The structure of groups of multigerm equivalences
arXiv:1110.1981
Abstract
We study the structure of classical groups of equivalences for smooth multigerms , and extend several known results for monogerm equivalences to the case of mulitgerms. In particular, we study the group $\A$ of source- and target diffeomorphism germs, and its stabilizer $\A_f$. For monogerms it is well-known that if is finitely $\A$-determined, then $\A_f$ has a maximal compact subgroup $MC(\A_f)$, unique up to conjugacy, and $\A_f/MC(\A_f)$ is contractible. We prove the same result for finitely $\A$-determined multigerms . Moreover, we show that for a ministable multigerm , the maximal compact subgroup $MC(\A_f)$ decomposes as a product of maximal compact subgroups $MC(\A_{g_i})$ for suitable representatives of the monogerm components of . We study a product decomposition of $MC(\A_f)$ in terms of and a group of target diffeomorphisms, and conjecture a decomposition theorem. Finally, we show that for a large class of maps, maximal compact subgroups are small and easy to compute.
24 pages, 1 figure