Orientation of a topological manifold

An orientation of a topological manifold chooses a generator of every local homology group $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$, compatibly on coordinate neighbourhoods. Each choice is a local orientation of a manifold. Compatibility means the generators are locally restrictions of the same relative orientation class. It can be checked by choosing oriented coordinate balls and requiring agreement wherever their local classes overlap.

For a connected closed manifold, an orientation gives a unique fundamental class in $H_n(M;\mathbb{Z})$ whose restriction at each point is the chosen local generator. In an oriented triangulation this class is the sum of compatibly signed top simplices: interior faces cancel in opposite pairs, while the local coefficient in every top simplex is one. Reversing every local generator negates the fundamental class.