In this section we denote by ${G}_{n}({\mathbb{R}}^{2n})$ the Grassmannian manifold of all *n*-dimensional subspaces of ${\mathbb{R}}^{2n}$ (cf. [12, §5]).
Let us recall that ${G}_{n}({\mathbb{R}}^{2n})$ is compact and that for every $n\in \mathbb{N}$ there is a canonical vector bundle ${\gamma}^{n}({\mathbb{R}}^{2n})$ over ${G}_{n}({\mathbb{R}}^{2n})$ which is called the *tautological bundle* and which has as total space

$\{(V,u)\in {G}_{n}({\mathbb{R}}^{2n})\times {\mathbb{R}}^{2n}:u\in V\}.$

In what follows we denote by $I=[0,1]$ the compact unit interval, and we let Λ be a compact manifold of dimension $m\ge 2$.
We denote by ${H}^{1}(I,{\mathbb{R}}^{n})$ the space of all absolutely continuous functions having a square integrable derivative with respect to the usual Sobolev norm

${\parallel u\parallel}_{{H}^{1}}^{2}:={\int}_{0}^{1}\u3008u,u\u3009\mathit{d}t+{\int}_{0}^{1}\u3008{u}^{\prime},{u}^{\prime}\u3009\mathit{d}t,$

and we let

$\phi :\mathrm{\Lambda}\times I\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$

be a continuous function such that each ${\phi}_{\lambda}:=\phi (\lambda ,\cdot ,\cdot ):I\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ is smooth, all its derivatives depend continuously on λ, and $\phi (\lambda ,t,0)=0$ for all $(\lambda ,t)\in \mathrm{\Lambda}\times I$. For a continuous map $b:\mathrm{\Lambda}\to {G}_{n}({\mathbb{R}}^{2n})$, we consider the family of differential equations

$\{\begin{array}{cc}& {u}^{\prime}(t)={\phi}_{\lambda}(t,u(t)),t\in [0,1],\hfill \\ & (u(0),u(1))\in b(\lambda ),\hfill \end{array}$(3.1)

and we note that $u\equiv 0$ is a solution for all $\lambda \in \mathrm{\Lambda}$.

#### Definition 3.1.

A parameter value ${\lambda}_{0}\in \mathrm{\Lambda}$ is called a bifurcation point of (3.1) if in every neighbourhood of $({\lambda}_{0},0)\in \mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{n})$ there is $(\lambda ,u)$ such that $u\not\equiv 0$ is a solution of (3.1).

In what follows we denote by $B\subset \mathrm{\Lambda}$ the set of all bifurcation points. The linearisation of (3.1) at 0 is

$\{\begin{array}{cc}& {u}^{\prime}(t)=({D}_{0}{\phi}_{\lambda})(t,\cdot )u(t),t\in [0,1],\hfill \\ & (u(0),u(1))\in b(\lambda ),\hfill \end{array}$(3.2)

where ${D}_{0}{\phi}_{\lambda}(t,\cdot ):{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ denotes the derivative of ${\phi}_{\lambda}(t,u)$ with respect to *u* at $0\in {\mathbb{R}}^{n}$.

We can now state our main theorem of this section.

#### Theorem 3.2.

*If there is some ${\lambda}_{\mathrm{0}}\mathrm{\in}\mathrm{\Lambda}$ such that (3.2) has only the trivial solution and*

$0\ne {b}^{\ast}\left({w}_{k}({\gamma}^{n}({\mathbb{R}}^{2n}))\right)\in {H}^{k}(\mathrm{\Lambda};{\mathbb{Z}}_{2})$

*for some $\mathrm{1}\mathrm{\le}k\mathrm{\le}m\mathrm{-}\mathrm{1}$, then the dimension of **B* is at least $m\mathrm{-}k$ and *B* is not a contractible topological space.

Let us note that ${G}_{n}({\mathbb{R}}^{2n})$ is itself a compact manifold, which is orientable as Grassmannians ${G}_{n}({\mathbb{R}}^{l})$ are orientable if and only if *l* is even (cf. [3]). We obtain the following corollary for $\mathrm{\Lambda}={G}_{n}({\mathbb{R}}^{2n})$ and *b* the identity.

