I think my adult daughter is lying to me

Nodding, smiling, staying friendly etc could all be seen, by either or both of them, as lack of support.
You need to tell each of them that you can't possibly be unbiased, you're finding the whole situation upsetting and that they need to talk it out with someone else, possibly a counselor.
That is especially the case if either of them is hoping the relationship can be saved.

When a marriage breaks up it's so emotional that facts get confused.
I couldn't count the number of times I lied to my parents when my first marriage went wrong. Or just avoided the truth. Mostly to protect them.
I agree with what others have said - don't probe too deep, just try to remain supportive of both.

There are 2 sides to every story, and in the end it makes no odds to you why they split, so stay friendly with both

They may well both be telling the truth, albeit from their own perspective. I hope, for their sakes, for yours and for the other parents' that they go for counselling and sort out their difficulties. You don't mention if children are involved.

They don't owe you any explanations and you don't need to figure out who's telling the truth. Please avoid asking them any further questions if that's what you've been doing. If one of them brings up the subject, change it as quickly as possible. If pushed to the wall, let whichever one know you're sure they made the best decisions for them and that they don't need to justify themselves to you.

They both need to talk to someone who can be unbiased. It would help if that was the same person e.g. a counsellor, but doesn't have to be.

There's only so far nodding and smiling can get you. It could be giving a completely wrong impression of agreeing with one or the other.

The best thing you can do is not take sides and allow them to sort their problems alone.

The advice here is sound, try not to get involved. It may help to avoid too much talk with either of them, the must sort this out, or separate, nothing you can do about it.

Sometimes, if you take sides , if they both get back together you can then be the 'baddy'.

I agree with Luckygirl. Stand back, don't take sides. If consulted nod, smile and make friendly comments to both. Just tell them both that it is something for them to sort out for themselves.

You cannot really help them, although I can understand how much you wish you could. They are adults - they must sort this out between them. Nod and smile and make friendly comments - but do not take sides - unless it is an abusive relationship that she has escaped.

How can you know that she or he are lying? No-one can be a fly on the wall to their relationship. They will have to sort it our or not in their own ways.

In the long term you need to maintain a good relationship with your DD, so should not even hint to her that you do not believe some of what she is saying.

Emotions are running high and a bit of exaggeration would not come as a surprise. She may back track on some of it when time has passed and the dust has settled. She needs you by her side, without vociferously decrying her partner.