3 corrected indexing

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

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

1. are these maps indeed equal (up to sign)?

2. is there a proof of this, which does not involve chasing differentials in the spectral sequence and the Hurewicz theorem?