Thus, If, e " e 1`e2 P Ü 1 pX 1 q and Ü´1 1 peq "ē 1`ē2 , then by (2.3) and (2.4) we obtain Therefore,ē 1 " e 1 and τ´1˝Pr F 2˝ϕ ÜÛ " e 2 and so This means, Then, pφ, Uq and pψ, Vq are the desired charts. Indeed, Thus, if we setφη and ϕ -Pr E 1˝ϕ ÜÛ˝Ü´1 1 .
Proof. Let x P ϕ´1pSq, then by Proposition 2.1 there are chart pφ, Uq around x and pψ, Vq around y " ϕpxq such that Thus, φpêq P F 1 for allê P ϕ´1pSq X U. Therefore, E 1 Ă φpϕ´1pSq X Uq, since for each e 1 P E 1 we have ϕ φψ pe 1 q " ϕpe 1 q`φ˝Pr E 2 pe 1 q " ϕpe 1 q P F 1 . Hence, ψ˝ϕ˝φ´1pe 1 q P F 1 implies that ϕ˝φ´1pe 1 q P ψ´1pF 1 q " S X V and so ϕ˝φ´1pe 1 q which means φ´1pe 1 q P ϕpSq X V that yields e 1 P ψpϕ´1pSq X Vq. Therefore, for x P ϕ´1pSq there is a chart pφ, Uq with φpUq " In the charts, we have T If S has finite co-dimension then F 2 has finite dimension and thus by Proposition 2.1, The proof of the last statement is standard.
As an immediate consequence we have: Corollary 2.1. Let ϕ : M Ñ N be an MC k -mapping, k ě 1. If q is a regular value of ϕ, then the level set ϕ´1pqq is a submanifold of M and its tangent space at p " ϕpqq is ker T p ϕ. Moreover, if q is a regular value of ϕ and ϕ is an MC k -Lipschitz-Fredholm mapping of index l, then dim ϕ´1pSq " l.
To prove the parametric transversality theorem we apply the following Sard's theorem. Let π M and π S be the local representatives of Pr M and Pr S , respectively. We show that π M and consequently π S are Lipschitz-Fredholm operators of index n´m.
Finite dimensionality of R n and closedness of S implies that K -S`pt0uˆR n q is closed in EˆR n . Also, codim K is finite because it contains the finite co-dimensional subspace S. Therefore K has a finite-dimensional complement K 1 Ă Eˆt0u, that is EˆR n " K ' K 1 . Let K 2 -S X t0uˆR n . Since K 2 Ă R n we can choose closed subspaces S 1 Ă R n and R 0 Ă t0uˆR n such that S " S 1 ' K 1 and t0uˆR n " The mapping π S | S 1 'K 2 : S 1 ' K 2 Ñ E is an isomorphism, K 1 " ker π S , and π M pK 2 q is a finite dimensional complement to π M pSq in R n . Thus, π M is a Lipschitz-Fredholm operator and we have Since, K 1 ' R 0 " t0uˆR n and R 0 ' K 2 is a complement to S in EˆR n and therefore its dimension is n, so the index of π M is n´m.
Now, we prove that if x is a regular value of Pr S if and only if ϕ x &S. From the definition of ϕ& we have @px, aq P S pT px,aq ϕqpT x MˆT a Aq`T ϕpx,aq S " T ϕpx,aq N, (2.6) and pT px,aq ϕq´1pT ϕ px,aq Sq splits in T x MˆT a A.
(2.7) Since A has finite dimension, it follows that the mapping a P A Þ Ñ ϕpx, aq for a fixed x P M is transversal to S if and only if @px, aq P S, T a ϕ x pT a Aq`T ϕpx,aq S " T ϕpx,aq S. (2.8) Since Pr S is a Lipschitz-Fredholm mapping, ker T Pr S splits at any point as its dimension is finite. Then x is a regular value of Pr S if and only if @px, aq P S, @v P T x M, Du P T a A : T pv,uq ϕpv, uq P T px,aq S. (2.9) Pick x P M and a P A such that px, aq P S and let w P T px,aq S. By (2.6) and (2.7) we obtain that there exist v P T a A, x 1 P T x M, y 1 P T px,aq S such that T px,aq ϕpv, x 1 q`y 1 " w. (2.10) Then, there exists x 2 P T x M such that T px,aq ϕpv, x 2 q P T ϕ px,aq S. Hence, w " T px,aq ϕpv, x 1 q´T px,aq ϕpv, x 2 q`T px,aq ϕpv, x 2 q`y 1 " T px,aq ϕp0, x 1´x2 q`T px,aq ϕpv, x 2 q`y 1 " T px,aq ϕp0, uq`T ϕpx,aq S`y 2 P T a ϕ x pT a Aq, where u " x 1´x2 and y 2 " T px,aq ϕpv, x 2 q`y 1 P T ϕ px,aq S. Thus, (2.8) holds. Now we show that (2.8) implies (2.9). Pick a P A, x P M such that px, aq P S. Let v P T x M, a 1 P T a A, y 1 P T ϕ px,aq S and set w -T px,aq ϕ pv,x 1 q`y1 . By (2.8) there exist a 2 P T a A and y 2 P T ϕ px,aq S such that w " T a ϕ x pa 2 q`y 2 . Then, 0 E " T px,aq ϕpv, a 1 q´T a ϕ x pa 2 q`y 1´y2 " T px,aq ϕpv, a 1´a2 q`y 1´y2 , so T px,aq ϕpv, a 1´a2 q " y 2´y1 P T ϕ px,aq S so (2.9) holds. Thus, we showed that if x is a regular value of Pr S if and only if ϕ x &S. Since Pr S : S Ñ M is a Lipschitz-Fredholm of class MC k with the index n´m and codim S " codim S " m and k ą t0, n´mu, the Sard's theorem 2.3 concludes the theorem.

