Assume first that is connected; otherwise the definition is made separately on each connected component. Choose a regular value and define the degree modulo two by
The following course results make this well-defined.
- By Sard theorem, regular values are dense, so one can choose .
- By the preimage theorem, is a zero-dimensional submanifold because . It is discrete and closed in compact , hence finite.
- The parity is independent of the regular value. Indeed, after choosing a path between two regular values transverse to , its inverse image is a compact one-dimensional manifold whose boundary is the disjoint union of the two fibres. Every compact one-manifold has an even number of boundary points.
The same transverse-preimage argument applied to a smooth homotopy proves homotopy invariance. Consequently the definition depends only on , rather than on the chosen regular value.
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