In Spanier AT, there is some construction of fiber bundle defined as follows
If $\xi$ is an $n$-sphere bundle, denote its total space by $\dot{E}_\xi$. The mapping cylinder of the bundle projection $p:\dot{E}_\xi\to B$ is the total space $E_\xi$ of a fiber bundle $(E_\xi,B,E^{n+1},p_\xi)$ where $p_\xi:E_\xi\to B$ is the retraction of the mapping cylinder to $B$ and $p_\xi|_{\dot{E}_\xi}:\dot{E}_\xi\to B$ is the original bundle projection.
The book didn't specify the space $E$. I want to know what $E$ is : If I let $x\in B\subset M_{p_\xi}$ then as $p^{-1}(x) = S^n$, I think $p_\xi^{-1}(x) = S^n\times I$. But it's not of the form $E^{n+1}$ for some space $E$. Could you help?
$E^n$ stands for the closed $n$-ball (see p.9 where Spanier introduces notation).
The mapping cylinder of $p$ is the adjunction space $B \sqcup_{p'} \dot E_\xi \times I$ where $p' : \dot E_\xi \times \{0\} \to B$ is the obvious map. The fiber $p_\xi^{-1}(b)$ over $b$ is $(p^{-1}(b) \times I)/(p^{-1}(b) \times \{0\})$ which is homeomorphic to $(S^n \times I)/(S^n \times \{0\})$, i.e. to $E^{n+1}$.