Let X and Y be Banach spaces, an open neighbourhood of and Λ a compact topological space. A family of -Fredholm maps is a map such that each is and the Fréchet derivatives are Fredholm operators of index 0, which depend continuously on with respect to the operator norm on the space of bounded linear operators . In what follows, we assume that for all , and we say that is a bifurcation point if every neighbourhood of in contains an element such that and . By the implicit function theorem, the linear operator
is non-invertible if is a bifurcation point, and so a non-trivial kernel of some 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  and  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 of Fredholm operators has an index bundle, which is a K-theory class in that generalises the integral index for Fredholm operators to families. Moreover, Atiyah introduced in the sixties the J-group and a natural epimorphism , which is called the generalised J-homomorphism. Pejsachowicz’ bifurcation theorem  reads as follows.
Let be a family of -Fredholm maps, parametrised by a connected finite CW-complex Λ, such that for all . If there is some such that is invertible and , then the family f has at least one bifurcation point from the trivial branch .
Let us point out that the non-triviality of 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 .
The paper  deals with the question if 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 of f. Let us recall that the covering dimension of a topological space is the minimal value of such that every finite open cover of has a finite open refinement in which no point is included in more than elements. In what follows, for , we denote by the Stiefel–Whitney classes and by the Wu classes of the index bundle , where in the latter case we assume that is orientable. Pejsachowicz’ bifurcation theorem , which is based on his joint paper  with Fitzpatrick, reads as follows.
Let Λ be a compact connected topological manifold of dimension and let be a continuous family of -Fredholm maps verifying , and such that there is some for which is invertible.
If Λ and are orientable and, for some odd prime p , there is such that , then the covering dimension of the set is at least .
If for some , then the dimension of is at least .
Moreover, either the set disconnects Λ or it cannot be deformed in Λ into a point.
Note that if is non-orientable, and so (i) cannot be applied, then and we obtain by (ii) the strongest result that Theorem 1.2 can yield. Moreover, as in , the characteristic classes of 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 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 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 the subspace of consisting of all Fredholm operators, and so the derivatives define a family 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 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 consisting of the union of the kernels of the maps , . As L is Fredholm of index 0, the dimensions of and V coincide. Hence, if stands for the product bundle , 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 is another compact topological space and a continuous map, then , defined by , is a family of Fredholm operators parametrised by 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 and . Moreover, E and F are stably fibrewise homotopy equivalent if and are fibrewise homotopy equivalent for some non-negative integers . The quotient of by the subgroup generated of all where E and F are stably fibrewise homotopy equivalent is denoted by , and the quotient map is called the generalised J-homomorphism (cf. ). Very little is known about as these groups are notoriously hard to compute.
Note that in order to apply Theorem 1.1 it is not necessary to know . All what is needed is a way to decide if is non-trivial, which can be done by spherical characteristic classes. We denote by the Stiefel–Whitney classes of real vector bundles E over Λ, and by the Wu classes of (real) orientable vector bundles E over Λ, where p is an odd prime (cf. ). By the naturality of characteristic classes, if is another compact space and is continuous, then
As and are invariant under addition of trivial bundles, they are well defined on . Moreover, they only depend on the stable fibrewise homotopy class of the associated sphere bundles, and so they even factorise through . In particular, they detect elements having a non-trivial image under the J-homomorphism, i.e., if or does not vanish for some k.
The aim of this article is to prove the following modification of Theorem 1.2.
Let Λ be a compact connected topological manifold of dimension , and let be a continuous family of -Fredholm maps verifying . We denote by the Fréchet derivative of at , and we assume that there is some for which is invertible.
If Λ and are orientable and, for some odd prime p , there is such that , then the covering dimension of the set is at least .
If for some , then the dimension of is at least .
Moreover, the set 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 is never a contractible space.
2.2 Proof of Theorem 2.1
In this section we denote by the Čech cohomology groups of a topological space with respect to the abelian coefficient group G. Moreover, we use without further reference the fact that if (cf. [8, §VIII.4.A]).
Proof of Theorem 2.1.
We split the proof into two parts depending on whether or not is connected.
Case 1: is not connected. In this case the reduced homology group is non-trivial as its rank is one less than the number of connected components of . 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 is non-trivial. As is closed, we obtain by Poincaré–Lefschetz duality (cf. [6, Corollary VI.8.4]) an isomorphism
and so . Consequently, as , is not contractible to a point and, moreover, we obtain , which is greater or equal to any of the lower bounds stated in Theorem 2.1.
Note that, as we use 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: 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 . A similar argument can also be found in .
As is a field, the duality pairing
is non-degenerate. Hence, there exists such that . Now we let be the Poincaré dual of α with respect to a fixed -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 . Because of the commutativity, the class is dual to and we now assume by contradiction the triviality of the latter one.
