Degree-d self-maps of a sphere have Lefschetz number 1+(-1)^n d

The trace computation. By [F1] the only nonzero rational homology groups are $H_0$ and $H_n$, with $f_{\ast }=\mathrm{i}\mathrm{d}$ on $H_0$ and $f_{\ast }=d$ on $H_n$; the defining alternating sum therefore has exactly the two terms $(-1{)}^0\mathrm{tr}(\mathrm{i}\mathrm{d})=1$ and $(-1{)}^{n}d$, so $L(f)=1+(-1{)}^{n}d$.

Consequences. For $n=1$ this is $1-d$, vanishing exactly for the degree-one circle maps such as rotations; for $n=2$ it is $1+d$, and this is nonzero exactly when $d\mathrm{\ne }-1$, so the fixed-point theorem forces a fixed point in that case; the degree-$-1$ antipodal map has no fixed points and Lefschetz number $1-1=0$, the standard sharpness example. When $f$ is smooth with nondegenerate fixed points the same number is the index sum by [L1], for instance for an integer $d\ge 2$ the map $z\mapsto z^d$ extends smoothly to $S^2$, with the coordinate expression $w\mapsto w^d$ at infinity. Its fixed points are $0$, $\mathrm{\infty }$, and the $d-1$ solutions of $z^{d-1}=1$. The derivative is zero at $0$ and infinity, and is complex multiplication by $d$ at those roots; hence $I-Df$ has positive real determinant at all $d+1$ points and every local index is $+1$ by The index of a nondegenerate fixed point is the sign of det(I-Df). A nonzero finite target value has $d$ regular preimages of positive orientation, proving the asserted degree $d$ by Regular-value formula for degree.