  This paper is concerned with problems in the context of the theoretical
foundation of adaptive (wavelet) algorithms for the numerical treatment of
operator equations. It is well-known that the analysis of such schemes
naturally leads to function spaces of Besov type. But, especially when dealing
with equations on non-smooth manifolds, the definition of these spaces is not
straightforward. Nevertheless, motivated by applications, recently Besov-type
spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise
smooth surfaces were defined and employed successfully. In the present paper,
we extend this definition (based on wavelet expansions) to a quite general
class of $d$-dimensional manifolds and investigate some analytical properties
(such as, e.g., embeddings and best $n$-term approximation rates) of the
resulting quasi-Banach spaces. In particular, we prove that different prominent
constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds
$\Gamma$ which admit a decomposition into smooth patches actually generate the
same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that
their univariate ingredients possess a sufficiently large order of cancellation
and regularity (compared to the smoothness parameter $\alpha$ of the space).
For this purpose, a theory of almost diagonal matrices on related sequence
spaces $b^\alpha_{p,q}(\nabla)$ of Besov type is developed.
  Keywords: Besov spaces, wavelets, localization, sequence spaces, adaptive
methods, non-linear approximation, manifolds, domain decomposition.
