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On the homotopy classification of proper Fredholm maps into a Hilbert space

Alberto Abbondandolo and Thomas O. Rot

Abstract

We classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli–Smale mod 2 degree and the absolute value of the oriented degree.

A Extension of Fredholm maps

The aim of this appendix is to prove Theorem 2.1 on the extension of Fredholm maps.

By the already cited theorem of Eells and Elworthy [11], every Hilbert manifold has an open embedding into its model. Therefore, we may assume that M and N are open subsets of . The trivializations of TM and TN which we use to write the differential of a Fredholm map between M and N as a Φ()-valued map on M are the ones induced by these embeddings.

Up to the regularization of A, see Lemma 1.3, and up to the choice of a smaller V, we may assume that the map A is smooth. For any xM set

δ0(x):=dist(g(x),Nc)>0,δ1(x):=dist(A(x),Φn()c)>0,

where Nc and Φn()c denote the complements of N and Φn() in and L(), respectively. We set δ0(x):=+ if N=. Now we choose r(x)>0 to be so small that:

  1. Br(x)(x)M,

  2. if Br(x)(x)U, then Br(x)(x)V,

  3. conv¯g(Br(x)(x)){zNz-g(x)<δ0(x)2},

  4. conv¯A(Br(x)(x)){TΦn()T-A(x)L()<δ1(x)2},

  5. r(x)sup{A(y)L()yBr(x)(x)}<δ0(x)4,

  6. r(x)sup{dA(y)L(,L())yBr(x)(x)}<δ1(x)2.

Here conv¯ denotes the closed convex hull. Let 𝒰={UjjJ} be a locally finite star-refinement of the open covering {Br(x)(x)xM}. That is,

(A.1)Uj1UjkUj1UjkBr(x)(x) for some xM.

Let {φjjJ} be a smooth partition of unity subordinated to 𝒰, and for every jJ fix a point xj in Uj. We define the smooth maps

f¯(x):=jJφj(x)(g(xj)+01A(xj+s(x-xj))(x-xj)𝑑s),xM,

and

h(t,x):=tf¯(x)+(1-t)g(x),(t,x)[0,1]×M.

Fix a point x in M and let Uj1,,Ujk be the elements of the covering 𝒰 which contain x, so that

i=1kφji(x)=1,

and φj(x)=0 if j is not in {j1,,jk}. By property (A.1), there is a point y in M such that

(A.2)xUj1UjkBr(y)(y).

Then x-xji<2r(y) for every i=1,,k, and the norm of the vector

h(t,x)-i=1kφji(x)(tg(xji)+(1-t)g(x))=ti=1kφji(x)01A(xji+s(x-xji))(x-xji)𝑑s

does not exceed the quantity

2r(y)supzBr(y)(y)A(z)L(),

which by (e) is smaller than δ0(y)2. On the other hand, (c) implies that

g(y)-i=1kφji(x)(tg(xji)+(1-t)g(x))<δ0(y)2.

Therefore,

h(t,x)-g(y)<δ0(y),

and by the definition of δ0 we conclude that h and, a fortiori, f¯ take values into N.

If, moreover, x belongs to U, then (b) guarantees that Br(y)(y) is contained in V, so by (A.2), (i) and (ii),

h(t,x)=tj=1kφji(x)(f(xji)+01𝑑f(xji+s(x-xji))(x-xji)𝑑s)+(1-t)f(x)
=tj=1kφji(x)(f(xji)+01ddsf(xji+s(x-xji))𝑑s)+(1-t)f(x)
=tj=1kφji(x)f(x)+(1-t)f(x)
=f(x),

hence f¯|U=f|U and h is a homotopy between f¯ and g relative U, proving (i’).

The differential of f¯ at x is the operator

df¯(x)=A1(x)+A2(x)+A3(x),

where

A1(x)u:=j=1kdφji(x)[u](g(xji)+01A(xji+s(x-xji))(x-xji)𝑑s),u,
A2(x):=i=1kφji(x)01A(xji+s(x-xji))𝑑s,
A3(x)u:=i=1kφji(x)01s𝑑A(xji+s(x-xji))[u](x-xji)𝑑s,u.

The operator A1(x) has rank at most k. By (A.2) and (d),

(A.3)A2(x)-A(y)L()<δ1(y)2.

By (A.2) and (f),

(A.4)A3(x)L()2r(y)supzBr(y)(y)dA(z)L(,L())01s𝑑s<δ1(y)2.

By (A.3) and (A.4),

A2(x)+A3(x)-A(y)L()<δ1(y),

so by the definition of δ1, A2(x)+A3(x) belongs to Φn(). Since A1(x) has finite rank, df¯(x) belongs to Φn() too. Hence, f¯ is a Fredholm map of index n.

