Difference between revisions of "Problems on retractions and M-polyfolds"

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== A useful toy retraction ==
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Fix a non-negative function  <math>\beta\in \mathcal{C}_0^\infty</math> for which <math>\|\beta\|_{E_0} = \|\beta\|_{L^2}=1</math>.
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We consider the sc-Banach space <math>\mathbb{E}= \bigl(H^{k, \delta_k}(\mathbb{R}, \mathbb{R}) \bigr)_{k\in\mathbb{N}_0}</math>
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with <math>\delta_0 = 0</math>. 
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Define a family of linear ''projections'' <math> \pi_t: E_0\to E_0</math> for <math>t\in\R</math> by <math>L^2</math>-projection onto the subspace spanned by <math>\beta_t:= \beta(e^{1/t}+\cdot)</math> for <math>t>0</math> respectively <math>\beta_t:=0</math> for <math>t \leq 0</math>.
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The corresponding retraction
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<center>
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<math>
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\mathbb{R}\times \mathbb{E} \to \mathbb{R}\times \mathbb{E}, \qquad (t, e)\mapsto (t, \pi_t(e)) =
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\begin{cases}
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\bigl( t, \langle f, \beta_t \rangle_{L^2} \beta_t \bigr)&; t>0\\
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(t, 0 ) &; t\leq 0
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\end{cases}
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</math>
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</center>
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is sc<math>^\infty</math> (see Lemma 1.23 in the HWZ sc-smoothness paper)) and a retraction (in fact, a splicing).
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'''Question''': What is the sc-retract of this sc-smooth retraction?  Describe this space (somewhat) geometrically.
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'''Question''': Find a subset of <math>\mathbb{R}^2</math> homeomorphic to this sc-retract. ''Hint'': It will contain the line <math>\mathbb{R}\times\{0\}</math>.
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'''Question''': Which paths of the form <math> t\mapsto (t, \gamma_t) \in \mathbb{R}\times \mathbb{E}</math> are contained in the sc-retract and are sc-smooth?  Can these paths be described via the above homeomorphism of the retract to <math>\mathbb{R}^2</math>?
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'''Question''': What is the tangent fiber of the sc-retract at the point <math>(0,0)</math>? Can all points in this fiber be achieved as tangent vectors to paths in the retract passing through <math>(0,0)</math>? Is the tangent vector at <math>(0,0)</math> of a path in the sc-retract always contained in the tangent fiber of the retract at <math>(0,0)</math>?
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== Independence of choice of retraction ==
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Let <math>\mathbb{E}</math> be some sc-Banach space, and suppose <math>r:\mathbb{E}\to \mathbb{E}</math> is an sc-smooth retraction, with <math>O = r(E_0)</math>. Recall that if <math>\mathbb{F}</math> is a sc-Banach space, then a function <math>f:O\to \mathbb{F}</math> is sc-smooth if and only if <math>f\circ r : \mathbb{E}\to \mathbb{F}</math> is sc-smooth.  Suppose <math>\rho:\mathbb{E}\to \mathbb{E}</math> is an sc-smooth retraction with the property that <math>\rho(E) = O</math>. 
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'''Question''': Is <math>f\circ \rho: \mathbb{E}\to \mathbb{F}</math> sc-smooth?
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'''Question''': The sc-differentiable structure on <math>O\subset E_0</math> is induced from <math>\mathbb{E}</math> and <math>r</math>, but to what extent does it depend on <math>r</math>?  Specifically, does <math>\rho</math> define a different sc-differentiable structure on <math>O</math>?
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== Subsets of retracts are retracts ==
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Suppose <math>\mathbb{E}</math> is a sc-Banach space, <math>U\subset E_0</math> an open set, and <math>r:U\to U</math> an sc-smooth retraction with <math>O=r(U)</math>.  Let <math>x\in O</math> be a point. 
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'''Question''': For each small open neighborhood (in the subspace topology) <math>O'\subset O</math> does there exist an open set <math>U'\subset E_0</math> such that <math>r\big|_{U'}: U' \to U'</math> and <math>r(U') = O'</math>?
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== Retracts in retracts are retracts ==
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Suppose <math>\mathbb{E}</math> is a sc-Banach space, <math>U\subset E_0</math> an open set, and <math>r:U\to U</math> an sc-smooth retraction with <math>O=r(U)</math>.  Suppose <math>\rho:O\to O</math> is sc-smooth and satisfies <math>\rho\circ\rho =\rho</math>. 
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'''Question''': Is <math>\rho(O)\subset E_0</math> a retract of an sc-smooth retraction defined on an open set in <math>E_0</math>?
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== The model lemmas ==
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Restate Proposition 2.17 from the [https://arxiv.org/pdf/1002.3381.pdf sc-Smoothness paper] in the context of Floer strips; i.e. maps in <math>H^{3+m, \delta_m}(\mathbb{R}^\pm \times [0, 1], \mathbb{R}^{2n})</math> with appropriate Lagrangian boundary conditions <math>u(\cdot, 0)\in \mathbb{R}^n\times \{0\}</math> and <math>u(\cdot, 1) \in \{0\}\times \mathbb{R}^n </math>.

