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Pullback of a differential form commutes with the exterior derivative, so $f^{\ast }$ gives a linear map on de Rham cohomology. Functoriality and $f\circ f=\mathrm{id}$ imply $(f^{\ast }{)}^2=I$. Over the real numbers, define the complementary projectionsThey obey $P_{\pm }^2=P_{\pm }$, $P_{+}{P}_{-}=0$ and $P_{+}+P_{-}=I$. Every cohomology class therefore decomposes asThe first term is fixed and the second negated by $f^{\ast }$. Their intersection is zero, since a class in both equals its own negative. ConsequentlyThis 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.