I was sent for training at another location, and I observed that the supervisor there doesn't like me. I still was able to learn what I went out there for without having issues with anyone. When I left, he wrote a very bad report about me, saying that some people in the rig we worked complained about me. I contacted those people and they denied saying those things. Now I want to write to the manager demanding punishment for the supervisor for lying against me and trying to get me fired. How should I approach management about this?
Demanding punishment is not a good strategy, it makes you look bad.
Just state the facts clearly, that should make sure that you are believed and not him.
So: “Supervisor claimed that people complained about me. That is not true. Nobody complained about me, and I gave nobody reason to complain.
Specifically, supervisor claimed that X complained. I asked X whether this was true, and X told me he made no complaint at all. And so on.
I also felt that supervisor tried to stop me from getting proper training. For example, and so on. “
Just calm, just stating the facts. That will work best.
First of all, you need to calm down, telling your manager that you demand he be punished is only going to backfire on you.
Talk to the people named in the supervisor's report and ask them if they are willing to give statements to your manager about the deceit. Odds are they will do so gladly since their names have been used in vain and their trust in the supervisor is likely broken.
Once you've done that, then go to your manager, state something to the tune of, "In regards to [supervisor's] report, I have spoken to [names]. They have told me that the statements in their name are untrue and are willing to give you statements and genuine feedback on my performance that day." You can tell your manager you believe the statements are defamatory (since you believe they will hinder your career) as well as untrue, because this combination will get the attention of the HR department, but do not demand anything. Stick to the facts.