# zbMATH — the first resource for mathematics

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)

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