The degree of Lipschitz-Fredholm mappings
In this section we construct the degree of MC k -Lipschitz-Fredholm mappings and apply it to prove an invariance of domain theorem, a rank theorem and a Bursuk-Ulam type theorem. The construction of the degree relies on the following transversality result.
Theorem 3.1. [2, Theorem 3.3] Let ϕ : M Ñ N be an MC k -Lipschitz-Fredholm mapping, k ě 1. Let ı : A Ñ N be an MC 1 -embedding of a finite dimension manifold A with k ą maxtInd ϕ`dim A, 0u. Then there exists an MC 1 fine approximation g of ı such that g is embedding and ϕ&g. Moreover, suppose S is a closed subset of A and ϕ&ıpSq, then g can be chosen so that ı " g on S.
We shall need the following theorem that gives the connection between proper and closed mappings. Let ϕ : M Ñ N be a non-constant closed Lipschitz-Fredholm mapping with index l ě 0 of class MC k such that k ą l`1. If q is a regular value of ϕ, then by Theorem 3.2 and Corollary 2.1 the preimage ϕ´1pqq is a compact submanifold of dimension l.
Let ı : r0, 1s ãÑ N be an MC 1 -embedding that connects two distinct regular values q 1 and q 2 . By Theorem 3.1 we may suppose ı is transversal to ϕ. Thus, by Theorem 2.2 the preimage M -ϕ´1pıpr0, 1sqq is a compact pl`1q-dimensional submanifold of M such that its boundary, BM, is the disjoint union of ϕ´1pq 1 q and ϕ´1pq 2 q, BM " ϕ´1pq 1 q > ϕ´1pq 2 q. Therefore, ϕ´1pq 1 q and ϕ´1pq 2 q are non-oriented cobordant which gives the invariance of the mapping. Following Smale [6] we associate to ϕ a degree, denoted by deg ϕ, defined as the non-oriented cobordism class of ϕ´1pqq for some regular value q. If l " 0, then deg ϕ P Z 2 is the number modulo 2 of preimage of a regular value.
Let O Ď M. Suppose ϕ : O Ñ N is a non-constant closed continuous mapping such that its restriction to O is an MC k`1 -Lipschitz-Fredholm mapping of index k, k ě 0. Let p P NzϕpBOq and let p a regular value of ϕ in the connected component of NzϕpBOq containing p, the existence of such regular value follows from Sard's theorem 2.3. Again, we associate to ϕ a degree, degpϕ, pq, defined as non-oriented class of k-dimensional compact manifold ϕ´1ppq. This degree does not depend on the choice of p.
The following theorem which presents the local representation of MC k -mappings is crucial for the rest of the paper. Suppose that D ϕpu 0 q has closed split image F 1 with closed topological complement F 2 and split kernel E 2 with closed topological complement E 1 . Then, there are two open sets U 1 Ď U and V Ď F 1 ' E 2 and an MC k -diffeomorphism Ψ : V Ñ U 1 , such that pϕ˝Ψqpf, eq " pf, ηpf, eqq for all pf, eq P V, where η : V Ñ E 2 is an MC k -mapping.
Theorem 3.4 (Rank theorem for MC k -mappings). Let ϕ : U Ď E Ñ F be an MC kmapping, k ě 1. Suppose u 0 P U and D ϕpu 0 q has closed split image F 1 with closed complement F 2 and split kernel E 2 with closed complement E 1 . Also, assume D ϕpUqpEq is closed in F and D ϕpuq| E 1 : E 1 Ñ D ϕpuqpEq is an MC k -isomorphism for each u P U. Then, there exist open sets U 1 Ď F 1 ' E 2 , U 2 Ď E, V 1 Ď F , and V 2 Ď F and there are MC k -diffeomorphisms φ : V 1 Ñ V 2 and ψ : U 1 Ñ U 2 such that pφ˝ϕ˝ψqpf, eq " pf, 0q, @pf, eq P U 1 .
