Degree of a smooth map

Let $M$ and $N$ be oriented smooth $n$ -manifolds without boundary, with $N$ connected, and let $f:M\to N$ be a proper smooth map. For a regular value $y\in N$ , the fiber $f^{-1}(y)$ is finite. At each $x\in f^{-1}(y)$ , the isomorphism $d f_x:T_xM\to T_yN$ has sign $+1$ or $-1$ according as it preserves or reverses orientation. The degree of $f$ is

\deg(f)=\sum_{x\in f^{-1}(y)}\operatorname{sgn}(d f_x).

This integer is independent of the regular value. The empty sum is zero.

Integral characterization

For every compactly supported $n$ -form $\omega$ on $N$ ,

\int_M f^*\omega=\deg(f)\int_N\omega.

This identity both recovers the signed-count definition and extends its computational reach. When $M$ and $N$ are compact and connected, degree is the scalar by which $f$ acts on top-dimensional real cohomology.

Homotopy and composition

Degree is invariant under proper smooth homotopies for which the combined map to $N$ is proper. For composable proper maps of oriented $n$ -manifolds,

\deg(g\circ f)=\deg(g)\deg(f).

An orientation-preserving diffeomorphism has degree $1$ , whereas an orientation-reversing one has degree $-1$ .

Examples and scope

The map $S^1\to S^1$ , $z\mapsto z^m$ , has degree $m$ , including negative $m$ . The constant map between positive-dimensional closed oriented manifolds has degree $0$ . Equal dimensions, orientations, properness, and the absence of boundary are part of the stated convention; relative degree and mod- $2$ degree are different variants when these hypotheses are changed.

References

Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint of the 1974 edition. AMS record. Relevant: Chapter 3, mapping degree and integration.
John W. Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, 1997. Publisher excerpt. Relevant: §§5–6, degree and oriented manifolds.