Does anyone know what kind of material is used for the interior roof of a car?

I have helped redo a few of these. Mostly an american car problem. The reason it has fallen is because it was originally glued to a very thin layer of foam rubber and that foam has deteriorated. If you remove the side trim pieces around the roofline, ussually clipped in place, remove the interior light, sun visors, etc. and the panel will come out. Scrape it with a putty knife until it is really clean, buy a piece of light weight cloth from any fabric store and you can use a very thin piece of foam again. The foam allows you a lot of leaway when gluing the cloth back on. Many fabric stores actually stock headliner material and the foam. Fabric store clerk will know exactly what you need. This is not expensive. Use the 3M spray on glue and glue the foam to the panael then glue the material to the foam. Hint, allow to dry in the sun for a few hours before reinstalling the headliner and don't soak the material or the foam with to much glue, a light coating will do. It is a project for a day off. Good Luck...

I would suggest that you talk to someone who fits sun-roofs. They will be the experts since they have to cut through the roof. They should know not only the outermost material but also any inner insulating material and the appropriate glues that will be needed. They might even do a full repair job.