Fighting doesn't help
A complaint came in yesterday again from the same woman who had complained about another cat. The caregiver in the area said that she seldom sees the cat and that she doesn't think the woman is telling the truth about what happened. This is quite possibly so, but the caregiver said she was going to 'fight' with the woman if she continued lying.
It doesn't help to assume the person is lying without trying to look into the situation. They may be - but if you keep an open mind, it'll allow you to really look into the situation. It also allows you to maintain credibility - let's say the person's allegation is true. Then you'll look as if you have no objectivity and that you are very biased. Your TC or Management Committee is going to be far less likely to ask you to look into complaints if that is the case.
In addition, fighting really doesn't help. It makes you seem very unreasonable - and the idea is to try and look reasonable and trustworthy. It also doesn't solve the problem. The TC might think that you are impossible to work with and decide to remove the cat instead of working with you.