Difference between revisions of "Global Polyfold Fredholm setup"
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* '''(proper):''' For every <math>x \in X</math>, there exists a neighborhood <math>V(x)</math> so that <math>t: s^{-1}\bigl(\overline{V(x)}\bigr) \to X</math> is proper. | * '''(proper):''' For every <math>x \in X</math>, there exists a neighborhood <math>V(x)</math> so that <math>t: s^{-1}\bigl(\overline{V(x)}\bigr) \to X</math> is proper. | ||
Note that '''(Lie)''' makes sense because '''(etale)''' hypothesis implies that <math>\mathbf X {}_s\times_t \mathbf X</math> inherits an M-polyfold structure. | Note that '''(Lie)''' makes sense because '''(etale)''' hypothesis implies that <math>\mathbf X {}_s\times_t \mathbf X</math> inherits an M-polyfold structure. | ||
| − | Moreover, '''(proper)''' implies that each isotropy group <math>\mathbf G(x) := \{g \ | + | Moreover, '''(proper)''' implies that each isotropy group <math>\mathbf G(x) := \{g \;|\; s(g) = t(g) = x\}</math> is finite. |
We denote the orbit space by <math>|X|</math>. | We denote the orbit space by <math>|X|</math>. | ||
A '''polyfold structure''' on a (paracompact, Hausdorff) space <math>Z</math> is simply <math>(X,\beta)</math> where <math>X</math> is an EP-groupoid and <math>\beta: |X| \to Z</math> is a homeomorphism. | A '''polyfold structure''' on a (paracompact, Hausdorff) space <math>Z</math> is simply <math>(X,\beta)</math> where <math>X</math> is an EP-groupoid and <math>\beta: |X| \to Z</math> is a homeomorphism. | ||
Revision as of 15:57, 2 June 2017
EP-groupoid basics
A polyfold should be an "M-polyfold with isotropy".
This is implemented via the language of EP-groupoids, which in finite dimensions reduce to orbifolds.
First, recall that a groupoid is a small category in which every morphism is invertible (hence a groupoid with a single object is the same thing as a group).
Now define an EP-groupoid to be a groupoid 
with morphism set 
satisfying these properties:
- (Lie): and are equipped with M-polyfold structures, with respect to which the source, target, multiplication, unit, and inversion maps
are sc-smooth.
- (etale):
and 
are surjective local sc-diffeomorphisms. - (proper): For every
, there exists a neighborhood 
so that 
is proper.
Note that (Lie) makes sense because (etale) hypothesis implies that 
inherits an M-polyfold structure.
Moreover, (proper) implies that each isotropy group 
is finite.
We denote the orbit space by 
.
A polyfold structure on a (paracompact, Hausdorff) space 
is simply 
where is an EP-groupoid and 
is a homeomorphism.
Example: We work out the EP-groupoid structure of 
in detail.
