There are no exotic actions of diffeomorphism groups on 1-manifolds

[摘要] Let MMM be a manifold and NNN a 111-dimensional manifold. Assuming that r≠dim⁡(M)+1r\neq\dim(M)+1r=dim(M)+1, we show that any nontrivial homomorphism ρ:Diffcr(M)→Homeo(N)\rho:\mathsf{Diff}^r_c(M)\rightarrow\mathsf{Homeo}(N)ρ:Diffcr​(M)→Homeo(N) has a standard form: necessarily MMM is 111-dimensional, and there are countably many embeddings ϕi:M→N\phi_i: M\rightarrow Nϕi​:M→N with disjoint images such that the action of ρ\rhoρ is conjugate (via the product of the ϕi\phi_iϕi​) to the diagonal action of Diffcr(M)\mathsf{Diff}^r_c(M)Diffcr​(M) on M×M×⋯M\times M\times\cdotsM×M×⋯ on ⋃iϕi(M)\bigcup_i\phi_i(M)⋃i​ϕi​(M), and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups Diffcr(M)\mathsf{Diff}^r_c(M)Diffcr​(M) have no countable index subgroups.

[学科分类] 神经科学

[关键词] Homeomorphism groups;diffeomorphism groups;group actions on a circle