Orientation of Fredholm maps between Hilbert manifolds
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Though one can define a notion of an oriention of a manifold in the setting of Banach manifolds (namely, for Fredholm manifolds) as was done in [Elworty&Tromba1970], it seems that the more natural concept is that of an orientation of a map between such manifolds. In this note we define a notion of an orientation of a Fredholm map between Hilbert manifolds. We don't claim any originality. Related notions were given in [Morava1968], [Fitzpatrick&Pejsachowicz&Rabier1994] and [Benevieri&Furi1998] for non-linear maps between Banach spaces. The definition of an orientation is given in terms of a line bundle, the determinant line bundle. Our construction of the determinant line bundle agrees with that of [Wang2005] but he defines orientation only for index 0 Fredholm maps (since he is interested in the degree of proper Fredholm maps). We also add some useful information like orientation of a composition of two maps and product orientations.
Contents |
1 Introduction
Given a Fredholm map we want to construct the determinant line bundle. The rough idea is to consider the tensor product of the determinant line bundle of the kernel "bundle" of the differential of
with the pullback of the determinant line bundle of the cokernel "bundle". Unless the dimension of
is constant, these are not bundles and thus one has to make sense of these constructions. We will do this by following Jänich's construction of an index bundle for a Fredholm map. This is similar to the construction of the determinant line bundle in [Quillen1985].
2 Definition
Let be a Fredholm map. We equip
and
with Riemannian metrics. For each
we will construct an open neighborhood
and a line bundle
over
depending on a choice. For different choices one has explicit isomorphisms which are compatible with restrictions to open subsets of
. These isomorphisms depend only on the metric and the differential of
and so one can glue these bundles together to obtain the desired global line bundle. Since the space of metrics is convex the resulting bundle is also independent of the metric.
To find such an open neighborhood of
and the desired line bundle over
we note that the arguments in [Jänich1965, Lemma 3] (he constructs for compact spaces a global object, but the first step is the local object, and that's all we need) implies that for each
in
there is an open neighborhood
and a finite dimensional subbundle
of
such that
for all
. Let
be the orthogonal projection to the orthogonal complement of
in
. Then
is a Fredholm bundle map and its kernel is the subbundle
[Jänich1965, p. 138]. Thus the cokernel bundle with fibre over
the cokernel of
is defined and denoted by
. To avoid signs, when
we define the product orientation and interchange the factors we always assume that
is even dimensional, which can be achieved by stabilizing
and by this
, if
necessary. We define a line bundle over
by
, where
denotes the determinant line bundle, and denote it by
. If
is open then
.
If is a subbundle, the differential of
and the Riemannian metric can be used to construct explicit isomorphisms
. For this we note that - using the Riemannian metric - we have an embedding of
into
and that Jänich shows ([Jänich1965], p. 138) that the differential induces an isomorphism
. Since for all
and
there is a common finite dimensional subbundle of
containing both one obtains isomorphisms
in general.
Now we glue all these line bundles over such open subsets in
together using the isomorphisms
and the fact that everything is compatible with restrictions and the construction of
based on the differential (and the global Riemannian metric) fulfills the cocycle conditions, since they are induced by differentials. This is our determinant line bundle
over
.
Definition 2.1.
An orientation of a Fredholm map is an orientation of
.
If we replace the orientation on
by the opposite orientation we call the corresponding Fredholm map
Lemma 2.2.
If has constant dimension then

Proof.
Take to be
globally.

3 Orientation of compositions
Given two Fredholm maps and
, it follows from the proof of [Jänich1965, Lemma 6] that there is an explicit isomorphism

Thus if both maps are oriented one obtains an induced orientation of the composition by using the isomorphism above to orient .
4 Product orientation
Given two oriented Fredholm maps
and
, we would like to orient the product map
. For this we consider the local construction of
and similarly
and obtain:
.
By interchanging the factors in the middle we identify this with
. Recall that we always assume the
is even
dimensional. Then we take the product orientation on
and equip
with the induced orientation given by the isomorphism
interchanging the middle factors.
5 Orientation of the cylinder and of the boundary of a Fredholm map
A useful thing for coboridsm is to orient the cylinder , where
is the projection to
. As explained before, this is an orientation of
, and since
, the trivial bundle, it is equivalent to an orientation of
and of
. If we change the orientation on
we obtain the opposite orientation. Using this and a collar we orient the boundary of a Fredholm map.
6 Orientation of maps in finite dimensions
This construction also works for maps between finite dimensional manifolds. In this case we can choose globally as
, then we get

If we denote by the infinite dimensional separable Hilbert space, then for the product map
we have

where is the projection from
to
, since in this case we note that we can choose
to be
.
Remark 6.1. The above implies that a map between oriented finite dimensional manifolds has a natural orientation in this sense.
7 References
- [Benevieri&Furi1998] T. P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory, Ann. Sci. Math. Québec 22 (1998), 131-148. MR1677235 (2000a:58024) Zbl 1058.58502
- [Elworty&Tromba1970] K. D. Elworthy and A. J. Tromba, Differential structures and Fredholm maps in Banach manifolds, Global analysis (Berkeley, 1968), Proc. Sympos. Pure Math., 15, pp. 45--94, Amer. Math. Soc., Providence, RI, 1970. MR0264708 (41 #9299) Zbl 0206.52504
- [Fitzpatrick&Pejsachowicz&Rabier1994] P. M. Fitzpatrick, J. Pejsachowicz, and P. J. Rabier, Orientability of Fredholm families and topological degree for orientable non-linear Fredholm mappings, J. Funct. Anal. 124 (1994), 1--39. MR1284601 Zbl 0802.47056
- [Jänich1965] T. K. Jänich, Vektorraumbündel und der Raum der Fredholm-Operatoren, Math. Ann. 161, 1965 129–142. MR0190946 (32 #8356) Zbl 0148.12401
- [Morava1968] J. J. Morava, Algebraic topology of Fredholm maps, Thesis (Ph.D.)–Rice University. 1968. MR2617315
- [Quillen1985] D. Quillen, Determinants of Cauchy–Riemann operators on Riemann surfaces, Funct. Anal. Appl., 19 (1) (1985), pp. 31–34. MR0783704 Zbl 0603.32016
- [Wang2005] S. Wang, On orientability and degree of Fredholm maps, Michigan Math. J. 53 (2005), no. 2, 419–428. MR2152708 Zbl 1093.58003