= Solution
<Pullback of a differential form> commutes with the <exterior derivative>, so $f^*$ gives a <linear map> on <de Rham cohomology>. Functoriality and $f\circ f=\operatorname{id}$ imply $(f^*)^2=I$. Over the <real numbers>, define the complementary projections
$$
P_+=\tfrac12(I+f^*),\qquad P_-=\tfrac12(I-f^*).
$$
They obey $P_\pm^2=P_\pm$, $P_+P_-=0$ and $P_++P_-=I$. Every <cohomology> class therefore decomposes as
$$
[\alpha]=\frac{[\alpha]+f^*[\alpha]}2+\frac{[\alpha]-f^*[\alpha]}2.
$$
The first term is fixed and the second negated by $f^*$. Their intersection is zero, since a class in both equals its own negative. Consequently
$$
\boxed{H^p_{\mathrm{dR}}(M)=H^p_+(M)\oplus H^p_-(M)\quad(p\geq0).}
$$
This is the <eigenspace decomposition of a linear involution>. The factor $1/2$ uses the real coefficient field; a corresponding argument over characteristic two would fail. At the level of <differential forms>, the same averaging projections give representatives of either parity whenever the class has that parity.
Back to article page
