The stratified structure of spaces of smooth orbifold mappings

Joseph E. Borzellino, Victor Brunsden

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Abstract

We consider four notions of maps between smooth C orbifolds $\mathcal{O}$, $\mathcal{P}$ with $\mathcal{O}$ compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of Cr maps between $\mathcal{O}$ and $\mathcal{P}$ with the Cr topology carries the structure of a smooth C Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of Cr maps between $\mathcal{O}$ and $\mathcal{P}$ with the Cr topology carries the structure of a smooth C Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The Cr orbifold maps between $\mathcal{O}$ and $\mathcal{P}$ is locally a stratified space with strata modeled on smooth C Banach (r finite)/Fréchet (r = ∞) manifolds while the set of Cr reduced orbifold maps between $\mathcal{O}$ and $\mathcal{P}$ locally has the structure of a stratified space with strata modeled on smooth C Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.

Original languageEnglish (US)
Article number1350018
JournalCommunications in Contemporary Mathematics
Volume15
Issue number5
DOIs
StatePublished - Oct 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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