Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, Section 2.5, a section space $\Gamma(W-M_0,W-M;X)$ is constructed. On the other hand, in the paper Homology of $\Omega^{n+1}\Sigma^{n+1}X$ and $C_{n+1}X$, $n>0$, Fred Cohen, Bull. Amer. Math. Soc. 1973, the author used $\Omega^m\Sigma^m X$ to take the place of $\Gamma (W, W-M;X)$ when $M=\mathbb{R}^m$, $M_0=\emptyset$, $m\geq 2$.
Question 1: When $M=\mathbb{R}^m$, $M_0=\emptyset$, $m\geq 2$, how to see that $\Gamma(W,W-M;X)$ reduces to $\Omega^m\Sigma^mX$?
I generalized the case $X=S^0$ of Question 1 to Question 2 below.
Let $B$ be a CW-complex (or a manifold) and $B_0$ a CW-subcomplex (or a submanifold) of $B$. Let $\xi=(E,p,B)$ be a fibre bundle with fibre $S^n$. Choose a basepoint $*\in S^n$. Let $\Gamma(\xi)$ be the space of cross-sections of $\xi$ equipped with compact-open topology, and define a subspace of $\Gamma(\xi)$ by $$ \Gamma(\xi;B,B_0)=\{X\in \Gamma(\xi)\mid X(p)=* \text{ for any } p\in B_0\}. $$ I want to write $\Gamma(\xi;B,B_0)$ in the form of a mapping space preserving basepoints.
Question 2: If $B$ is a manifold, $B_0$ a submanifold of $B$, and $\xi$ the fibre-wise one-point compactification of the tangent bundle $TB$, does $$ \Gamma(\xi;B,B_0)\cong \text{Map}_*(B/B_0,S^n)? $$ How to prove it?
Question 3: For general case, do we have $$ \Gamma(\xi;B,B_0)\cong \text{Map}_*(B/B_0,S^n)? $$