Then, by exactness, there exists such that . Moreover, since homology is compactly supported (cf. [11, §20.4]), there exist a compact connected CW-complex P and a map such that for some .
Let us recall that by assumption there is some such that is invertible. As by the implicit function theorem and is connected, we can deform g such that belongs to its image. Clearly, this does not affect the property that , and so we can assume without loss of generality that . We now set and consider
Then for all and the linearisation of at is . Moreover, as is in the image of g, there is some such that is invertible. Clearly, sends bifurcation points of to bifurcation points of f, and as , we see that the family has no bifurcation points. Consequently, , by Theorem 1.1, showing that . From (2.1) and (2.2), we obtain
which is a contradiction to the choice of α.
Consequently, is non-trivial, and so is non-trivial, as well. Therefore, and, moreover, is not contractible to a point as . ∎
Let us point out that the estimate of the dimension of in Theorem 2.1 (ii) was obtained by Bartsch in  for semilinear Fredholm maps f by different methods. His argument is also based on Poincaré–Lefschetz duality and diagram (2.3) for . However, instead of the J-homomorphism and Theorem 1.1, he used the codegree as a numerical invariant that measures the non-triviality of in .
3 An example
In this section we denote by the Grassmannian manifold of all n-dimensional subspaces of (cf. [12, §5]). Let us recall that is compact and that for every there is a canonical vector bundle over which is called the tautological bundle and which has as total space
In what follows we denote by the compact unit interval, and we let Λ be a compact manifold of dimension . We denote by 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 is smooth, all its derivatives depend continuously on λ, and for all . For a continuous map , we consider the family of differential equations
and we note that is a solution for all .
In what follows we denote by the set of all bifurcation points. The linearisation of (3.1) at 0 is
where denotes the derivative of with respect to u at .
We can now state our main theorem of this section.
If there is some such that (3.2) has only the trivial solution and
for some , then the dimension of B is at least and B is not a contractible topological space.
Let us note that is itself a compact manifold, which is orientable as Grassmannians are orientable if and only if l is even (cf. ). We obtain the following corollary for and b the identity.
If and there is some such that (3.2) has only the trivial solution, then B is a non-contractible space which has at most codimension 1 in .
and we note that each one of the following maps is :
and we leave it to the reader to check that each is a Fredholm operator of index 0.
Note that we cannot apply Theorem 2.1 to the operators as they are not defined on a single Banach space. To overcome this technical issue, we need the following lemma.
is a Hilbert subbundle of the product bundle .
Let be a family of projections in such that is the fibre of over V for . We consider the family
where are the projections on the first and last n components in , respectively, and is the identity in . The reader can easily check that and , i.e., is a projection in onto . Hence,
is a subbundle of , 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 for some separable Hilbert space H. Moreover, by [10, Theorem VII.3.1] we can assume without loss of generality that each is orthogonal and so an isometry. Composing ψ and f, we obtain a family of maps
such that for all , and each is Fredholm of index 0. Consequently, we can now apply Theorem 2.1 to the family , and so we need to compute , where , . By the chain rule, we get
Consequently, if we introduce two Hilbert bundle morphisms
then , .
The construction of the index bundle, which we recalled in Section 2.1, generalises almost verbatim to Fredholm morphisms of Banach bundles (cf. ). Accordingly, if , are Banach bundles over a compact base Λ, and is a bundle morphism such that every is Fredholm, then
where 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 is compact for all and so, by [21, Corollary 7],
Our next aim is to compute . Let us denote by the n-dimensional subspace of constant functions in and
Clearly, and, as every function can be written as
we see that . If , then
is an element of which is mapped to v. Consequently, for all , and we get that
where denotes the product bundle with fibre over Λ. Hence,
by the assumption of Theorem 3.2. Moreover, as (3.2) has only the trivial solution for , we see that the operator is injective and so invertible as it is Fredholm of index 0. Since is invertible, we conclude that is invertible as well. Consequently, from Theorem 2.1, we obtain that the family of -Fredholm maps has a bifurcation point , i.e., there is a sequence converging to such that and for all . We now set for . As is an isometry, it follows that as , where is the norm of H. Since is a solution of (3.1) by the definition of f, we obtain that 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 . Indeed, if we consider the tautological bundle over the infinite Grassmannian , then by [12, Theorem 5.6], for every n-dimensional vector bundle ξ, there is a bundle map . Hence, , where is the map between the base space of ξ and that is induced by g. Hence, if , then would be trivial for every n-dimensional bundle ξ. However, this is wrong as if we set , where is the product bundle with fibre over , then
and the latter class is non-trivial as is isomorphic to the Möbius bundle over , which is non-orientable and so has a non-trivial first Stiefel–Whitney class.
If we now let be the canonical inclusion map, then is isomorphic to and we obtain
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About the article
Published Online: 2016-07-12
Published in Print: 2018-08-01