As an immediate consequence we have the following: The following theorem gives the openness property of the set of Lipschitz-Fredholm mappings.  Proof. This is a local problem so we assume M is an open set in E and N is an open set in F . Let s P Singpϕq be arbitrary and U an open neighborhood of s in Singpϕq. For each n P N Y t0u define S n -tm P M | dim D ϕpmq ě nu . Then, M " M 0 Ą M 1 Ą¨¨¨, therefore, is a unique n 0 such that M " M n 0 ‰ M n 0`1 . Let m 0 P M n 0 zM n 0`1 such that dim ker D ϕpm 0 q " n 0 . By Theorem 3.5, there exists an open neighborhood V of m 0 in U such that for all v P V we have dim ker D ϕpvq ď n 0 and hence dim D ϕpvq " n 0 ě 1. By Corollary 3.1, there is a local representative ϕ around zero such that ψ˝ϕ˝φ´1pf, eq " pf, 0q for pf, eq P A 1 ' R n 0 which contradicts the injectivity of ϕ, therefore, Singpϕq contains a nonempty open set. The closedness of Singpϕq is obvious in virtue of Theorem 3.5. Proof. Let p P U Ď M and q " ϕppq. The point p has a connected open neighborhood U Ď M such that ϕ | U : U Ñ N is proper and injective. Whence q R ϕpBUq and ϕpBUq is closed in N. Let V be a connected component of NzϕpBUq containing q which is its open neighborhood. Since U is connected it implies that ϕpUq Ă V. It follows from ϕpBUqXV " H that U X ϕ´1pVq " U and so ϕ | U : U Ñ N is proper and injective. By Theorem 3.6 there is a point x P M such that the tangent map T x ϕ is injective and since Ind ϕ " 0 it is surjective too. Therefor, y " ϕpxq is a regular value with ϕ´1pyq " txu and so deg ϕ " 1. It follows that ϕ is surjective, because if it is not , then any point in NzϕpMq is regular and deg ϕ " 0 which is contradiction. Then, V " ϕpUq is the open neighborhood of q.
Corollary 3.2 (Nonlinear Fredholm alternative). Let ϕ : M Ñ N be an MC k -Lipschitz-Fredholm mapping of index zero, k ą 1. If N is connected and ϕ is locally injective, then ϕ is surjective and finite covering mapping. If M is connected and N is simply connected, then ϕ is a homeomorphism.
The following theorem is a generalization of the Bursuk-Ulam theorem, the proof is slight modification of the Banach case.
Theorem 3.8. Let ϕ : U Ñ F be a non-constant closed Lipschitz-Fredholom mapping of class MC 2 with index zero, where U Ď F is symmetric. If ϕ is odd and for u 0 P U we have u 0 R ϕpBUq. Then degpϕ, u 0 q " 1 mod 2.
Proof. Since D ϕpu 0 q is a Lipschitz-Fredholm mapping with index zero F " F 1 ' ker ϕ " F 2 ' Img ϕ and dim F 2 " dim ker ϕ. The image ϕpBUq is closed as ϕ is closed, hence a " ̺pϕpUq, u 0 q ą 0 because u 0 R ϕpBUq.
Therefore, u 0 R Φ φ pBUq. We obtain degpϕ, u 0 q " degpΦ φ , u 0 q as the mapping ψ : r0, 1sˆU Ñ F defined by pt, uq Ñ ϕpuq`tφpuq is proper and u 0 R ψpBUq for all t. Considering the fact that ψp´uq "´ψpuq, we may use the perturbation by compact operators to find the degree of ϕ. Let C be a set of global Lipschitz-compact linear operators φ : F Ñ F with Lippφq ă b ă a{k. Let p φ P C be such that its restriction to F 1 equals u 0 and p φ | ker D ϕpu 0 q : ker D ϕpu 0 q Ñ F 2 is an MC 1 -isomorphism. Therefore, D ϕpu 0 q`p φ and consequently D ϕpu 0 q is an MC 1isomorphism. Now define the mapping Ψ : UˆC Ñ F by pu, φq " Φ φ puq. For sufficiently small b the differential D Ψpu, φqpv, ψq " pD ϕpuq`φqv`ψpuq is surjective at u 0 as D ϕpu 0 q is an MC 1 -isomorphism. Also, it is clear that it is surjective at the other points. Then, the mapping Ψ satisfies the assumption of Theorem 2.4, therefore, Ψ´1pu 0 q is a submanifold and the mapping Π : Ψ´1pu 0 q Ñ C induced by the projection onto the second order is Lipschitz-Fredholm of index zero. By employing the local version of Sard's theorem we may find a regular point φ of Π, and from the proof of the Theorem 2.4 it follows that u 0 is a regular value of Φ φ and consequently u 0 is a regular value of ϕ. Thus, properness and ϕp´uq "´ϕpuq imply that ϕ´1pu 0 q " tu 0 , f 1 ,´f 1 ,¨¨¨f m ,´f m u and therefore degpϕ, u 0 q " 1 mod 2.
Topology lab. Institute of Mathematics of National Academy of Sciences of Ukraine, Tereshchenkivska st. 3, Kyiv, 01601 Ukraine Email address: kaveh@imath.kiev.ua