Difference between revisions of "Global Polyfold Fredholm setup"

Latest revision as of 10:16, 3 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 now illustrate the concept of an EP-groupoid in the following

[Expand]

Example: the EP-groupoid structure of $DM(1,2)$ .

@@ Line 17: / Line 17: @@
 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.
-We will construct polyfold structures on spaces of maps by the following process:
+We now illustrate the concept of an EP-groupoid in the following
-# blah
-# blah
 <div class="toccolours mw-collapsible mw-collapsed">
-'''Example''': We work out the EP-groupoid structure of <math>DM(1,2)</math> in detail.
+'''Example''': the EP-groupoid structure of <math>DM(1,2)</math>.
 <div class="mw-collapsible-content">
 blah blah
 </div></div>

Difference between revisions of "Global Polyfold Fredholm setup"

Latest revision as of 10:16, 3 June 2017

EP-groupoid basics

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