Lefschetz fixed point theorem

Prove the contrapositive, assuming $f$ has no fixed point. If $M$ is empty, its homology groups and $L(f)$ are zero. If $\dim M=0$, compactness makes the discrete manifold finite. Every singular simplex is constant; in each point summand its boundary is multiplication by ${\sum }_{j=0}^k(-1{)}^j$, which is one for positive even $k$ and zero for odd $k$. Thus homology is zero in positive degrees and $H_0$ has the point basis. The matrix of $f_{\ast }$ on this basis has a diagonal one exactly at a fixed point, so its trace is zero. Hence $L(f)=0$. Assume now $M\mathrm{\ne }\mathrm{\varnothing }$ and $\dim M\ge 1$.

Embed $M$ by $e$ as $S\subset {\mathbb{R}}^N$ and take $r:U\to S$ from [F1]. Write $F=e\circ f$. The function $x\mapsto \mathrm{\parallel }e(x)-F(x)\mathrm{\parallel }$ is continuous and positive, so [F3] gives a minimum $d>0$. Choose $b>0$ such that the closed $b$-neighbourhood $K$ of $S$ lies in $U$: finitely many open balls whose doubled balls lie in $U$ cover compact $S$, and the minimum of their radii supplies such a $b$ after shrinking. The set $K$ is compact by Euclidean closedness and boundedness. Continuity of $r$ on $K$ supplies $\eta >0$, with $\eta <b$, such that $\mathrm{\parallel }y-z\mathrm{\parallel }<\eta$ for $y,z\in K$ implies $\mathrm{\parallel }r(y)-r(z)\mathrm{\parallel }<d\mathrm{/}2$: cover $K$ by neighbourhood balls on which oscillation is less than $d\mathrm{/}4$, take a finite cover by their half-sized balls, and use the minimum half-radius.

By [F2], choose smooth $A:M\to {\mathbb{R}}^N$ with $\mathrm{\parallel }A(x)-F(x)\mathrm{\parallel }<\eta$ for every $x$. The segments $(1-t)F(x)+tA(x)$ lie in $K\subset U$, so $H(t,x)=e^{-1}r((1-t)F(x)+tA(x))$ is a continuous homotopy from $f$ to the smooth self-map $g=e^{-1}rA$. Moreover $\mathrm{\parallel }e(g(x))-F(x)\mathrm{\parallel }=\mathrm{\parallel }r(A(x))-r(F(x))\mathrm{\parallel }<d\mathrm{/}2$ since $r(F(x))=F(x)$. Thus $\mathrm{\parallel }e(x)-e(g(x))\mathrm{\parallel }>d\mathrm{/}2$, and $g$ is fixed-point-free.

By [F4], $L(g)=0$ and $L(f)=L(g)$, proving the contrapositive and hence the fixed-point theorem for every closed smooth manifold. AC supplies the approximation and embedding hypotheses as well as those of the index formula.