Extending fibrations after cell attachments

Question

Assume $p:E\rightarrow B$ is a fibration of connected CW-complexes. Let $x\in p_*(\pi_n(E))$ and assume $x$ is represented by a map $f:S^n \rightarrow B$. Attach $n+1$ dimensional cell along the map $f$ to kill the element $x$ in $\pi_n(B)$. Is it possible to attach some cells to $E$ so that the map $p$ can be extended to a fibration over $B \cup$(the attached $n+1$-cell)

Answer 1

I'm going to assume that in the original fibration the fiber over the basepoint is a subcomplex. As well, I'm going to assume we are trying to extend it to a fibration again where the fiber over the basepoint is a subcomplex. I do not know whether or not you can reduce the general question to this case.

Take the trivial fibration $S^1 \times I \rightarrow S^1$. Attach a disk along the identity loop of $S^1$. This is a contractible space, so by the long exact sequence of a fibration the homotopy groups of the fiber of a fibration over this space are the homotopy groups of the total space.

Now since the 2-skeleton of the total space you are looking for is the cylinder, the fundamental group of such a total space must be $\mathbb{Z}$. Hence, the fundamental group of the fiber is $\mathbb{Z}$. However, the fiber must have a contractible 2-skeleton since the 2-skeleton of the fiber over the basepoint is $[0,1]$. This means the fundamental group of the fiber is trivial, meaning no such total space can exist.