In the page 129 of the book Bott&Tu, they give an exercise that prove the Lefschetz fixed - point theorem.
More percisely, for a compact orientable manifold $M$ with dimension $n$, they assume the map $f:M \to M$ is transversal and set $\sigma_p=\mathrm{sgn} det(Df|_p-I)$ for a fix point $p$. Then they ask reader to prove that $\int_{\Delta}{}\eta _{\Gamma}=L(f)=\Sigma_p \sigma_p$, here $\Delta$ is the diagonal of $M \times M$, and $\eta_{\Gamma}$ the poincare dual of $\Gamma_f$, which is the graph $(x,f(x))$, here $x \in M$, $L(f)$ the lefschetz number of map $f$.
Actually I can prove this theorem by using the duality $H^k(M) \to H_{n-k}(M)$ and $H_i(M) \times H_j(M) \to H_{n-i-j}(M)$. Is there any approach that only use some knowledge mentioned in the book Bott&Tu ? Thanks!