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Advances in Nonlinear Analysis

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A remark on bifurcation of Fredholm maps

Nils Waterstraat
  • School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom
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Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/anona-2016-0067


We modify an argument for multiparameter bifurcation of Fredholm maps by Fitzpatrick and Pejsachowicz to strengthen results on the topology of the bifurcation set. Furthermore, we discuss an application to families of differential equations parametrised by Grassmannians.

Keywords: Bifurcation; Fredholm maps; index bundle

MSC 2010: 58E07; 58J20; 19L20

1 Introduction

Let X and Y be Banach spaces, 𝒪X an open neighbourhood of 0X and Λ a compact topological space. A family of C1-Fredholm maps is a map f:Λ×𝒪Y such that each fλ:=f(λ,):𝒪Y is C1 and the Fréchet derivatives Dufλ(X,Y) are Fredholm operators of index 0, which depend continuously on (λ,u)Λ×𝒪 with respect to the operator norm on the space of bounded linear operators (X,Y). In what follows, we assume that f(λ,0)=0 for all λΛ, and we say that λΛ is a bifurcation point if every neighbourhood of (λ,0) in Λ×𝒪 contains an element (λ,u) such that u0 and f(λ,u)=0. By the implicit function theorem, the linear operator


is non-invertible if λ is a bifurcation point, and so a non-trivial kernel of some Lλ=D0fλ is a necessary assumption for the existence of a bifurcation point. Lots of efforts have been made to obtain sufficient criteria for the existence of bifurcation points by topological methods (cf., e.g., [9, 1, 4, 5]). Here we focus on the articles [16] and [17] of Pejsachowicz, which are the state of the art of a line of research on which he has worked for at least three decades (cf. [13, 7, 14, 15, 18]). In order to explain these results briefly, let us recall that every family {Lλ}λΛ of Fredholm operators has an index bundle, which is a K-theory class indL in KO~(Λ) that generalises the integral index for Fredholm operators to families. Moreover, Atiyah introduced in the sixties the J-group J(Λ) and a natural epimorphism J:KO~(Λ)J(Λ), which is called the generalised J-homomorphism. Pejsachowicz’ bifurcation theorem [16] reads as follows.

Theorem 1.1.

Let f:Λ×OY be a family of C1-Fredholm maps, parametrised by a connected finite CW-complex Λ, such that f(λ,0)=0 for all λΛ. If there is some λ0Λ such that Lλ0 is invertible and J(indL)0, then the family f has at least one bifurcation point from the trivial branch Λ×{0}.

Let us point out that the non-triviality of J(indL) can be obtained by characteristic classes, what paves the way for obtaining bifurcation invariants by methods from global analysis. Pejsachowicz has obtained striking results for bifurcation of families of nonlinear elliptic boundary value problems along these lines in [16].

The paper [17] deals with the question if J(indL) not only provides information about the existence of a single bifurcation point, but if we can also get results about the dimension and topology of the set of all bifurcation points B(f)Λ of f. Let us recall that the covering dimension dim(𝒳) of a topological space 𝒳 is the minimal value of n such that every finite open cover of 𝒳 has a finite open refinement in which no point is included in more than n+1 elements. In what follows, for k, we denote by wk(indL)Hk(Λ;2) the Stiefel–Whitney classes and by qk(indL)H2(p-1)k(Λ;p) the Wu classes of the index bundle indLKO~(Λ), where in the latter case we assume that indL is orientable. Pejsachowicz’ bifurcation theorem [17], which is based on his joint paper [7] with Fitzpatrick, reads as follows.

Theorem 1.2.

Let Λ be a compact connected topological manifold of dimension n2 and let f:Λ×OY be a continuous family of C1-Fredholm maps verifying f(λ,0)=0, and such that there is some λ0Λ for which Lλ0 is invertible.

  • (i)

    If Λ and indL are orientable and, for some odd prime p , there is 1kn-12(p-1) such that 0qk(indL)H2(p-1)k(Λ;p) , then the covering dimension of the set B(f) is at least n-2(p-1)k.

