### 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) |
---|---|

Article number | 1350018 |

Journal | Communications in Contemporary Mathematics |

Volume | 15 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2013 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Contemporary Mathematics*,

*15*(5), [1350018]. https://doi.org/10.1142/S0219199713500181

}

*Communications in Contemporary Mathematics*, vol. 15, no. 5, 1350018. https://doi.org/10.1142/S0219199713500181

**The stratified structure of spaces of smooth orbifold mappings.** / Borzellino, Joseph E.; Brunsden, Victor W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The stratified structure of spaces of smooth orbifold mappings

AU - Borzellino, Joseph E.

AU - Brunsden, Victor W.

PY - 2013/10/1

Y1 - 2013/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84878663635&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878663635&partnerID=8YFLogxK

U2 - 10.1142/S0219199713500181

DO - 10.1142/S0219199713500181

M3 - Article

VL - 15

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 5

M1 - 1350018

ER -