Difference between revisions of "Global Polyfold Fredholm setup"

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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.
  
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We will construct polyfold structures on spaces of maps by the following process:
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# blah
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# blah
  
 
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Revision as of 15:05, 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 X with morphism set {\mathbf X} satisfying these properties:

  • (Lie): X and {\mathbf X} are equipped with M-polyfold structures, with respect to which the source, target, multiplication, unit, and inversion maps
s:{\mathbf X}\to X,\quad t:{\mathbf X}\to X,\quad m:{\mathbf X}{}_{s}\times _{t}{\mathbf X}\to X,\quad u:X\to {\mathbf X},\quad i:{\mathbf X}\to {\mathbf X}

are sc-smooth.

  • (etale): s and t are surjective local sc-diffeomorphisms.
  • (proper): For every x\in X, there exists a neighborhood V(x) so that t:s^{{-1}}{\bigl (}\overline {V(x)}{\bigr )}\to X is proper.

Note that (Lie) makes sense because (etale) hypothesis implies that {\mathbf X}{}_{s}\times _{t}{\mathbf X} inherits an M-polyfold structure. Moreover, (proper) implies that each isotropy group {\mathbf G}(x):=\{g\;|\;s(g)=t(g)=x\} is finite. We denote the orbit space by |X|. A polyfold structure on a (paracompact, Hausdorff) space Z is simply (X,\beta ) where X is an EP-groupoid and \beta :|X|\to Z is a homeomorphism.

We will construct polyfold structures on spaces of maps by the following process:

  1. blah
  2. blah

Example: We work out the EP-groupoid structure of DM(1,2) in detail.

blah blah

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