  • (ii)

    If 0wk(indL)Hk(Λ;2) for some 1kn-1 , then the dimension of B(f) is at least n-k.

Moreover, either the set B(f) disconnects Λ or it cannot be deformed in Λ into a point.

Note that if indL is non-orientable, and so (i) cannot be applied, then w1(indL)0 and we obtain by (ii) the strongest result that Theorem 1.2 can yield. Moreover, as in [16], the characteristic classes of indL in Theorem 1.2 can be computed for families of elliptic boundary value problems by methods from global analysis.

A little downside of the final statement on the topology of B(f) in Theorem 1.2 is that there is no a priori knowledge about which of the alternatives hold. The aim of this short article is to show that some simple modifications in Pejsachowicz’ proof of Theorem 1.2 show that actually B(f) is never a contractible topological space. Note that this is slightly weaker than the second alternative in Theorem 1.2 as every contractible subspace of Λ can also be deformed in Λ into a point.

After having stated and proved our version of Theorem 1.2 in the following section, we discuss an application to families of boundary value problems of ordinary differential equations.

2 Theorem and proof

2.1 Statement of the theorem

We denote by Φ(X,Y) the subspace of (X,Y) consisting of all Fredholm operators, and so the derivatives Lλ=D0fλ:XY define a family L:ΛΦ(X,Y) parametrised by the compact space Λ. Atiyah and Jänich introduced independently the index bundle for families of Fredholm operators (cf., e.g., [2, 21]). Accordingly, there is a finite dimensional subspace VY such that


Hence, if we denote by P the projection onto V, then we obtain a family of exact sequences of the form


and so a vector bundle E(L,V) consisting of the union of the kernels of the maps (IY-P)Lλ, λΛ. As L is Fredholm of index 0, the dimensions of E(L,V) and V coincide. Hence, if Θ(V) stands for the product bundle Λ×V, then we obtain the reduced KO-theory class


which is called the index bundle of L. Let us mention for later reference that the index bundle is natural, i.e., if Λ is another compact topological space and f:ΛΛ a continuous map, then fL:ΛΦ(X,Y), defined by (fL)λ=Lf(λ), is a family of Fredholm operators parametrised by Λ and


Two vector bundles E and F over Λ are called fibrewise homotopy equivalent if there is a fibre preserving homotopy equivalence between their sphere bundles S(E) and S(F). Moreover, E and F are stably fibrewise homotopy equivalent if EΘ(m) and FΘ(n) are fibrewise homotopy equivalent for some non-negative integers m,n. The quotient of KO~(Λ) by the subgroup generated of all [E]-[F] where E and F are stably fibrewise homotopy equivalent is denoted by J(Λ), and the quotient map J:KO~(Λ)J(Λ) is called the generalised J-homomorphism (cf. [2]). Very little is known about J(Λ) as these groups are notoriously hard to compute.

Note that in order to apply Theorem 1.1 it is not necessary to know J(Λ). All what is needed is a way to decide if J(indL) is non-trivial, which can be done by spherical characteristic classes. We denote by wk(E)Hk(Λ;2) the Stiefel–Whitney classes of real vector bundles E over Λ, and by qk(E)H2(p-1)k(Λ;p) the Wu classes of (real) orientable vector bundles E over Λ, where p is an odd prime (cf. [12]). By the naturality of characteristic classes, if Λ is another compact space and f:ΛΛ is continuous, then


As wk and qk are invariant under addition of trivial bundles, they are well defined on KO~(Λ). Moreover, they only depend on the stable fibrewise homotopy class of the associated sphere bundles, and so they even factorise through J(Λ). In particular, they detect elements having a non-trivial image under the J-homomorphism, i.e., J(indL)0 if wk(indL) or qk(indL) does not vanish for some k.

The aim of this article is to prove the following modification of Theorem 1.2.

Theorem 2.1.

Let Λ be a compact connected topological manifold of dimension n2, and let f:Λ×OY be a continuous family of C1-Fredholm maps verifying f(λ,0)=0. We denote by Lλ:=D0fλ the Fréchet derivative of fλ at 0O, and we assume that there is some λ0Λ for which Lλ0 is invertible.

