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We don't claim any originality. 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 [ Wang ] 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.
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.
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This is similar to the construction of the determinant line bundle in [ Quillen ]. 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.
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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.
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Let be the orthogonal projection to the orthogonal complement of in. 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.