# Generalised homology of a split fibration

Let $E, X$ be path-connected and suppose I have a fibration $p\colon E\to X$ which admits a section $s$. For a generalised homology theory $\mathcal{E}_\ast$, there is a splitting $\mathcal{E}_\ast (E)\cong \mathcal{E}_\ast(X)\oplus R_\ast$ induced by $p$ and $s$.

Choosing a basepoint $x\colon \ast \to X$, we have a homotopy fibre sequence $$\begin{array}{ccc} F & \to & E \\ \downarrow & & \downarrow \\ \ast & \to & X \end{array}$$ which implies a map $\psi\colon \mathcal{E}_\ast(F) \to \mathcal{E}_\ast(X)\oplus R_\ast$. Commutativity of the diagram implies that projecting $\psi$ to the first factor results in the zero map.

Question: is the map $\mathcal{E}_\ast (F)\to R_\ast$ an isomorphism?

This is not true. Consider the following (split) homotopy fiber sequence $$S^1\to S^1\times S^1\to S^1$$ Then, by a standard argument, we have $$\Sigma(S^1\times S^1)=S^2\vee S^3\vee S^2$$ so for every spectrum $E$ $$E_*(S^1\times S^1)=E_*(S^1)\oplus E_*(S^2)\oplus E_*(S^1)$$ In particular the sequence $$E_*(S^1)\to E_*(S^1\times S^1)\to E_*(S^1)$$ is not exact (the first map is the inclusion of the first summand and the second one is the projection onto the third summand).
• As far as ordinary homology with field coefficients goes, the splitting implies the collapse of Eilenberg-Moore spectral sequence at $E_2$, so the homology of the fiber injects to that of the total space. – user43326 Jan 7 '18 at 10:07