  • (i)

    If Λ and indL are orientable and, for some odd prime p , there is 1kn-12(p-1) such that 0qk(indL)H2(p-1)k(Λ;p) , then the covering dimension of the set B(f) is at least n-2(p-1)k.

  • (ii)

    If 0wk(indL)Hk(Λ;2) for some 1kn-1 , then the dimension of B(f) is at least n-k.

Moreover, the set B(f) is not contractible to a point.

Let us point out once again that the novelty in our approach to this theorem is the result that B(f) is never a contractible space.

2.2 Proof of Theorem 2.1

In this section we denote by Hˇk(𝒳;G) the Čech cohomology groups of a topological space 𝒳 with respect to the abelian coefficient group G. Moreover, we use without further reference the fact that dim(𝒳)k if Hˇk(𝒳;G)0 (cf. [8, §VIII.4.A]).

Proof of Theorem 2.1.

We split the proof into two parts depending on whether or not ΛB(f) is connected.

Case 1: ΛB(f) is not connected. In this case the reduced homology group H~0(ΛB(f);2) is non-trivial as its rank is one less than the number of connected components of ΛB(f). By the long exact sequence of reduced homology (cf. [6, §IV.6])


where we used that Λ is connected by assumption, it follows that there is a surjective map


which shows that H1(Λ,ΛB(f);2) is non-trivial. As B(f) is closed, we obtain by Poincaré–Lefschetz duality (cf. [6, Corollary VI.8.4]) an isomorphism


and so Hˇn-1(B(f);2)0. Consequently, as n2, B(f) is not contractible to a point and, moreover, we obtain dimB(f)n-1, which is greater or equal to any of the lower bounds stated in Theorem 2.1.

Note that, as we use 2 coefficients, Λ does not need to be orientable in this part of the proof and so the argument works under either of the assumptions (i) or (ii).

Case 2: ΛB(f) is connected. We only give the proof under the assumption (i), as the argument for (ii) is very similar and even simpler, since we do not need to take orientability into account. Let us point out that this second part of the proof follows quite closely [17]. A similar argument can also be found in [19].

As p is a field, the duality pairing


is non-degenerate. Hence, there exists αH2(p-1)k(Λ;p) such that 0qk(indL),αp. Now we let ηHn-2(p-1)k(Λ;p) be the Poincaré dual of α with respect to a fixed p-orientation of Λ. According to [6, Corollary VI.8.4], there is a commutative diagram


where the vertical arrows are isomorphisms given by Poincaré–Lefschetz duality and the lower horizontal sequence is part of the long exact homology sequence of the pair (Λ,ΛB(f)). Because of the commutativity, the class iη is dual to πα and we now assume by contradiction the triviality of the latter one.

Then, by exactness, there exists βH2(p-1)k(ΛB(f);p) such that α=jβ. Moreover, since homology is compactly supported (cf. [11, §20.4]), there exist a compact connected CW-complex P and a map g:PΛB(f) such that β=gγ for some γH2(p-1)k(P;p).

Let us recall that by assumption there is some λ0Λ such that Lλ0 is invertible. As λ0B(f) by the implicit function theorem and ΛB(f) is connected, we can deform g such that λ0 belongs to its image. Clearly, this does not affect the property that β=gγ, and so we can assume without loss of generality that λ0img. We now set g¯=jg and consider


Then f¯(p,0)=0 for all pP and the linearisation of f¯p at 0X is L¯p=Lg¯(p). Moreover, as λ0 is in the image of g, there is some p0P such that L¯p0 is invertible. Clearly, g¯ sends bifurcation points of f¯ to bifurcation points of f, and as g¯(P)B(f)=, we see that the family f¯ has no bifurcation points. Consequently, J(indL¯)=0J(P), by Theorem 1.1, showing that qk(indL¯)=0H2(p-1)k(P;p). From (2.1) and (2.2), we obtain


which is a contradiction to the choice of α.

