Your daughter’s father has shown no interest in having a relationship with you or your daughter. But, as you say, it is useful to keep the door open.
Also, you might not want this man, who you know very little about, in your daughter’s life. He might be horrible. Be careful what you wish for.
In your circumstances, I think I’d send him an email on daughter’s birthday and/or at Xmas each year with a short note – daughter started school, can ride a bike, has learnt to swim etc – and maybe a photo.
Leave it at that. Don’t chase, don’t expect a reply. When your daughter asks, you could say ‘daddy’s name is Fred, he has brown eyes and likes cycling, but he didn’t enjoy being a daddy so that’s why we don’t live with him.’
That gives her enough to tell her little classmates, because they will ask. It’s true and it gives her something to imagine.
I’d leave trying to build a connection between them until her teens or later. She needs to be old enough to deal with rejection and that’s very tough. ☹️
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