# The stratified structure of spaces of smooth orbifold mappings

Joseph E. Borzellino, Victor W. Brunsden

Research output: Contribution to journalArticle

3 Citations (Scopus)

### 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 C r maps between $\mathcal{O}$ and $\mathcal{P}$ with the C r 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 C r maps between $\mathcal{O}$ and $\mathcal{P}$ with the C r topology carries the structure of a smooth C Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The C r 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 C r 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 language English (US) 1350018 Communications in Contemporary Mathematics 15 5 https://doi.org/10.1142/S0219199713500181 Published - Oct 1 2013

### Fingerprint

Orbifold
Stefan Banach
Stratified Space
Topology
Lie groups
Diffeomorphism Group
Pullback
Topological group
Necessary

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics

### Cite this

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title = "The stratified structure of spaces of smooth orbifold mappings",
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 C r maps between $\mathcal{O}$ and $\mathcal{P}$ with the C r topology carries the structure of a smooth C ∞ Banach (r finite)/Fr{\'e}chet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of C r maps between $\mathcal{O}$ and $\mathcal{P}$ with the C r topology carries the structure of a smooth C ∞ Banach (r finite)/Fr{\'e}chet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The C r orbifold maps between $\mathcal{O}$ and $\mathcal{P}$ is locally a stratified space with strata modeled on smooth C ∞ Banach (r finite)/Fr{\'e}chet (r = ∞) manifolds while the set of C r 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)/Fr{\'e}chet (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)/Fr{\'e}chet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fr{\'e}chet Lie groups.",
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The stratified structure of spaces of smooth orbifold mappings. / Borzellino, Joseph E.; Brunsden, Victor W.

In: Communications in Contemporary Mathematics, Vol. 15, No. 5, 1350018, 01.10.2013.

Research output: Contribution to journalArticle

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