Consequently, πα is non-trivial, and so iηHˇn-2(p-1)k(B(f);p) is non-trivial, as well. Therefore, dimB(f)n-2(p-1)k and, moreover, B(f) is not contractible to a point as n-2(p-1)k1. ∎

Remark 2.2.

Let us point out that the estimate of the dimension of B(f) in Theorem 2.1 (ii) was obtained by Bartsch in [5] for semilinear Fredholm maps f by different methods. His argument is also based on Poincaré–Lefschetz duality and diagram (2.3) for p=2. However, instead of the J-homomorphism and Theorem 1.1, he used the codegree as a numerical invariant that measures the non-triviality of indL in KO~(Λ).

3 An example

In this section we denote by Gn(2n) the Grassmannian manifold of all n-dimensional subspaces of 2n (cf. [12, §5]). Let us recall that Gn(2n) is compact and that for every n there is a canonical vector bundle γn(2n) over Gn(2n) which is called the tautological bundle and which has as total space


In what follows we denote by I=[0,1] the compact unit interval, and we let Λ be a compact manifold of dimension m2. We denote by H1(I,n) the space of all absolutely continuous functions having a square integrable derivative with respect to the usual Sobolev norm


and we let


be a continuous function such that each φλ:=φ(λ,,):I×nn is smooth, all its derivatives depend continuously on λ, and φ(λ,t,0)=0 for all (λ,t)Λ×I. For a continuous map b:ΛGn(2n), we consider the family of differential equations


and we note that u0 is a solution for all λΛ.

Definition 3.1.

A parameter value λ0Λ is called a bifurcation point of (3.1) if in every neighbourhood of (λ0,0)Λ×H1(I,n) there is (λ,u) such that u0 is a solution of (3.1).

In what follows we denote by BΛ the set of all bifurcation points. The linearisation of (3.1) at 0 is


where D0φλ(t,):nn denotes the derivative of φλ(t,u) with respect to u at 0n.

We can now state our main theorem of this section.

Theorem 3.2.

If there is some λ0Λ such that (3.2) has only the trivial solution and


for some 1km-1, then the dimension of B is at least m-k and B is not a contractible topological space.

Let us note that Gn(2n) is itself a compact manifold, which is orientable as Grassmannians Gn(l) are orientable if and only if l is even (cf. [3]). We obtain the following corollary for Λ=Gn(2n) and b the identity.

Corollary 3.3.

If n2 and there is some V0Gn(R2n) such that (3.2) has only the trivial solution, then B is a non-contractible space which has at most codimension 1 in Gn(R2n).

Let us point out that the restriction which we impose on the dimension n in Corollary 3.3 is satisfied for all equations (3.1) that are first-order reductions of second-order scalar equations.

In the remainder of this section we will be concerned with the proofs of Theorem 3.2 and Corollary 3.3. We set


and we note that each one of the following maps is C1:




and we leave it to the reader to check that each Dvfλ:λL2(I,n) is a Fredholm operator of index 0.

Note that we cannot apply Theorem 2.1 to the operators fλ as they are not defined on a single Banach space. To overcome this technical issue, we need the following lemma.

Lemma 3.4.

The set


is a Hilbert subbundle of the product bundle Λ×H1(I,Rn).


Let P~:Gn(2n)×2n2n be a family of projections in 2n such that im(P~V) is the fibre of γn(2n) over V for VGn(2n). We consider the family


defined by


where P1,P2:2nn are the projections on the first and last n components in 2n, respectively, and I2n is the identity in 2n. The reader can easily check that Pλ2=Pλ and im(Pλ)=λ, i.e., Pλ is a projection in H1(I,n) onto λ. Hence,


is a subbundle of Λ×H1(I,2n), by [10, §III.3]. ∎

As every Hilbert bundle over a finite CW-complex is trivial, by Kuiper’s Theorem and [20, p. 54], there exists a bundle isomorphism ψ:Λ×H for some separable Hilbert space H. Moreover, by [10, Theorem VII.3.1] we can assume without loss of generality that each ψλ:Hλ is orthogonal and so an isometry. Composing ψ and f, we obtain a family of C1 maps


such that f(λ,0)=0 for all λΛ, and each f~λ:HL2(I,n) is Fredholm of index 0. Consequently, we can now apply Theorem 2.1 to the family f~, and so we need to compute ind(L~)KO~(Λ), where L~λ=D0f~λ, λΛ. By the chain rule, we get


Consequently, if we introduce two Hilbert bundle morphisms




then L~λ=LλMλ, λΛ.

