The following definition of "essential manifold" is in this wiki page:
A closed $n$-manifold $M$ is called essential if its fundamental class $[M]$ defines a nonzero element in the homology of its fundamental group $\pi$, or more precisely in the homology of the corresponding Eilenberg–MacLane space $K(\pi, 1)$, via the natural homomorphism $$\displaystyle H_{n}(M) \to H_{n}(K(\pi ,1)).$$ Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo $2$, otherwise.
My question is: what is this "natural homomorphism" in the definition?
Added:
In the original paper of Gromov (Gromov, M.: "Filling Riemannian manifolds," J. Diff. Geom. 18 (1983), 1–147.), he defines an essential manifold apparently in a different way:
He takes $K$ to be an arbitrary $K(\pi, 1)$ or $\pi = \pi_1(M,m)$? Question: are both definitions equivalent?