 Solution

A closed manifold is a compact manifold without boundary. Here $M$ is Hausdorff and every point has a neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^n$; the usual countability condition is included in the manifold convention. Compactness also gives a finite atlas and hence second countability.

A local orientation at $x$ is a choice of generator of $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$. An orientation of a manifold is a compatible choice of these generators as $x$ varies: locally they are induced by the same coordinate-ball orientation, or equivalently by a local relative homology class. For an oriented closed manifold, the fundamental class $[M]\in H_n(M;\mathbb{Z})$ is the unique class mapping to the chosen generator at every point. On disconnected closed manifolds this is the sum of the fundamental classes of their finitely many components.

For a triangulated or Delta complex description, orient the top-dimensional simplices compatibly with the local orientation and sum them with their corresponding signs. Each codimension-one face appears twice with opposite induced orientations, so the sum is a cycle. At a point inside any top simplex its image is the prescribed local generator, so this cycle represents $[M]$. In the connected case $H_n(M)=\mathbb{Z}$ generated by this class.

The homology conclusions below use the opening closed-manifold hypothesis for $M_1,M_2$. That hypothesis is necessary if their later description is read independently: removing a ball from ${\mathbb{R}}^n$ for $n\ge 2$ creates a nonzero $(n-1)$st homology group, so the claimed isomorphism would fail without closedness.