THE STRATIFIED STRUCTURE OF SPACES OF SMOOTH ORBIFOLD MAPPINGS
Abstract
We consider four notions of maps between smooth C∞ orbifolds 
, 
with 
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 
and 
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 
and 
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 
and 
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 
and 
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.
References
- R. Abraham, Lectures of Smale on differential topology, Lecture Notes, Columbia University (1962) . Google Scholar
-
A. Adem , J. Leida and Y. Ruan , Orbifolds and Stringy Topology ,Cambridge Tracts in Mathematics 171 ( Cambridge University Press , Cambridge , 2007 ) . Crossref, Google Scholar -
A. Banyaga , The Structure of Classical Diffeomorphism Groups ,Mathematics and its Applications 400 ( Kluwer Academic Publishers Group , Dordrecht , 1997 ) . Crossref, Google Scholar - Math. Scand. 20 , 249 ( 1967 ) . Crossref, Google Scholar
-
J. E. Borzellino , Riemannian Geometry of Orbifolds ( University of California , Los Angeles , 1992 ) . Google Scholar - Indiana Univ. Math. J. 42(1), 37 (1993). Crossref, ISI, Google Scholar
J. E. Borzellino and V. Brunsden , Infinite Dimensional Lie Groups in Geometry and Representation Theory (World Scientific Publishing, River Edge, NJ, 2002) pp. 116–137. Link, Google Scholar- J. Lie Theory 18(4), 979 (2008). ISI, Google Scholar
- Compositio Math. 30 , 1 ( 1975 ) . ISI, Google Scholar
- Commun. Contemp. Math. 8(5), 569 (2006). Link, ISI, Google Scholar
W. Chen and Y. Ruan , Orbifolds in Mathematics and Physics,Contemporary Mathematics 310 (American Mathematical Society, Providence, RI, 2002) pp. 25–85. Crossref, Google Scholar- Bull. Amer. Math. Soc. 72 , 751 ( 1966 ) . Crossref, ISI, Google Scholar
- Bull. Amer. Math. Soc. 76 , 1314 ( 1970 ) . Crossref, ISI, Google Scholar
A. E. Fischer , Relativity (Plenum, New York, 1970) pp. 303–357. Crossref, Google ScholarJ. Franks , Studies, in Analysis,Advances in Mathematics Supplementary Studies 4 (Academic Press, New York, 1979) pp. 271–290. Google Scholar- C. R. Acad. Sci. Paris Sér. I Math. 293(2), 125 (1981). Google Scholar
-
A. Frölicher and A. Kriegl , Linear Spaces and Differentiation Theory ,Pure and Applied Mathematics ( Wiley , Chichester , 1988 ) . Google Scholar - Bull. Amer. Math. Soc. (N.S.) 7(1), 65 (1982). Crossref, ISI, Google Scholar
- Topology Appl. 159(5), 1489 (2012). Crossref, ISI, Google Scholar
-
H. H. Keller , Differential Calculus in Locally Convex Spaces ,Lecture Notes in Mathematics 417 ( Springer-Verlag , Berlin , 1974 ) . Crossref, Google Scholar - Monatsh. Math. 95(4), 287 (1983). Crossref, ISI, Google Scholar
-
A. Kriegl and P. W. Michor , The Convenient Setting of Global Analysis ,Mathematical Surveys and Monographs 53 ( American Mathematical Society , Providence, RI , 1997 ) . Crossref, Google Scholar - Topology 6 , 263 ( 1967 ) . Crossref, Google Scholar
-
P. W. Michor , Manifolds of Differentiable Mappings ,Shiva Mathematics Series 3 ( Shiva Publishing Ltd. , Nantwich , 1980 ) . Google Scholar I. Moerdijk , Orbifolds in Mathematics and Physics,Contemporary Mathematics 310 (American Mathematical Society, Providence, RI, 2002) pp. 205–222. Crossref, Google Scholar- K-Theory 12(1), 3 (1997). Crossref, ISI, Google Scholar
-
Z. Nitecki , Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms ( The MIT Press , Cambridge, MA , 1971 ) . Google Scholar -
R. S. Palais , Foundations of Global Non-Linear Analysis ( W. A. Benjamin , New York , 1968 ) . Google Scholar - J. Geom. Symmetry Phys. 1 , 54 ( 2004 ) . Google Scholar
-
W. Thurston , The geometry and topology of 3-manifolds ,Lecture Notes ( Mathematics Department , Princeton University , 1978 ) . Google Scholar
| Remember to check out the Most Cited Articles! |
|---|
|
Be inspired by these NEW Mathematics books for inspirations & latest information in your research area! |
