The (generalized) Hopf fibrations $S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ have the property that their total spaces are more highly connected than their fibers.

Are there examples of fibrations $X \to E \to B$ with $X$ a finite CW-complex so that $X$ is $(j-1)$-connected but not $j$-connected, $E$ is $j$-connected, and $j \neq 1, 3, 7$?

Some thoughts:

If we drop the assumption that $X$ is finite, examples exist for any $j$: the fiber sequence $K(\mathbf Z,j) \to \ast \to K(\mathbf Z,j+1)$ can be realized as such a fibration.

For $X$ finite, however, one can convince oneself that no examples exist unless:

$\bullet$ The number $j$ is odd. Sketch: if $j$ is even, pull the fibration back along a map $S^{j} \to B$ and make use of the polynomial nature of $H^{\ast}(\Omega S^d;\mathbf Q)$.

$\bullet$ The Euler characteristic $\chi(X)$ is zero (transfer argument).

But what about $j = 5$ or $j > 7$ odd and $\chi(X) = 0$. Are there any examples?

The (generalized) Hopf fibrations $S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ have the property that their total spaces are more highly connected than their fibers.

Are there examples of fibrations $X \to E \to B$ with $X$ a connected finite CW-complex so that $X$ is $(j-1)$-connected but not $j$-connected, and $E$ is $j$-connected for some $j \neq 1, 3, 7$?

Some thoughts on this:

If we drop the assumption that $X$ is finite, such examples exist for any $j$: the fiber sequence $K(\mathbf Z,j) \to \ast \to K(\mathbf Z,j+1)$ can be realized as such a fibration.

For $X$ finite, however, one can convince oneself that no examples exist unless:

$\bullet$ the number $j$ is odd. Sketch: if $j$ is even, pull the fibration back along a map $S^{j} \to B$ and make use of the polynomial nature of $H^{\ast}(\Omega S^d;\mathbf Q)$,

and

$\bullet$ the Euler characteristic $\chi(X)$ is zero (transfer argument).

But what about $j = 5$ or $j > 7$ odd and $\chi(X) = 0$? Are there any examples?