Let $V$ and $W$ be finite dimensional real vector spaces with an inner product, and let $mor(V,W)$ be the set of linear maps from $V$ to $W$ which preserves the inner product i.e. $f \in mor(V,W)$ is a linear isometry.
Let $\gamma_n(V,W)$ be the vector bundle over $mor(V,W)$ consisting of pairs $(f,x)$ such that $f \in mor(V,W)$ and $x \in \Bbb{R}^n \otimes \text{coker}(f) $.
Then let $mor_n(V,W)$ be the Thom space of $\gamma_n(V,W)$.
I was wondering if anyone could explain the proof of the following proposition.
Proposition: The reduced mapping cone of the restriction $mor_0(\Bbb{R}\oplus V, W) \to mor_0(V,W)$ is homeomorphic to $mor_1(V,W)$.
The proof (which can be found here (Proposition 1.1)) says that
1) if $\dim(V) \geq dim(W)$ the homeomorphism is obvious. How?
For this is understand when $dim(V) < dim(W)$, $mor_0(V,W) = *$ But I don't see how this makes the homeomorphism between the mapping cylinder and $mor_1(V,W)$ obvious. For $dim(V)=dim(W)$ I see that $\gamma_n(V,V)=0$ so $mor_n(V,V)= \gamma_n(V,V)_+ = O(V)_+$. Again, I'm not seeing how this makes the required homeomorphism between the mapping cylinder and $mor_1(V,V)$ obvious. My guess is that it has something to do with $mor_0(V,V) \cong mor_1(V,V)$ but I'm not sure how.
2) It gives a proof for $\dim(V) < dim(W)$, in which it says that the mapping cone is a quotient of the space $[0, \infty] \times mor_o(\Bbb{R}\oplus V,W)$. How?
3)Moreover it says that the restriction map is onto. How? [Answered]