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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?

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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).

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  • $\begingroup$ Thanks! It's good to have such a clear counterexample. A follow-up question: in this example the homology of the homotopy fibre injects into the homology of the total space. Is this something that occurs generally for split fibrations? (for arbitrary fibrations it is, of course, false) $\endgroup$
    – VBM
    Commented Jan 6, 2018 at 9:49
  • $\begingroup$ @VBM I don't think it is true, but no counterexample immediately springs to mind (maybe the sphere bundle obtained by fiberwise 1-point compactifying some vector bundle can do the trick). You should try to consider the Eilenberg-Moore spectral sequence for ordinary homology, that should clarify the relationship between the homology of the total space and the homology of the fiber. $\endgroup$
    – Denis Nardin
    Commented Jan 6, 2018 at 9:56
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    $\begingroup$ 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. $\endgroup$
    – user43326
    Commented Jan 7, 2018 at 10:07

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