By using again (A.2) and (d),

tA2(x)+(1-t)A(x)-A(y)L()<δ1(y)2,t[0,1].

Together with (A.4) and the fact that A1(x) has finite rank, this implies that the homotopy

H(t,x):=tdf¯(x)+(1-t)A(x)=tA1(x)+(tA2(x)+(1-t)A(x))+tA3(x)

takes values into Φn(). If, moreover, x belongs to U, then A(x)=df(x)=df¯(x). Therefore,

H(t,x)=df(x),xU,

and H is a homotopy relative U, proving (ii’).

B Proper extension of Fredholm maps

The aim of this appendix is to prove Theorem 3.1 on the extension of proper Fredholm maps. The proof follow closely the proof of [12, Lemma 4.2], which has slightly stronger assumptions.

We start by recalling the following result of Bessaga [4]. See also [5, Section III.6].

Theorem B.1.

There exists a diffeomorphism d1:HH{0} which is the identity outside the ball of radius 12.

Let Br(x) be the open ball in with center x and radius r, and let S:=B1(0) be the unit sphere in . A famous corollary of Bessaga’s theorem is that S is diffeomorphic to .

Corollary B.2.

There exists a diffeomorphism d2:SH.

Proof.

Let x0S. We first construct a diffeomorphism SS{x0} by applying d1 in a chart around x0. The stereographic projection ϕ:S{x0}x0 defined by

ϕ(x)=x-x,x0x01-x,x0

is a diffeomorphism onto the hyperplane x0 orthogonal to x0, which in turn is diffeomorphic to . By composition, we obtain a diffeomorphism d2:S. ∎

Bessaga’s theorem allows us to construct various other useful maps, for example an involution on which exchanges the interior with the exterior of S:

Corollary B.3.

There is a diffeomorphism J:HH such that

J|S=id,J2=id,𝑎𝑛𝑑J(B1(0))=B¯1(0).

The map J is Fredholm homotopic to id relative S.

Proof.

Consider the diffeomorphism j:{0}{0} given by

j(x)=xx2.

Then j2=id and the diffeomorphism J: defined by J=d1-1jd1 also satisfies J2=id. Since d1 maps the interior of the unit ball to itself, and the complement of the unit ball to itself, we have J(B1(0))=B¯1(0).

For every t[0,1] let jt:{0}{0} be defined by jt(x)=λt(x)x, with the smooth function λt:{0} defined by

λt(x)=1-t+tx2.

Since λt(x)>0 for every t[0,1] and x0, this map is well defined and Fredholm: indeed, its differential

djt(x)=xdλt(x)+λt(x)id

is a Fredholm operator, being a perturbation of rank at most one of the isomorphism λt(x)id. Then Jt=d1-1jtd1 is the required Fredholm homotopy. For each xS we see that jt(x)=x, so Jt is a homotopy relative S. ∎

We will need the following extension result for proper Fredholm maps into the unit sphere:

Lemma B.4.

Let M be a Hilbert manifold and f:MS a Fredholm map into the infinite-dimensional sphere of H. Let AM be a closed set and V a neighborhood of A such that f|V¯ is proper (A and V might be empty). Then there exists a smooth function λ:M[1,+) such that Aλ-1(1)V and the map g:MH defined by

g(x):=λ(x)f(x)

is proper and Fredholm of index indf-1.

Proof.

Since Fredholm maps are locally proper (see e.g. [24, Theorem 1.6]), every xM has a neighborhood U such that f|U¯ is proper. Since M is paracompact, we can find a countable locally finite open cover 𝒰={Uj}j of MA such that the restriction f|Uj¯ is proper. Set U0:=V and consider the covering 𝒰{U0} of M. Let {ϕj}j0, 0={0}, be a smooth partition of unity subordinated to the latter covering such that

(B.1)Aϕ0-1(1).

Consider the function

λ:M[1,+),λ:=j=02jϕj,

and set g:=λf. By (B.1) we have Aλ-1(1). If xV=U0, then ϕ0(x)=0 and hence λ(x)2; this implies that λ-1(1)V. For every n the set λ-1([1,2n]) is contained in the union U0U1Un, because on the complement of this union all ϕj with 0jn vanish and hence λ2n+1. The restrictions g|U¯j are proper, because g(M)B1(0) and for every KB1(0) there holds

(g|Uj¯)-1(K)(f|Uj¯)-1(π(K)),

where π:{0}S is the radial projection onto the sphere. If K is compact, it is in particular closed and the set (g|Uj¯)-1(K) is a closed subset of a compact set as f is proper when restricted to Uj¯, thus (g|Uj¯)-1(K) is compact.