#### Corollary 3.3.

*If $n\mathrm{\ge}\mathrm{2}$ and there is some ${V}_{\mathrm{0}}\mathrm{\in}{G}_{n}\mathit{}\mathrm{(}{\mathrm{R}}^{\mathrm{2}\mathit{}n}\mathrm{)}$ such that (3.2) has only the trivial solution, then **B* is a non-contractible space which has at most codimension 1 in ${G}_{n}\mathit{}\mathrm{(}{\mathrm{R}}^{\mathrm{2}\mathit{}n}\mathrm{)}$.

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

${\mathcal{\mathscr{H}}}_{\lambda}:=\{u\in {H}^{1}(I,{\mathbb{R}}^{n}):(u(0),u(1))\in b(\lambda )\},$

and we note that each one of the following maps is ${C}^{1}$:

${f}_{\lambda}:{\mathcal{\mathscr{H}}}_{\lambda}\to {L}^{2}(I,{\mathbb{R}}^{n}),{f}_{\lambda}(u)={u}^{\prime}-{\phi}_{\lambda}(\cdot ,u).$

Moreover,

$({D}_{v}{f}_{\lambda})u={u}^{\prime}-({D}_{v}{\phi}_{\lambda})u,$

and we leave it to the reader to check that each ${D}_{v}{f}_{\lambda}:{\mathcal{\mathscr{H}}}_{\lambda}\to {L}^{2}(I,{\mathbb{R}}^{n})$ is a Fredholm operator of index 0.

Note that we cannot apply Theorem 2.1 to the operators ${f}_{\lambda}$ 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*

$\mathcal{\mathscr{H}}:=\{(\lambda ,u)\in \mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{n}):u\in {\mathcal{\mathscr{H}}}_{\lambda}\}$

*is a Hilbert subbundle of the product bundle $\mathrm{\Lambda}\mathrm{\times}{H}^{\mathrm{1}}\mathit{}\mathrm{(}I\mathrm{,}{\mathrm{R}}^{n}\mathrm{)}$.*

#### Proof.

Let $\stackrel{~}{P}:{G}_{n}({\mathbb{R}}^{2n})\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^{2n}$ be a family of projections in ${\mathbb{R}}^{2n}$ such that $\mathrm{im}({\stackrel{~}{P}}_{V})$ is the fibre of ${\gamma}^{n}({\mathbb{R}}^{2n})$ over *V* for $V\in {G}_{n}({\mathbb{R}}^{2n})$. We consider the family

$P:\mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{n})\to {H}^{1}(I,{\mathbb{R}}^{n})$

defined by

$({P}_{\lambda}u)(t)=u(t)-(1-t){P}_{1}({I}_{{\mathbb{R}}^{2n}}-{\stackrel{~}{P}}_{b(\lambda )})(u(0),u(1))-t{P}_{2}({I}_{{\mathbb{R}}^{2n}}-{\stackrel{~}{P}}_{b(\lambda )})(u(0),u(1)),$

where ${P}_{1},{P}_{2}:{\mathbb{R}}^{2n}\to {\mathbb{R}}^{n}$ are the projections on the first and last *n* components in ${\mathbb{R}}^{2n}$, respectively, and ${I}_{{\mathbb{R}}^{2n}}$ is the identity in ${\mathbb{R}}^{2n}$.
The reader can easily check that ${P}_{\lambda}^{2}={P}_{\lambda}$ and $\mathrm{im}({P}_{\lambda})={\mathcal{\mathscr{H}}}_{\lambda}$, i.e., ${P}_{\lambda}$ is a projection in ${H}^{1}(I,{\mathbb{R}}^{n})$ onto ${\mathcal{\mathscr{H}}}_{\lambda}$.
Hence,

$\{(\lambda ,u)\in \mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{2n}):u\in \mathrm{im}({P}_{\lambda})\}=\mathcal{\mathscr{H}}$

is a subbundle of $\mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{2n})$, 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 $\psi :\mathrm{\Lambda}\times H\to \mathcal{\mathscr{H}}$ for some separable Hilbert space *H*.
Moreover, by [10, Theorem VII.3.1] we can assume without loss of generality that each ${\psi}_{\lambda}:H\to {\mathcal{\mathscr{H}}}_{\lambda}$ is orthogonal and so an isometry.
Composing ψ and *f*, we obtain a family of ${C}^{1}$ maps

