Some retraction problems

A useful toy retraction

1. State 2. Investigate smooth paths 3. Investigate Tangent space at strange point

Retractions instead of sc-Banach manifolds

1. Transversal constraint construction as retraction 2. Target manifold as ${\mathbb {R}}^{N}$ 3. Retractions on retracts

The $\overline {\partial _{J}}$ operator as sc-Fredholm

1. Use the original definition of HWZ for sc-Fredholm to prove dbar is. 2. Show that the linearized d-bar operator at a constant solution is an isomorphism

Scale calculus was developed to make reparametrization actions on function spaces smooth. All the key issues and ideas can already be seen at the example of $S^{1}=\mathbb{R} /\mathbb{Z }$ acting on a space of nonconstant functions on $S^{1}$, for example the shift action

$\sigma :S^{1}\times {\mathcal {C}}_{{{\rm {id}}}}^{1}\;\to \;{\mathcal {C}}_{{{\rm {id}}}}^{1},\quad (\tau ,u)\mapsto u(\tau +\cdot )$

${\text{on}}\qquad \qquad \qquad \qquad {\mathcal {C}}_{{{\rm {id}}}}^{1}:=\{u:S^{1}\to S^{1}\;|\;u\in {\mathcal {C}}^{1}\;{\text{and homotopic to the identity}}\}.\qquad \qquad \qquad \qquad .$

The purpose of the following exercises is to give the quotient of S^1-valued functions homotopic to the identity, modulo reparametrization by shifts, ${\mathcal {B}}:={\mathcal {C}}_{{{\rm {id}}}}^{1}/S^{1}$ the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.

A subtitle

Compute the directional derivatives of the shift map $\sigma$, first at $(\tau ,u)=(0,0)$, then at $(\tau ,u)\in S^{1}\times {\mathcal {C}}_{{{\rm {id}}}}^{2}$.

Next, understand how to identify $T{\mathcal {C}}_{{{\rm {id}}}}^{1}\simeq {\mathcal {C}}^{1}(S^{1},\mathbb{R} )$, and write down a conjectural formula for the differential $d\sigma :\mathbb{R} \times {\mathcal {C}}^{1}(S^{1},\mathbb{R} )\to {\mathcal {C}}^{1}(S^{1},\mathbb{R} )$.

Then recall the definition of differentiability in terms of uniform linear approximation and see whether it can be satisfied at any base point $(\tau ,u)$.

YOUR SOLUTION WANTS TO BE HERE

Another subtitle

Verify that ${\mathcal {B}}$ has local Banach manifold charts modeled on $E:=\{\xi \in {\mathcal {C}}^{1}(S^{1},\mathbb{R} )\,|\,\xi (0)=0\}$. To set up a chart near a given point, pick a representative $[u_{0}]\in {\mathcal {B}}$ with ${\dot u}_{0}(0)\neq 0$ and consider the map

$\phi _{{u_{0}}}:\{\xi \in E\,|\,\|\xi \|_{{{\mathcal {C}}^{1}}}<\epsilon \}\to {\mathcal {B}},\qquad \xi \mapsto [u_{0}+\xi ].$

YOUR SOLUTION WANTS TO BE HERE