It follows that the restriction of g to any finite union of the sets Uj¯ is proper, and from the fact that for every R1 the set λ-1([1,R]) is contained in finitely many of the sets Uj¯ we conclude that g is proper. At a point xM we have

dg(x)=f(x)dλ(x)+λ(x)df(x).

The map f(x)dλ(x) has rank at most one, and since λ(x) does not vanish, it follows that λ(x)df(x):TxM is a Fredholm operator whose index equals the index of the differential df(x):TxMTf(x)S minus 1. Hence g is a Fredholm map with index indf-1. ∎

The main step in the proof of Theorem 3.1 is contained in the following result. The difference with respect to [12, Lemma 4.2] is that the point z is not required to belong to f(U), an assumption which turns out to be superfluous.

Lemma B.5.

Let M be a Hilbert manifold and let U,V be open subsets of M such that U¯V. Let f:MH be a Fredholm map of index n such that f|V¯ is proper. Moreover, assume that there exists a point z in Hf(U). Then there exists a proper Fredholm map f¯:MH of index n such that f¯|U=f|U and

  1. f¯ and f are Fredholm homotopic relative U,

  2. there exists a neighborhood Z of z such that f¯-1(Z)=f-1(Z)U.

Proof.

By Corollary B.2 we can replace the Hilbert space by the Hilbert sphere S in this statement: f is a Fredholm map of index n from M to S such that f|V¯ is proper, z is a point in Sf(U) and we must find a proper Fredholm map f¯:MS of the same index such that f¯|U=f|U and (a) and (b) hold.

A proper map into a metric space is closed and hence f(U) is closed. Let YS be an open neighborhood of f(U) such that z does not belong to Y¯. Let V1 be an open neighborhood of U such that f(V1)Y and V1¯V. Then the open set V0:=UV1 satisfies

U¯V0V0¯V

and is such that z does not belong to f(V0¯U). Let A be a closed neighborhood of U¯ contained in V0. By Lemma B.4 there exists a smooth function λ:M[1,+) such that

Aλ-1(1)V0

and g1:=λf:M is a proper Fredholm map. The index of g1 is n-1 and its image is contained in B1(0). Let i:S denote the inclusion mapping. Then g1 is Fredholm homotopic to if relative U via the homotopy

h~t:=tg1+(1-t)if=(tλ+1-t)if,t[0,1].

Indeed, the formula

dh~t(x)=tf(x)dλ(x)+(tλ(x)+1-t)df(x)

shows that the operator dh~t(x) is Fredholm, being a finite rank perturbation of the Fredholm operator (tλ(x)+1-t)df(x).

By construction, z does not belong to g1(MU) and since the latter set is closed, there exists an open ball B containing z such that Bg1(MU)=. Set

g2:=Jg1:M,

with J as in Corollary B.3. The image of g2 is contained in the closed unit ball of . Since J is a diffeomorphism, g2 is Fredholm homotopic to g1 relative U by the corresponding property of J stated in Corollary B.3. Therefore, g2 is Fredholm homotopic to if relative U.

Since z is fixed by J, we can find an open ball B0 containing z and such that B0J(B). It follows that B0g2(MU)=. Let

r:B¯1(0){z}B¯1(0){z}

be the retraction onto S{z} which is defined by mapping each xB¯1(0){z} into the intersection of the ray emanating from z in the direction of x and the set S{z}. This map is Fredholm and is Fredholm homotopic to the identity mapping on B¯1(0){z} relative S{z} by convex interpolation. The retraction r does not extend continuously to z. However, since g2 maps a neighborhood of U into S{z}, the map

g3:M,g3(x)={r(g2(x))if xMU,g2(x)if xU,

is smooth. The map g3 is Fredholm of index n-1 and is proper, because g2 is proper and r is proper on B¯1(0)B0. Moreover, the fact that r is Fredholm homotopic to the identity relative S{z} implies that g3 is Fredholm homotopic to g2 relative U. Therefore, g3 is Fredholm homotopic to if relative U. Denote by

h:[0,1]×M

such a Fredholm homotopy. Since the image of g3 is contained in S, we can restrict its codomain and obtain a proper Fredholm map f¯:MS of index n. The image of [0,1]×U by the map h is contained in S. Let π:{0}S be the radial projection and d1:{0} the diffeomorphism of Theorem B.2. Then

πd1h:[0,1]×MS

is a Fredholm homotopy, relative U, between f=πd1if and f¯=πd1g3. Therefore, the proper Fredholm map f¯ satisfies f¯|U=f|U and (a). It also satisfies (b) with Z:=B0S. ∎

We can finally prove Theorem 3.1

Proof of Theorem 3.1.