$\stackrel{~}{f}:=f\circ \psi :\mathrm{\Lambda}\times H\to {L}^{2}(I,{\mathbb{R}}^{n})$

such that $f(\lambda ,0)=0$ for all $\lambda \in \mathrm{\Lambda}$, and each ${\stackrel{~}{f}}_{\lambda}:H\to {L}^{2}(I,{\mathbb{R}}^{n})$ is Fredholm of index 0.
Consequently, we can now apply Theorem 2.1 to the family $\stackrel{~}{f}$, and so we need to compute $\mathrm{ind}(\stackrel{~}{L})\in \stackrel{~}{KO}(\mathrm{\Lambda})$, where ${\stackrel{~}{L}}_{\lambda}={D}_{0}{\stackrel{~}{f}}_{\lambda}$, $\lambda \in \mathrm{\Lambda}$.
By the chain rule, we get

${\stackrel{~}{L}}_{\lambda}=({D}_{{\psi}_{\lambda}(0)}{f}_{\lambda})({D}_{0}{\psi}_{\lambda})=({D}_{0}{f}_{\lambda}){\psi}_{\lambda}.$

Consequently, if we introduce two Hilbert bundle morphisms

$M:\mathrm{\Lambda}\times H\to \mathcal{\mathscr{H}},{M}_{\lambda}u={\psi}_{\lambda}u,$

and

$L:\mathcal{\mathscr{H}}\to \mathrm{\Lambda}\times {L}^{2}(I,{\mathbb{R}}^{n}),{L}_{\lambda}u=({D}_{0}{f}_{\lambda})u={u}^{\prime}-({D}_{0}{\phi}_{\lambda})u,$

then ${\stackrel{~}{L}}_{\lambda}={L}_{\lambda}{M}_{\lambda}$, $\lambda \in \mathrm{\Lambda}$.

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 $\mathcal{\mathcal{X}}$, $\mathcal{\mathcal{Y}}$ are Banach bundles over a compact base Λ, and $A:\mathcal{\mathcal{X}}\to \mathcal{\mathcal{Y}}$ is a bundle morphism such that every ${A}_{\lambda}:{\mathcal{\mathcal{X}}}_{\lambda}\to {\mathcal{\mathcal{Y}}}_{\lambda}$ is Fredholm, then

$\mathrm{ind}(A)=[E(A,\mathcal{\mathcal{V}})]-[\mathcal{\mathcal{V}}]\in \stackrel{~}{KO}(\mathrm{\Lambda}),$

where $\mathcal{\mathcal{V}}\subset \mathcal{\mathcal{Y}}$ is a finite dimensional subbundle such that

$\mathrm{im}({A}_{\lambda})+{\mathcal{\mathcal{V}}}_{\lambda}={\mathcal{\mathcal{Y}}}_{\lambda},\lambda \in \mathrm{\Lambda}.$

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

$\mathrm{ind}(\stackrel{~}{L})=\mathrm{ind}(LM)=\mathrm{ind}(L)+\mathrm{ind}(M)=\mathrm{ind}(L)\in \stackrel{~}{KO}(\mathrm{\Lambda}).$(3.3)

Let us now consider the Fredholm morphism

$\widehat{L}:\mathcal{\mathscr{H}}\to \mathrm{\Lambda}\times {L}^{2}(I,{\mathbb{R}}^{n}),{\widehat{L}}_{\lambda}u={u}^{\prime}.$

Then ${L}_{\lambda}-{\widehat{L}}_{\lambda}:{\mathcal{\mathscr{H}}}_{\lambda}\to {L}^{2}(I,{\mathbb{R}}^{n})$ is compact for all $\lambda \in \mathrm{\Lambda}$ and so, by [21, Corollary 7],

$\mathrm{ind}(L)=\mathrm{ind}(\widehat{L})\in \stackrel{~}{KO}(\mathrm{\Lambda})$(3.4)

Our next aim is to compute $\mathrm{ind}(\widehat{L})$. Let us denote by ${Y}_{1}$ the *n*-dimensional subspace of constant functions in ${L}^{2}(I,{\mathbb{R}}^{n})$ and

