The professor seemed positive but the students feared retaliation, so they signed a petition to investigate false allegations against him.
The missing words in the sentence are:
1. The professor seemed very positive but the students were afraid he would retaliate, which is why they signed the petition to investigate the false allegations made against him.
2. The students believed that the professor was being unfair or biased towards them. They were worried that he would take some form of revenge against them if they voiced their concerns openly.
3. By signing the petition, the students wanted to bring attention to the false accusations made against the professor and demand a thorough investigation into the matter.
The probable question may be:
Why did the students sign a petition to investigate false allegations against the professor, even though he seemed positive?