By Corollary 2.4 we can find a Fredholm map g:M such that g|V=f|V and

  1. dg:MΦn() is homotopic to A relative V.

Then g|V¯=f|V¯ is proper and z belongs to g(U). By Lemma B.5 applied to g we can find a proper Fredholm map f¯:M which coincides with g|U=f|U on U and such that:

  1. f¯ and g are Fredholm homotopic relative U,

  2. there exists a neighborhood Z of z such that

    f¯-1(Z)=g-1(Z)U=f-1(Z)U.

Then the proper Fredholm extension f¯ of f|U satisfies (i’) by (a) and (b), and (ii’) by (c). ∎

C Weakening the hypotheses

In many applications to nonlinear partial differential equations, it is useful to weaken some of the assumptions that we made in this article: On the one hand, one would like to work with maps with finite regularity, on the other hand one would like to replace the Hilbert space by more general Banach spaces. In this concluding appendix, we briefly discuss these two issues.

All the notions which we have introduced above make sense when the Fredholm maps are just of class C1. Higher regularity is needed in the proofs when one wants to apply the Sard–Smale theorem, see Theorem 1.1. By taking into account the minimal regularity needed in this theorem, we can make the following observations: Theorems 1 and 2 hold for Fredholm maps of class Cn+2, Theorem 3 just requires C1 maps, and Theorem 4 needs C2 regularity.

Using regularization by smooth partitions of unity, we can show that any C1 Fredholm map f:MN is Fredholm homotopic to a smooth Fredholm map f~:MN. Indeed, one can smoothen f and keep the smoothing Fredholm by seeing N as an open subset of (see [11] again) and by setting

f~(x):=nϕn(x)(f(xj)+df(xj)[x-xj]),xM,

where {ϕn}n is a suitable smooth partition of unity on M. Using this fact, one can easily show that Theorem 1 actually holds for Fredholm maps of class C1.

Unfortunately, we do not know whether it is possible to regularize proper Fredholm maps by keeping also the properness, and hence get the validity of Theorems 2 and 4 for C1 maps just by regularization. In the literature, various alternative approaches have been developed in order to deal with this difficulty, starting with [23] and [16]. A particularly interesting approach based on finite-dimensional reduction is developed in [28]. It produces a degree theory for C1 proper Fredholm maps of index zero between Banach manifolds, and also homotopy invariants for proper Fredholm maps of positive degree. It would be interesting to compare these invariant with the complete ones presented here.

Now we turn to the question of replacing by a more general Banach space 𝔼. Among the properties of the model space that we use, a couple stand out as crucial:

  1. 𝔼 is a Kuiper space, i.e. the general linear group GL(𝔼) is contractible. It is known that not all infinite-dimensional Banach spaces are Kuiper. For an overview of Kuiper and non-Kuiper Banach spaces, we refer to [20].

  2. 𝔼 is stable. This means that 𝔼 is isomorphic to ×𝔼. Non-stable Banach spaces exist. Gowers [15] was the first to construct an example of such a space.

  3. 𝔼 is diffeomorphic to its unit sphere. In [3] it is shown that any infinite-dimensional Banach space with Ck norm is Ck diffeomorphic to its unit sphere.

  4. For the explicit computation of the homotopy classes of proper Fredholm maps of non-positive index we use that the homotopy type of the space of Fredholm operators is known, namely it equals ×BO. Koschorke [17] has shown that the space of Fredholm operators of index zero of a infinite-dimensional separable Kuiper Banach space is homotopy equivalent to BO. Moreover, if one further assumes 𝔼 to be stable, the homotopy type of the space of all Fredholm operators on 𝔼 equals ×BO.

An inspection to our proofs and the above results show that it is possible to replace by a real separable Banach space 𝔼 which is Kuiper, stable and whose norm is sufficiently regular. Indeed, when working on Banach spaces, or Banach manifolds, an extra difficulty is the lack of smooth partitions of unity. Due to this fact, the proofs of Theorems 1, 2, 3 and 4 extend to paracompact manifolds modeled on a separable, stable and Kuiper Banach space 𝔼 when 𝔼 admits a norm of class Cn+2, respectively Cn+2, C1 and C2. Lowering the regularity assumptions to C1 as discussed above is hence relevant also for the extension to the Banach setting. We expect Theorems 1, 2, 3 and 4 to hold for Fredholm maps of class C1 on paracompact manifolds modeled on a separable, stable and Kuiper Banach space 𝔼 admitting a norm of class C1.

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Received: 2016-11-15
Revised: 2017-09-14
Published Online: 2018-01-23
Published in Print: 2020-02-01

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