Revision as of 14:09, 13 June 2017

A useful toy retraction

Fix a non-negative function \beta \in {\mathcal {C}}_{0}^{\infty } for which \|\beta \|_{{E_{0}}}=\|\beta \|_{{L^{2}}}=1. We consider the sc-Banach space {\mathbb {E}}={\bigl (}H^{{k,\delta _{k}}}({\mathbb {R}},{\mathbb {R}}){\bigr )}_{{k\in {\mathbb {N}}_{0}}} with \delta _{0}=0. Define a family of linear projections \pi _{t}:E_{0}\to E_{0} for t\in \mathbb{R} by L^{2}-projection onto the subspace spanned by \beta _{t}:=\beta (e^{{1/t}}+\cdot ) for t>0 respectively \beta _{t}:=0 for t\leq 0. The corresponding retraction

{\mathbb {R}}\times {\mathbb {E}}\to {\mathbb {R}}\times {\mathbb {E}},\qquad (t,e)\mapsto (t,\pi _{t}(e))={\begin{cases}{\bigl (}t,\langle f,\beta _{t}\rangle _{{L^{2}}}\beta _{t}{\bigr )}&;t>0\\(t,0)&;t\leq 0\end{cases}}

is sc^{\infty } (see Lemma 1.23 in the HWZ sc-smoothness paper)) and a retraction (in fact, a splicing).

Question: What is the sc-retract of this sc-smooth retraction? Describe this space (somewhat) geometrically.

Question: Find a subset of {\mathbb {R}}^{2} homeomorphic to this sc-retract. Hint: It will contain the line {\mathbb {R}}\times \{0\}.

Question: Which paths of the form t\mapsto (t,\gamma _{t})\in {\mathbb {R}}\times {\mathbb {E}} are contained in the sc-retract and are sc-smooth? Can these paths be described via the above homeomorphism of the retract to {\mathbb {R}}^{2}?

Question: What is the tangent fiber of the sc-retract at the point (0,0)? Can all points in this fiber be achieved as tangent vectors to paths in the retract passing through (0,0)? Is the tangent vector at (0,0) of a path in the sc-retract always contained in the tangent fiber of the retract at (0,0)?

Independence of choice of retraction

Let {\mathbb {E}} be some sc-Banach space, and suppose r:{\mathbb {E}}\to {\mathbb {E}} is an sc-smooth retraction, with O=r(E_{0}). Recall that if {\mathbb {F}} is a sc-Banach space, then a function f:O\to {\mathbb {F}} is sc-smooth if and only if f\circ r:{\mathbb {E}}\to {\mathbb {F}} is sc-smooth. Suppose \rho :{\mathbb {E}}\to {\mathbb {E}} is an sc-smooth retraction with the property that \rho (E)=O.

Question: Is f\circ \rho :{\mathbb {E}}\to {\mathbb {F}} sc-smooth?

Question: The sc-differentiable structure on O\subset E_{0} is induced from {\mathbb {E}} and r, but to what extent does it depend on r? Specifically, does \rho define a different sc-differentiable structure on O?

Subsets of retracts are retracts

Suppose {\mathbb {E}} is a sc-Banach space, U\subset E_{0} an open set, and r:U\to U an sc-smooth retraction with O=r(U). Let x\in O be a point.

Question: For each small open neighborhood (in the subspace topology) O'\subset O does there exist an open set U'\subset E_{0} such that r{\big |}_{{U'}}:U'\to U' and r(U')=O'?

Retracts in retracts are retracts

Suppose {\mathbb {E}} is a sc-Banach space, U\subset E_{0} an open set, and r:U\to U an sc-smooth retraction with O=r(U). Suppose \rho :O\to O is sc-smooth and satisfies \rho \circ \rho =\rho .

Question: Is \rho (O)\subset E_{0} a retract of an sc-smooth retraction defined on an open set in E_{0}?

The model lemmas

Restate Proposition 2.17 from the sc-Smoothness paper in the context of Floer strips; i.e. maps in H^{{3+m,\delta _{m}}}({\mathbb {R}}^{\pm }\times [0,1],{\mathbb {R}}^{{2n}}) with appropriate Lagrangian boundary conditions u(\cdot ,0)\in {\mathbb {R}}^{n}\times \{0\} and u(\cdot ,1)\in \{0\}\times {\mathbb {R}}^{n}.

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