The construction of the index bundle, which we recalled in Section 2.1, generalises almost verbatim to Fredholm morphisms of Banach bundles (cf. [21]). Accordingly, if 𝒳, 𝒴 are Banach bundles over a compact base Λ, and A:𝒳𝒴 is a bundle morphism such that every Aλ:𝒳λ𝒴λ is Fredholm, then


where 𝒱𝒴 is a finite dimensional subbundle such that


As the index bundle is additive under compositions of bundle morphisms, and it is trivial for bundle isomorphisms, we obtain


Let us now consider the Fredholm morphism


Then Lλ-L^λ:λL2(I,n) is compact for all λΛ and so, by [21, Corollary 7],


Our next aim is to compute ind(L^). Let us denote by Y1 the n-dimensional subspace of constant functions in L2(I,n) and


Clearly, Y1Y2={0} and, as every function uL2(I,n) can be written as


we see that L2(I,n)=Y1Y2. If vY2, then


is an element of λ which is mapped to v. Consequently, im(L^λ)+Y1=L2(I,n) for all λΛ, and we get that


As this vector bundle is isomorphic to the pullback b(γn(2n)) of the tautological bundle γn(2n) over Gn(2n) by b, we finally obtain, by (3.3) and (3.4),


where Θn denotes the product bundle with fibre n over Λ. Hence,


by the assumption of Theorem 3.2. Moreover, as (3.2) has only the trivial solution for λ=λ0, we see that the operator Lλ0:λ0L2(I,n) is injective and so invertible as it is Fredholm of index 0. Since ψλ0:Hλ0 is invertible, we conclude that L~λ0 is invertible as well. Consequently, from Theorem 2.1, we obtain that the family of C1-Fredholm maps f~ has a bifurcation point λΛ, i.e., there is a sequence {(λn,un)}nΛ×H converging to (λ,0) such that un0 and f~λn(un)=fλn(ψλnun)=0 for all n. We now set vn:=ψλnun0λn for n. As ψλn:Hλn is an isometry, it follows that vnH1=unH0 as n, where H is the norm of H. Since vn0 is a solution of (3.1) by the definition of f, we obtain that λ is a bifurcation point of (3.1) in the sense of Definition 3.1, and so Theorem 3.2 is proved.

Let us now focus on the proof of Corollary 3.3. By Theorem 2.1 we only need to recall the well-known fact that 0w1(γn(2n))H1(Gn(2n);2). Indeed, if we consider the tautological bundle γn() over the infinite Grassmannian Gn(), then by [12, Theorem 5.6], for every n-dimensional vector bundle ξ, there is a bundle map g:ξγn(). Hence, w1(ξ)=g¯w1(γn()), where g¯ is the map between the base space of ξ and Gn() that is induced by g. Hence, if w1(γn())=0, then w1(ξ) would be trivial for every n-dimensional bundle ξ. However, this is wrong as if we set ξ=γ1(2)Θn-1, where Θn-1 is the product bundle with fibre n-1 over G1(2), then


and the latter class is non-trivial as γ1(2) is isomorphic to the Möbius bundle over S1, which is non-orientable and so has a non-trivial first Stiefel–Whitney class.

If we now let ι:Gn(2n)Gn() be the canonical inclusion map, then ι(γn()) is isomorphic to γn(2n) and we obtain


As ι:H1(Gn();2)H1(Gn(2n);2) is an isomorphism for n2 by [12, Problem 6-B], we conclude that w1(γn(2n)) is non-trivial, and so Corollary 3.3 is proved.


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About the article

Received: 2016-03-20

Revised: 2016-06-04

Accepted: 2016-06-05

Published Online: 2016-07-12

Published in Print: 2018-08-01

Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 3, Pages 285–292, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0067.

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