Long story short, I’ve been fighting with my landlord because my neighbor has been smoking all day and in all hours of the night (2-6 am). We share ventilation so when they smoke all their smoke comes into our room. And our room has no window and we cannot turn on the AC due to shared vents (otherwise we blow more smoke in). So in short we have been living in smoke for hours every day. My landlord went into their room once and caught them smoking. They told me if they caught them they would evict them as it is written in the contract. However they gave them another chance and when I went back to tell my landlord they still smoke, my landlord changed the tenants that she caught smoking to my other side neighbor . I told her I was sure I saw her go in to the other neighbors room, but she keeps denying it and to this day they are still smoking because she is lying and will not evict the proper neighbor.
Can I Sue my landlord if my landlord lied about which tenant they caught smoking?
It appears that you are entitled to injunctive relief under a legal theory of landlord's breach of contract or something akin to breach of the implied warranty of habitability. Claims of constructive eviction require that you as a result of recurrent nuisance(s) leave the premises within reasonable time, which so far is not the case. Only the non-smoker neighbor would have standing to sue the landlord for his false statements of fact.
From a contract-law standpoint, any terms of your lease prohibiting smoking create a reasonable presumption that the landlord would enforce similar terms in neighboring units in order to protect you from that nuisance.
Even if your lease is devoid of language supporting the aforementioned presumption, you may sue the landlord on grounds akin to Sholberg v. Truman, 852 N.W.2d 89, 92 (2014) "[a] defendant held liable for the nuisance must have possession or control of the land", more so that now the landlord is resorting to false statements for the purpose of eluding his assurance(s) or duty to you. The landlord obviously knows that the existence of shared ventilation subjects you to adverse conditions and consequences of the tenant's recurrent violation of his lease.
You might also want to inform your non-smoker neighbor about the landlord's false statements. To avoid a situation of unverifiable "he said, she said" statements, hopefully you have (or can get) evidence in writing of the landlord's lies. In turn, you should ask the non-smoker neighbor to execute an affidavit rebutting the landlord's false statements. That affidavit will strengthen the merits for injunctive relief.
At this point the non-smoker neighbor would obtain only an award of nominal damages if he were to sue the landlord for defamation. However, this neighbor's awareness of the landlord's conduct might help the neighbor to preempt landlord's other misconduct that in the future could be detrimental to him, the non-smoker neighbor. After all, you yourself would like to become aware of false statements that other is spreading about you.