${Y}_{2}:=\{u\in {L}^{2}(I,{\mathbb{R}}^{n}):{\int}_{0}^{1}u(t)\mathit{d}t=0\}.$

Clearly, ${Y}_{1}\cap {Y}_{2}=\{0\}$ and, as every function $u\in {L}^{2}(I,{\mathbb{R}}^{n})$ can be written as

$u(t)=\left(u(t)-{\int}_{0}^{1}u(s)\mathit{d}s\right)+{\int}_{0}^{1}u(s)\mathit{d}s,$

we see that ${L}^{2}(I,{\mathbb{R}}^{n})={Y}_{1}\oplus {Y}_{2}$.
If $v\in {Y}_{2}$, then

$u(t)={\int}_{0}^{t}v(s)\mathit{d}s,t\in I,$

is an element of ${\mathcal{\mathscr{H}}}_{\lambda}$ which is mapped to *v*. Consequently, $\mathrm{im}({\widehat{L}}_{\lambda})+{Y}_{1}={L}^{2}(I,{\mathbb{R}}^{n})$ for all $\lambda \in \mathrm{\Lambda}$, and we get that

$E(\widehat{L},{Y}_{1})=\{(\lambda ,u)\in \mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{n}):u\in {\mathcal{\mathscr{H}}}_{\lambda},{\widehat{L}}_{\lambda}u\in {Y}_{1}\}$$=\{(\lambda ,u)\in \mathrm{\Lambda}\times {H}^{1}(I,{\mathbb{R}}^{n}):u(t)=(1-t)a+tb,(a,b)\in b(\lambda )\}$$\cong \{(\lambda ,a,b)\in \mathrm{\Lambda}\times {\mathbb{R}}^{2n}:(a,b)\in b(\lambda )\}.$

As this vector bundle is isomorphic to the pullback ${b}^{\ast}({\gamma}^{n}({\mathbb{R}}^{2n}))$ of the tautological bundle ${\gamma}^{n}({\mathbb{R}}^{2n})$ over ${G}_{n}({\mathbb{R}}^{2n})$ by *b*, we finally obtain, by (3.3) and (3.4),

$\mathrm{ind}(\stackrel{~}{L})=\mathrm{ind}(L)=\mathrm{ind}(\widehat{L})=[E(\widehat{L},{Y}_{1})]-[{Y}_{1}]=[{b}^{\ast}({\gamma}^{n}({\mathbb{R}}^{2n}))]-[{\mathrm{\Theta}}^{n}]\in \stackrel{~}{KO}(\mathrm{\Lambda}),$

where ${\mathrm{\Theta}}^{n}$ denotes the product bundle with fibre ${\mathbb{R}}^{n}$ over Λ. Hence,

${w}_{k}(\mathrm{ind}(\stackrel{~}{L}))={w}_{k}\left({b}^{\ast}({\gamma}^{n}({\mathbb{R}}^{2n}))\right)={b}^{\ast}{w}_{k}({\gamma}^{n}({\mathbb{R}}^{2n}))\ne 0\in {H}^{k}(\mathrm{\Lambda};{\mathbb{Z}}_{2}),$

