I am currently trying to apply Ljusternik-Schnirelman theory to a problem in geometric calculus of variations and have the following problem: I consider an energy functional $\mathcal{E}$ on a submanifold $\mathcal{M}_s\subset W^{1+s,2}(I,\mathbb{R}^n)$ for some open interval $I$ and $s>\frac{1}{2}$ (Sobolev-Slobodeckij space). The latter space is compactly embedded into $C^1(I,\mathbb{R}^n)$. Now I can essentially define the same manifold $\mathcal{M}_1\subset C^1$, such that $\mathcal{M}_s\subset \mathcal{M}_1$.
I know the homotopy type of $\mathcal{M}_1$ and therefore know the LS-category, but I need to know the category of $M_s$. Because of the continuous embedding I know that $cat(\mathcal{M}_s,\mathcal{M}_1)\le cat(\mathcal{M}_1,\mathcal{M}_1)$ but I need equality to get a useful lower bound for the number of critical points of $\mathcal{E}$ on $\mathcal{M}_s$. So my question is the following:
Is there a deformation retract between $C^1$ and $W^{1+s,2}$, or more precisely between $\mathcal{M}_1$ and $\mathcal{M}_s$, that could allow me to use the knowledge about the homotopy type of $\mathcal{M}_1$ in my setting? Any result in that direction would be helpful.
Every help is much appreciated, and I thank you in advance.