Problems on retractions and M-polyfolds

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 $u(\cdot ,1)\in \{0\}\times {\mathbb {R}}^{n}$.