by the assumption of Theorem 3.2. Moreover, as (3.2) has only the trivial solution for $\lambda ={\lambda}_{0}$, we see that the operator ${L}_{{\lambda}_{0}}:{\mathcal{\mathscr{H}}}_{{\lambda}_{0}}\to {L}^{2}(I,{\mathbb{R}}^{n})$ is injective and so invertible as it is Fredholm of index 0.
Since ${\psi}_{{\lambda}_{0}}:H\to {\mathcal{\mathscr{H}}}_{{\lambda}_{0}}$ is invertible, we conclude that ${\stackrel{~}{L}}_{{\lambda}_{0}}$ is invertible as well.
Consequently, from Theorem 2.1, we obtain that the family of ${C}^{1}$-Fredholm maps $\stackrel{~}{f}$ has a bifurcation point ${\lambda}^{\ast}\in \mathrm{\Lambda}$, i.e., there is a sequence ${\{({\lambda}_{n},{u}_{n})\}}_{n\in \mathbb{N}}\subset \mathrm{\Lambda}\times H$ converging to $({\lambda}^{\ast},0)$ such that ${u}_{n}\ne 0$ and ${\stackrel{~}{f}}_{{\lambda}_{n}}({u}_{n})={f}_{{\lambda}_{n}}({\psi}_{{\lambda}_{n}}{u}_{n})=0$ for all $n\in \mathbb{N}$.
We now set ${v}_{n}:={\psi}_{{\lambda}_{n}}{u}_{n}\ne 0\in {\mathcal{\mathscr{H}}}_{{\lambda}_{n}}$ for $n\in \mathbb{N}$.
As ${\psi}_{{\lambda}_{n}}:H\to {\mathcal{\mathscr{H}}}_{{\lambda}_{n}}$ is an isometry, it follows that ${\parallel {v}_{n}\parallel}_{{H}^{1}}={\parallel {u}_{n}\parallel}_{H}\to 0$ as $n\to \mathrm{\infty}$, where $\parallel \cdot {\parallel}_{H}$ is the norm of *H*.
Since ${v}_{n}\ne 0$ is a solution of (3.1) by the definition of *f*, we obtain that ${\lambda}^{\ast}$ 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 $0\ne {w}_{1}({\gamma}^{n}({\mathbb{R}}^{2n}))\in {H}^{1}({G}_{n}({\mathbb{R}}^{2n});{\mathbb{Z}}_{2})$.
Indeed, if we consider the tautological bundle ${\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}})$ over the infinite Grassmannian ${G}_{n}({\mathbb{R}}^{\mathrm{\infty}})$, then by [12, Theorem 5.6], for every *n*-dimensional vector bundle ξ, there is a bundle map $g:\xi \to {\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}})$.
Hence, ${w}_{1}(\xi )={\overline{g}}^{\ast}{w}_{1}({\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}}))$, where $\overline{g}$ is the map between the base space
of ξ and ${G}_{n}({\mathbb{R}}^{\mathrm{\infty}})$ that is induced by *g*.
Hence, if ${w}_{1}({\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}}))=0$, then ${w}_{1}(\xi )$ would be trivial for every *n*-dimensional bundle ξ.
However, this is wrong as if we set $\xi ={\gamma}^{1}({\mathbb{R}}^{2})\oplus {\mathrm{\Theta}}^{n-1}$, where ${\mathrm{\Theta}}^{n-1}$ is the product bundle with fibre ${\mathbb{R}}^{n-1}$ over ${G}_{1}({\mathbb{R}}^{2})$, then

${w}_{1}(\xi )={w}_{1}({\gamma}^{1}({\mathbb{R}}^{2}))+{w}_{1}({\mathrm{\Theta}}^{n-1})={w}_{1}({\gamma}^{1}({\mathbb{R}}^{2})),$

and the latter class is non-trivial as ${\gamma}^{1}({\mathbb{R}}^{2})$ is isomorphic to the Möbius bundle over ${S}^{1}$, which is non-orientable and so has a non-trivial first Stiefel–Whitney class.

If we now let $\iota :{G}_{n}({\mathbb{R}}^{2n})\hookrightarrow {G}_{n}({\mathbb{R}}^{\mathrm{\infty}})$ be the canonical inclusion map, then ${\iota}^{\ast}({\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}}))$ is isomorphic to ${\gamma}^{n}({\mathbb{R}}^{2n})$ and we obtain

${\iota}^{\ast}{w}_{1}({\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}}))={w}_{1}\left({\iota}^{\ast}({\gamma}^{n}({\mathbb{R}}^{\mathrm{\infty}}))\right)={w}_{1}({\gamma}^{n}({\mathbb{R}}^{2n}))\in {H}^{1}({G}_{n}({\mathbb{R}}^{2n});{\mathbb{Z}}_{2}).$

As ${\iota}^{\ast}:{H}^{1}({G}_{n}({\mathbb{R}}^{\mathrm{\infty}});{\mathbb{Z}}_{2})\to {H}^{1}({G}_{n}({\mathbb{R}}^{2n});{\mathbb{Z}}_{2})$ is an isomorphism for $n\ge 2$ by [12, Problem 6-B], we conclude that ${w}_{1}({\gamma}^{n}({\mathbb{R}}^{2n}))$ is non-trivial, and so Corollary 3.3 is proved.

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