help with Eames chaise longue Billy Wilder  

?
What exactly is black support?

black support
Mark,
I did a google image search for black support and this is what appeared.

It's the like leather fabric under the leather cushions
to support the body when lies down. I'm very sorry for my bad english

just joking
jas,
Do not concern yourself with the spelling, I was just joking with Mark. Is the attached photo the Herman Miller piece you are discussing?
There are many people on this forum with a great amount of knowledge of Eames pieces and I am sure that you will get the answers you desire.

Exactly. This is the Chaise I...
Exactly. This is the Chaise I Refer. Under the leather cushions (pillows) there are a fabric, cover with black leatherette which is unglued by the use and abuse.

my best suggestion
jas,
All the "experts" seem to know about the Eames® lounge and ottoman and the inexpensive chairs. Should you not get any answers here regarding your Billy Wlider chaise lounge (this and the la Chaise are my favorites) I would suggest that you contact Herman Miller or Vitra (dependent on your country) to get the information you require. Best of luck.

Thanks niceguy
I will, but I hope that some connaisseur of this Forum, can help me in this matter.

Herman Miller official web page call this material fabric sling
Please, where I can find this kind of material

now
jas,
Now I belive that it is about time for you to do more of your own "homework". Call Herman Miller for the Home 800-646-4400 (this is also on their site) and ask them.