For the unit sphere $S^n \subset \mathbb{R}^{n+1}$
let $f : S^n \to S^n$ be the map reversing the signs of all
but one coordinate,
$$f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n):$$
(a) Compute the Lefschetz number $L(f)$.
My attempt to this question is Lefschetz number.
(b) For which values of $n$ is $f$ homotopic to a map without fixed points?
First consider
Poincare-Hopf Index Theorem. If $\vec{v}$ is a smooth vector field on the compact, oriented manifold $X$ with only finitely many zeros, then the global sum of the indices of $\vec{v}$ equals the Euler characteristic of $X$.
The Euler characteristic of an $n$-sphere is $1 + (-1)^n$. Hence, the degree of $f$ is $2$ if $k$ is even, and $0$ otherwise by Poincare-Hopf Index Theorem.
The Hopf Degree Theorem. Two maps of a compact, connected, oriented $k$-manifold $X$ into $S^k$ are homotopic if and only if they have the same degree.
We attempt a homotopy with $f$ and the antipodal map. They both are compact, connected, oriented $k$-manifold $X$ into $S^k$. So they are homotopic if and only if they have the same degree.
Based on Jared's absolutely worth reading answer The degree of antipodal map. is $(-1)^{k+1}$.
So, I am completely wrong here!
Alternatively,
Consider the homotopy $f_t = f + t(-2x_1)$, so that $f_0 = f$, but $f_1$ is the antipodal map.
So it is irrelevant to $n$! This also can't be right..
Best Answer
To compute the Lefschetz number, you just look at the induced map on homology. $H_0(S^n)=H_n(S^n)=\mathbb Z$ and all other homology groups are zero. The induced map $H_0(f)\colon \mathbb Z\to\mathbb Z$ is the identity, as is always the case for a map from a connected space to itself. Since $f$ is invertible, $H_n(f)=\pm\mathrm{id}$. To figure out which one, we need to figure out whether it preserves or reverses orientation. If you write down charts and an orientation in these charts, you can see that $f$ preserves orientation iff $n$ is even. (In fact, $f$ is the suspension of the antipodal map on $S^{n-1}$.) So $H_n(f)=(-1)^n\mathrm{id}$. Thus the Lefschetz number is $$L(f)=(-1)^0Tr(H_0(f))+(-1)^nTr(H_n(f))=1+(-1)^n(-1)^n=2.$$