the mapping degree
How many times does a map wrap one closed manifold around another? A map from a circle to a circle might go around once, twice, or backwards; a map from a sphere to a sphere might cover it twice with a fold. The mapping degree is a single integer (or, in the unoriented setting, a single bit) that captures this net wrapping number, and remarkably it depends only on the map up to continuous deformation, not on any details.
Let f: M -> N be a smooth map between closed (compact, boundaryless) connected n-manifolds. Pick a regular value q in N — Sard guarantees one exists. The preimage f^{-1}(q) is then a finite set of points, and near each the map is a local diffeomorphism. The mod-2 degree is simply the parity of the number of preimage points, an element of Z/2; it is independent of the regular value q and invariant under homotopy. If both M and N are oriented, do better: count each preimage point with a sign, +1 if df_p preserves orientation and -1 if it reverses, and the signed total is the oriented degree, an integer that is likewise independent of q and a homotopy invariant.
Degree is the prototype of an intersection-theoretic invariant and the engine behind a string of classical theorems: the fundamental theorem of algebra (a degree-n complex polynomial, viewed as a self-map of the sphere, has degree n, hence n roots), the hairy-ball theorem, the Brouwer fixed-point theorem, and the Poincare-Hopf theorem relating the index sum of a vector field to the Euler characteristic. Honest caveats: the oriented integer degree requires both manifolds oriented and the map between equal dimensions; drop orientability and you keep only the Z/2 degree. And the well-definedness genuinely rests on Sard (regular values exist) and on the homotopy invariance proof — it is not obvious from the definition that different regular values give the same count, that is a theorem.
The map z -> z^2 on the unit circle (writing z = e^{i theta}, sending it to e^{2 i theta}) has degree 2: a generic target point has exactly two preimages, both traversed in the same direction, so the signed count is 1 + 1 = 2. The map z -> z-bar has degree -1, since it reverses orientation.
z -> z^2 wraps the circle twice: degree 2.
The integer (oriented) degree needs both manifolds oriented and equidimensional; without orientation only the mod-2 degree survives. It is a theorem, not a definition, that the count is the same at every regular value.