Degree formulas for maps with nonintegrable Jacobian
The well-known formula \(\deg (f;X,Y) = \int_X f^\sharp \omega\) \((\int_Y \omega = 1)\) for the degree of a \(C^1\)-map \(f : X \to Y\) between closed \(n\)-dimensional manifolds is generalized under the additional assumption of nonvanishing homologies \(H^\ell (Y) \neq 0\) for certain \(\ell\) \((1 \leq \ell \leq n-1)\) as follows. Let \(X\), \(Y\) be equipped with smooth Riemannian structures, and \(\mu \in H^\ell (Y)\) be a harmonic field with \(\int_Y |\mu|^2 = 1\). Then the formula \[ \deg (f;X,Y) = \lim \int_X (f^\sharp \mu \mid f^\flat\mu)/|f^\sharp \mu|^\varepsilon |f^\flat\mu|^\varepsilon \qquad (\varepsilon \downarrow 0) \] generalizes the degree for each map \(f \in {\mathcal C}^n (X,Y)\). Here \({\mathcal C}^n(X,Y)\) is a certain class of mappings (which cannot be specified here) including some Sobolev-Orlicz spaces broader than \(W^{1,p} (X,Y)\), and \(f^\sharp\) is the pull-back, \(f^\flat = (-1)^{\ell n - \ell} *f^\sharp *\) the composition with the Hodge operator.
scientific article; zbMATH DE number 906857