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 , 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 and N as a -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 set
where and denote the complements of N and in and , respectively. We set if . Now we choose to be so small that:
if , then ,
Here denotes the closed convex hull. Let be a locally finite star-refinement of the open covering . That is,
Let be a smooth partition of unity subordinated to , and for every fix a point in . We define the smooth maps
Fix a point x in M and let be the elements of the covering which contain x, so that
and if j is not in . By property (A.1), there is a point y in M such that
Then for every , and the norm of the vector
does not exceed the quantity
which by (e) is smaller than . On the other hand, (c) implies that
and by the definition of we conclude that h and, a fortiori, take values into N.
If, moreover, x belongs to U, then (b) guarantees that is contained in V, so by (A.2), (i) and (ii),
hence and h is a homotopy between and g relative U, proving (i’).
The differential of at x is the operator
The operator has rank at most k. By (A.2) and (d),
By (A.2) and (f),
so by the definition of , belongs to . Since has finite rank, belongs to too. Hence, is a Fredholm map of index n.
By using again (A.2) and (d),
Together with (A.4) and the fact that has finite rank, this implies that the homotopy
takes values into . If, moreover, x belongs to U, then . Therefore,
and H is a homotopy relative U, proving (ii’).
B Proper extension of Fredholm maps
There exists a diffeomorphism which is the identity outside the ball of radius .
Let be the open ball in with center x and radius r, and let be the unit sphere in . A famous corollary of Bessaga’s theorem is that S is diffeomorphic to .
There exists a diffeomorphism .
Let . We first construct a diffeomorphism by applying in a chart around . The stereographic projection defined by
is a diffeomorphism onto the hyperplane orthogonal to , which in turn is diffeomorphic to . By composition, we obtain a diffeomorphism . ∎
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:
There is a diffeomorphism such that
The map J is Fredholm homotopic to relative S.
Consider the diffeomorphism given by
Then and the diffeomorphism defined by also satisfies . Since maps the interior of the unit ball to itself, and the complement of the unit ball to itself, we have .
For every let be defined by , with the smooth function defined by
Since for every and , this map is well defined and Fredholm: indeed, its differential
is a Fredholm operator, being a perturbation of rank at most one of the isomorphism . Then is the required Fredholm homotopy. For each we see that , so is a homotopy relative S. ∎
We will need the following extension result for proper Fredholm maps into the unit sphere:
Let M be a Hilbert manifold and a Fredholm map into the infinite-dimensional sphere of . Let be a closed set and V a neighborhood of A such that is proper (A and V might be empty). Then there exists a smooth function such that and the map defined by
is proper and Fredholm of index .
Since Fredholm maps are locally proper (see e.g. [24, Theorem 1.6]), every has a neighborhood U such that is proper. Since M is paracompact, we can find a countable locally finite open cover of such that the restriction is proper. Set and consider the covering of M. Let , , be a smooth partition of unity subordinated to the latter covering such that
Consider the function
and set . By (B.1) we have . If , then and hence ; this implies that . For every the set is contained in the union , because on the complement of this union all with vanish and hence . The restrictions are proper, because and for every there holds
where is the radial projection onto the sphere. If K is compact, it is in particular closed and the set is a closed subset of a compact set as f is proper when restricted to , thus is compact.
It follows that the restriction of g to any finite union of the sets is proper, and from the fact that for every the set is contained in finitely many of the sets we conclude that g is proper. At a point we have
The map has rank at most one, and since does not vanish, it follows that is a Fredholm operator whose index equals the index of the differential minus 1. Hence g is a Fredholm map with index . ∎
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 , an assumption which turns out to be superfluous.
Let M be a Hilbert manifold and let be open subsets of M such that . Let be a Fredholm map of index n such that is proper. Moreover, assume that there exists a point z in . Then there exists a proper Fredholm map of index n such that and
and f are Fredholm homotopic relative U,
there exists a neighborhood of z such that .
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 is proper, z is a point in and we must find a proper Fredholm map of the same index such that and (a) and (b) hold.
A proper map into a metric space is closed and hence is closed. Let be an open neighborhood of such that z does not belong to . Let be an open neighborhood of such that and . Then the open set satisfies
and is such that z does not belong to . Let A be a closed neighborhood of contained in . By Lemma B.4 there exists a smooth function such that
and is a proper Fredholm map. The index of is and its image is contained in . Let denote the inclusion mapping. Then is Fredholm homotopic to relative U via the homotopy
Indeed, the formula
shows that the operator is Fredholm, being a finite rank perturbation of the Fredholm operator .
By construction, z does not belong to and since the latter set is closed, there exists an open ball containing z such that . Set
with J as in Corollary B.3. The image of is contained in the closed unit ball of . Since J is a diffeomorphism, is Fredholm homotopic to relative U by the corresponding property of J stated in Corollary B.3. Therefore, is Fredholm homotopic to relative U.
Since z is fixed by J, we can find an open ball containing z and such that . It follows that . Let
be the retraction onto which is defined by mapping each into the intersection of the ray emanating from z in the direction of x and the set . This map is Fredholm and is Fredholm homotopic to the identity mapping on relative by convex interpolation. The retraction r does not extend continuously to z. However, since maps a neighborhood of into , the map
is smooth. The map is Fredholm of index and is proper, because is proper and r is proper on . Moreover, the fact that r is Fredholm homotopic to the identity relative implies that is Fredholm homotopic to relative U. Therefore, is Fredholm homotopic to relative U. Denote by
such a Fredholm homotopy. Since the image of is contained in S, we can restrict its codomain and obtain a proper Fredholm map of index n. The image of by the map h is contained in S. Let be the radial projection and the diffeomorphism of Theorem B.2. Then
is a Fredholm homotopy, relative U, between and . Therefore, the proper Fredholm map satisfies and (a). It also satisfies (b) with . ∎
We can finally prove Theorem 3.1
Proof of Theorem 3.1.
By Corollary 2.4 we can find a Fredholm map such that and
is homotopic to A relative V.
Then is proper and z belongs to . By Lemma B.5 applied to g we can find a proper Fredholm map which coincides with on U and such that:
and g are Fredholm homotopic relative U,
there exists a neighborhood of z such that
Then the proper Fredholm extension of 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 . 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 , Theorem 3 just requires maps, and Theorem 4 needs regularity.
Using regularization by smooth partitions of unity, we can show that any Fredholm map is Fredholm homotopic to a smooth Fredholm map . Indeed, one can smoothen f and keep the smoothing Fredholm by seeing N as an open subset of (see  again) and by setting
where 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 .
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 maps just by regularization. In the literature, various alternative approaches have been developed in order to deal with this difficulty, starting with  and . A particularly interesting approach based on finite-dimensional reduction is developed in . It produces a degree theory for 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 . Among the properties of the model space that we use, a couple stand out as crucial:
is a Kuiper space, i.e. the general linear group 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 .
is stable. This means that is isomorphic to . Non-stable Banach spaces exist. Gowers  was the first to construct an example of such a space.
is diffeomorphic to its unit sphere. In  it is shown that any infinite-dimensional Banach space with norm is diffeomorphic to its unit sphere.
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 . Koschorke  has shown that the space of Fredholm operators of index zero of a infinite-dimensional separable Kuiper Banach space is homotopy equivalent to . Moreover, if one further assumes to be stable, the homotopy type of the space of all Fredholm operators on equals .
An inspection to our proofs and the above results show that it is possible to replace by a real separable Banach space 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 when admits a norm of class , respectively , and . Lowering the regularity assumptions to 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 on paracompact manifolds modeled on a separable, stable and Kuiper Banach space admitting a norm of class .
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