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Remarks on the degree theory. (English) Zbl 0822.55003
The authors present a unified approach to the notion of degree for rectifiable currents and for approximately differentiable mappings with an \(L^ 1\) Jacobian determinant. Let \(T = \tau({\mathcal M}, \theta, {\mathbf T})\) be a rectifiable \(n\)-dimensional current in \(\mathbb{R}^ k \times \mathbb{R}^ n\), call \(\pi : \mathbb{R}^ k \times \mathbb{R}^ n \to \mathbb{R}^ k\) and \(\widehat {\pi} : \mathbb{R}^ k \times \mathbb{R}^ n \to \mathbb{R}^ n\) the orthogonal projections, let \((\text{e}_ 1, \dots, \text{e}_ k)\) and \((\varepsilon_ 1,\dots, \varepsilon_ n)\) be the canonical bases in \(\mathbb{R}^ k\) and \(\mathbb{R}^ n\) and write \({\mathbf T}(x,y) = \sum_{| \alpha| + | \beta | = n} T^{\alpha \beta} (x, y) \text{e}_ \alpha \wedge \varepsilon_ \beta\). Then one defines the degree of \(T\) at \(y\) with respect to \(\widehat {\pi}\) as \[ \text{deg} (T, \widehat {\pi}, y) = \sum_{x \in \widehat {\pi}^{-1} (\{y\}) \cap {\mathcal M}} \theta (x,y) \text{sign }T^{0(1,\dots, n)} (x,y). \] So \(\text{deg} (T, \widehat{\pi}, y)\) is an integer whenever it exists. Moreover, \(y \mapsto \text{deg} (T, \widehat {\pi}, y)\) is in \(L^ 1 (\mathbb{R}^ n)\) and \(\widehat {\pi}_ \# T(\phi) = \int \phi (y) \text{deg } (T, \widehat{\pi}, y)dy\). If \(\Omega \subset \mathbb{R}^ n\) is open and \(u : \Omega \to \mathbb{R}^ n\) is approximately differentiable with Jacobian minors in \(L^ 1\) one denotes by \(Du\) the approximate gradient of \(u\) and defines \(\text{deg} (u,\Omega, y) = \sum \text{sign det }Du(x)\) where the summation extends over those \(x \in u^{-1}(\{y\})\) where \(u\) is approximately differentiable. This fits into the context of currents since one may define, for \(i = 1, \dots, n, (M(Du))_ i = ({\mathbf e}_ 1 + D_ 1 u^ i \varepsilon_ i) \wedge \cdots \wedge ({\mathbf e}_ n + D_ n u^ i \varepsilon_ i)\). If then \(\omega\) is an \(n\)-form on \(\Omega \times \mathbb{R}^ n\), let \(G_ u(\omega) = \int_ \Omega \langle \omega, M(Du)\rangle dx\). Then \(G_ u\) is an \(n\)-dimensional current and it turns out that \(\text{deg}(u, \Omega, y) = \text{deg} (G_ u, \widehat {\pi}, y)\). The authors derive the usual properties of a degree function for \(n\)-dimensional rectifiable currents and they show how one can recover the classical degree theory from their approach.
Reviewer: C.Fenske (Gießen)

55M25 Degree, winding number
58C30 Fixed-point theorems on manifolds
47H11 Degree theory for nonlinear operators
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