Degree of a map between oriented closed manifolds

Let $f:M\to N$ be continuous, where $M,N$ are nonempty connected closed integrally oriented $n$-manifolds of the same dimension. Closed means compact and boundaryless. Write $[M],[N]$ for their specified fundamental classes from Fundamental class of a compact oriented manifold. The degree of $f$ is the integer $d=\mathrm{deg}(f)$ determined by $f_{\ast }[M]=d[N]\text{in }{H}_n(N;\mathbb{Z})\mathrm{.}$

This integer exists and is unique. By Top homology of a connected manifold, restriction at a point of $N$ identifies $H_n(N;\mathbb{Z})$ with its infinite cyclic local stalk, carrying $[N]$ to the prescribed generator. Thus every class in $H_n(N;\mathbb{Z})$ is a unique integer multiple of $[N]$. The continuous map induces a homomorphism $f_{\ast }$ by Singular chains and singular homology are covariantly functorial, so this applies to $f_{\ast }[M]$. The same theorem identifies $[M]$ as a generator of the source group. These facts define $d$ from classes and a canonical induced homomorphism, so no choice of cycle representatives affects it.

The orientations are part of the data. Replacing $[M]$ by $-[M]$ multiplies its image by $-1$, and replacing $[N]$ by $-[N]$ replaces the coordinate of a fixed target class by its negative. In either case changing exactly one orientation changes $d$ to $-d$; changing both leaves it unchanged. The fundamental-class definition proves these sign changes from the local orientation values. A degree-zero map means precisely that $f_{\ast }[M]=0$; uniqueness follows even in this case because $[N]$ has infinite order.

When $n=0$, a nonempty connected zero-manifold is a single point, as proved in the top-homology theorem. Write its source and target orientation classes as ${ϵ}_M[x]$ and ${ϵ}_N[y]$, with ${ϵ}_M,{ϵ}_N\in \{1,-1\}$. The unique point map sends $[x]$ to $[y]$, so $\mathrm{deg}(f)={ϵ}_M{ϵ}_N\mathrm{.}$ In particular the identically oriented point map has degree $1$, and opposite orientations give degree $-1$. Higher degenerate simplices in the point complex do not alter this computation of $H_0$.

Empty manifolds are excluded: their homology group $H_n$ is zero, so the equation would not determine a unique integer if the target were empty. More directly, $[\mathrm{\varnothing }]=0$ has no generator coordinate. Disconnected manifolds instead have several top-class coordinates and are outside this scalar definition. Coefficients here are exactly $\mathbb{Z}$, not the zero ring or an arbitrary ring. The definition and sign calculations use no AC.