Degree theory and oriented counts

First take a smooth map f:M→N between connected closed oriented n-manifolds, with n positive. Closed means compact and without boundary. At a regular target value y, its preimage is zero dimensional. It is also closed in the compact source, so it is a finite set. Each derivative at a preimage is an isomorphism between oriented tangent spaces and receives a sign according to whether it preserves orientation. The sum of these signs defines the degree.

The sign can be computed from the determinant in positively oriented source and target charts. Changing either chart within its orientation leaves the sign unchanged. Reversing the orientation of exactly one manifold reverses the degree. Counting only the number of points discards this information: one positive and one negative preimage cancel in degree even though both are actual solutions. A regular value outside the image has an empty signed sum, so it gives degree zero.

The degree does not depend on which regular value is used and is invariant under smooth homotopies of maps between these closed manifolds. One geometric proof makes an appropriate homotopy transverse to a target point. Its preimage is then a compact oriented one-manifold whose boundary contains the signed fibers at the two ends. Each interval has oppositely signed endpoints, giving equality of the total counts. This is a proof mechanism, not a claim that every unmodified homotopy is already transverse.

Degree is multiplicative under composition of maps between closed connected oriented manifolds of the same dimension. A nonzero degree forces surjectivity, because any omitted point would be regular with empty fiber and degree zero. The converse need not hold: cancellation can give degree zero even for a surjective map. A degree counts a global oriented covering effect and does not guarantee injectivity, local invertibility everywhere, or an exact unsigned number of solutions at critical values.

For a smooth proper map between oriented boundaryless n-manifolds with connected target, regular fibers are compact and discrete, hence finite, and a global degree can again be defined. Proper means every compact target set has compact inverse image. A homotopy must itself be proper as a map from the parameter interval times the source if one wants this version of homotopy invariance. Checking isolated slices or a few finite values is not enough to prevent solutions from escaping to infinity.

For a map from a bounded region in Euclidean space, another degree convention fixes a target value away from the image of the region's boundary. That relative situation has its own hypotheses and is not automatically the closed-manifold theorem. In applications, state the source, target, orientations, regular target value, and compactness or properness mechanism before adding signs. Those choices explain both why the sum is finite and why changing an auxiliary choice will not change the invariant.