We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $sigma=(sigma_1,dots,sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $overline{f}
colon {mathbb R}^m to V$ such that $f = sigma circ overline{f}$ is of class $C^{d-1,1}$ is locally of Sobolev class $W^{1,p}$ for all $1 le p<d/(d-1)$. In the case $m=1$ there always exists a continuous choice $overline{f}$ for given $fcolon {mathbb R} to sigma(V) subseteq {mathbb C}^n$. We give uniform bounds for the $W^{1,p}$-norm of $overline{f}$ in terms of the $C^{d-1,1}$-norm of $f$. The result is optimal: in general a lifting $overline{f}$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger Holder class.
Mosers theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular
we obtain Mosers theorem on simplices. The proof is based on Banyagas paper (1974), where Mosers theorem is proven for manifolds with boundary. A cohomological interpretation of Banyagas operator is given, which allows a proof of Lefschetz duality using differential forms.
Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficie
nts an analog of the weight filtration for complex algebraic varieties.
