Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am looking for an easy proof.
The first map is by the following procedure:
$$ H_{p}(\Omega X) \cong H_{p+1}(\Sigma \Omega X) \to H_{p+1}(X); $$
the first is the suspension isomorphism (o.k., I should assume $p>0$), the second is given by the evaluation $\Sigma \Omega X \to X$. The other map is given in terms of the homological Leray-Serre spectral sequence for the path-loop fibration $\Omega X \to PX \to X$ ($PX$=path space, $PX \simeq pt$). Note that the differential $d^{p+1}:E^{p+1}_{0,p}\to E^{p+1}_{p+1,0}$ is an isomorphism because $E^{p+2}_{0,p}= E^{p+2}_{p+1,0}=0$.
Now consider the composition
$$H_{p}(\Omega X) \to H_{0}(X, H_p (\Omega X))=E^{2}_{0,p} \to \ldots \to E^{p+1}_{0,p} \stackrel{(d^{p+1})^{-1}}{\to} E^{p+1}_{0,p+1} \subset E^{2}_{0,p+1}$$$$H_{p}(\Omega X) \to H_{0}(X, H_p (\Omega X))=E^{2}_{0,p} \to \ldots \to E^{p+1}_{0,p} \stackrel{(d^{p+1})^{-1}}{\to} E^{p+1}_{p+1,0} \subset E^{2}_{p+1,0}$$ $$= H_{p+1} (X, H_0(\Omega X)) \to H_{p+1}(X).$$
(everything makes perfect sense for nonsimply connected $X$, using local coefficient systems).
Here is my question:
are these maps indeed equal (up to sign)?
is there a proof of this, which does not involve chasing differentials in the spectral sequence and the Hurewicz theorem?
