## About the injective property from Morse-Novikov cohomology to de Rham-Witten cohomology

Let $\pi:\tilde{M}\rightarrow M$ be a smooth covering mapping,where $\tilde{M}$ is a compact manifold and $M$ is a manifold.

We define two operators:

(1) $d_{\theta}:A^{k}(M)\longrightarrow{A^{k+1}(M)}$, $\varphi\longmapsto{d\varphi-\theta\wedge\varphi}$, where $\theta\in A^{1}(M)$ and $d\theta=0$; So we get a cochain complex $(A^{k}(M),d_{\theta})$, Morse-Novikov cohomology $H^{k}_\theta(M)=\frac{\ker({d_{\theta}}:A^{k}(M)\rightarrow{A^{k+1}(M)})}{d_{\theta}(A^{k-1}(M))}$.

(2)Witten defined $d_{-1}=e^{\tilde{f}}\circ{d}\circ{e^{-\tilde{f}}}:A^{k}(\tilde{M})\longrightarrow{A^{k+1}(\tilde{M})}$,where $\tilde{f}\in C^{\infty}(\tilde{M})$ and $\pi^{\ast}\theta=d\tilde{f}$.So we get a cochain complex $(A^{k}(\tilde{M}),d_{-1})$, de Rham-Witten cohomology $H^{k}_{-1}(M)=\frac{\ker({d_{-1}}:A^{k}(\tilde{M})\rightarrow{A^{k+1}(\tilde{M})})}{d_{-1}(A^{k-1}(\tilde{M}))}$.

We know that the pullback $\pi^\ast:A^{k}(M)\rightarrow A^{k}(\tilde{M})$ induced homomorphism $\pi^\ast:H^{k}_{\theta}(M)\rightarrow H^{k}_{-1}(\tilde{M})$.

Question: Is the homomorphism $\pi^\ast:H^{k}_{\theta}(M)\rightarrow H^{k}_{-1}(\tilde{M})$ injective?

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