For a connect smooth manifold $M$, there is an orientation covering $\pi:\tilde M\to M$, where $M$ is a two-copy of $M$ if $M$ is orientable and is a connected orientable manifold if $M$ is nonorientable. Anyway, the Deck transformation group of $\pi$ is $\{\mathrm{Id},\tau\}\simeq\mathbb Z/2\mathbb Z$, which acts also on the space of differential forms $\Omega^\bullet(\tilde M)$ by pullback. Then $\Omega^\bullet(\tilde M)$ can be decomposed into sum of two subspaces: $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm{odd}}^\bullet(\tilde M)$, where $$\Omega_{\mathrm {even}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = \omega\},$$ $$\Omega_{\mathrm {odd}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = -\omega\}.$$ We can identify ''odd forms'' and ''even forms'' in the sense of de Rham with elements in $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm {odd}}^\bullet(\tilde M)$, respectively.

For a connect smooth manifold $M$, there is an orientation covering $\pi:\tilde M\to M$, where $M$ is a two-copy of $M$ if $M$ is orientable and is a connected orientable manifold if $M$ is nonorientable. Anyway, the Deck transformation group of $\pi$ is $\{\mathrm{Id},\tau\}\simeq\mathbb Z/2\mathbb Z$, which acts also on the space of differential forms $\Omega^\bullet(\tilde M)$ by pullback. Then $\Omega^\bullet(\tilde M)$ can be decomposed into direct sum of two subspaces: $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm{odd}}^\bullet(\tilde M)$, where $$\Omega_{\mathrm {even}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = \omega\},$$ $$\Omega_{\mathrm {odd}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = -\omega\}.$$ We can identify ''even forms'' and ''odd forms'' in the sense of de Rham with elements in $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm {odd}}^\bullet(\tilde M)$, respectively.

For a connect smooth manifold $M$, there is an orientation covering $\pi:\tilde M\to M$ called, where $M$ is a two-copy of $M$ if $M$ is orientable and is a connected orientable manifold if $M$ is nonorientable. Anyway, the Deck transformation group of $\pi$ is $\{\mathrm{Id},\tau\}\simeq\mathbb Z/2\mathbb Z$, which acts also on the space of differential forms $\Omega^\bullet(\tilde M)$ by pullback. Then $\Omega^\bullet(\tilde M)$ can be decomposed into sum of two subspaces: $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm{odd}}^\bullet(\tilde M)$, where $$\Omega_{\mathrm {even}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = \omega\},$$ $$\Omega_{\mathrm {odd}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = -\omega\}.$$ We can identify ''odd forms'' and ''even forms'' in the sense of de Rham with elements in $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm {odd}}^\bullet(\tilde M)$, respectively.

For a connect smooth manifold $M$, there is an orientation covering $\pi:\tilde M\to M$, where $M$ is a two-copy of $M$ if $M$ is orientable and is a connected orientable manifold if $M$ is nonorientable. Anyway, the Deck transformation group of $\pi$ is $\{\mathrm{Id},\tau\}\simeq\mathbb Z/2\mathbb Z$, which acts also on the space of differential forms $\Omega^\bullet(\tilde M)$ by pullback. Then $\Omega^\bullet(\tilde M)$ can be decomposed into sum of two subspaces: $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm{odd}}^\bullet(\tilde M)$, where $$\Omega_{\mathrm {even}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = \omega\},$$ $$\Omega_{\mathrm {odd}}^\bullet(\tilde M) = \{\omega\in\Omega^\bullet(\tilde M)|\tau^*\omega = -\omega\}.$$ We can identify ''odd forms'' and ''even forms'' in the sense of de Rham with elements in $\Omega_{\mathrm {even}}^\bullet(\tilde M)$ and $\Omega_{\mathrm {odd}}^\bullet(\tilde M)$, respectively.