On Orientability and Degree of Fredholm Maps
On Orientability and Degree of Fredholm Maps
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Fredholm 地图的定向性和度数
DOI：
10.1307/mmj/1123090776
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发表时间：
2005
影响因子：
0.9
通讯作者：
Shuguang Wang
中科院分区：
文献类型：
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作者：
Shuguang Wang
流形间映射的度在许多数学领域中起着重要的作用。为了理解度的概念，总是需要一定的可定向性。在有限维不可定向流形的情况下，这可以追溯到Hopf，Olum和Steenrod，在Brouwer对可定向流形的开创性工作之后（参见。[9]和其中的参考文献）。Elworthy和Tromba [4]首先研究了无限维Banach流形的情况，在那里他们引入了可定向Fredholm流形的度。然而，流形上的可定向性限制往往过于严格和不自然。这是菲茨帕特里克，Pejsachowicz和拉比耶[6]谁明确指出，唯一的要求是定向的地图涉及，而不是流形。(The有限维的版本是在Olum的工作。他们的方法是基于路径奇偶性的概念，这使得它在处理穿越奇异层的问题时特别有用。事实上，这通常是检验地图可定向性的唯一实用方法。最近，Benevieri和Furi [1]采用了另一种方法来定向Fredholm映射，这种方法在概念上更清晰，似乎更自然，因为它直接来自所有Fredholm算子的逐点定向。本文所采用的方法具有更多的几何风味，也提供了一个实例，几何和分析很好地相互作用。使用的行列式线丛，产生于几何链接方便的概念Benevieri-Furi和菲茨帕特里克-Pejsachowicz-Rabier。事实上，[1]、[2]和[6]中的许多性质通过我们的新方法变得更容易理解。相反，几何方法允许我们将泛函分析工具应用于规范理论中涉及真实的结构的一些问题，其中相关的流形通常是不可定向的或没有自然定向的，因此有必要将相关的映射定向。更多细节将在[10]中出现。
The degree of a map between two manifolds has played important roles in various mathematical areas. Certain orientability is always required in order to make sense of the concept of degree. In the case of finite-dimensional nonorientable manifolds, this goes back to Hopf, Olum, and Steenrod, after Brouwer’s pioneering work on orientable manifolds (cf. [9] and references therein). Elworthy and Tromba [4] took the first study in the case of infinite-dimensional Banach manifolds, where they introduced the degree on orientable Fredholm manifolds. This orientability restriction on manifolds is, however, often too severe and unnatural. It was Fitzpatrick, Pejsachowicz, and Rabier [6] who pointed out explicitly that the only requirement was the orientability of maps involved rather than that of manifolds. (The finitedimensional version was in Olum’s work.) Their approach is based on the concept of parity of paths, which makes it particularly useful in problems dealing with crossing singular strata. Indeed this is often the only practical way to check the orientability of a map. More recently, Benevieri and Furi [1] took another approach to orienting Fredholm maps that is conceptually more clear and seems more natural, since it comes directly from pointwise orientations of all Fredholm operators. The approach taken in this paper has a more geometric flavor and also provides an instance where geometry and analysis interact nicely. The use of a determinant line bundle that arises from geometry links conveniently the notions of Benevieri– Furi and Fitzpatrick–Pejsachowicz–Rabier. In fact, many properties in [1], [2], and [6] become much easier to understand through our new approach. Conversely, the geometric approach allows us to apply functional analysis tools to some problems in gauge theory involving a real structure, where the relevant manifolds are often nonorientable or with no natural orientation, hence making it necessary to orient relevant maps instead. More details will appear in [10].