The ncatlab page on Poincaré duality states the following:
Let $X$ be a Poincaré duality space of dimension $d$. Then there is a quasi-isomorphism
$$C^*(X) \rightarrow \Sigma^{-d}C_*(X)$$
between the chain complex of ordinary cohomology of $X$ with the $d$-fold de-suspension of the chain complex of ordinary homology of $X$. This is such that the image in homology is (up to sign) the traditional Poincaré duality isomorphism.
The reference they give for this result is this paper of E. J. Malm, Theorem 2.5.2.
Malm does not appear to provide a proof of this result, but instead states it in his background section, and I cannot find a reference of the result therein.
I find even the statement of this result in the paper by Malm hard to digest. Some knowledge is assumed, such as a canonical action of $C_*(\Omega X)$ on a ground ring $k$, and notation such as $k[\pi_1(X)]$ is used, which is unfamiliar to me.
Where can I find a proof of this result?
Also, what is the action of $C_*(\Omega X)$ on $k$, and what is $k[\pi_1(X)]$?