Hey Elliot,
The vac hose will come off the petcock and go to the carbs. It supplies vacuum from the carb to open a valve in the petcock, allowing fuel flow. If it's kinked, or leaking, etc. you get no fuel.

My question is this:
Is your bike a California model with the required P.A.I.R system? This is a pulse secondary air injection system that is essentially a maze of vacuum lines that are routed all over the dam bike, and it CAN be a major PITA. I'm all the way out here in PA and I managed to get one.
I removed all the P.A.I.R crap from mine a while back, but if yours is so equipped, you may not be able to find a good diagram. I know I had a couple issues along the way. I may be able to help. I can look at mine tonight and see where I have everything routed, and maybe post some stuff up.

 

Shadow is correct - the two identical fittings have overflow or vent lines attached. The tubes run down behind the carbs between them and the airbox and dump by the swingarm.

The suggestion to replace your battery is a good one. Just a little weak and the bike doesn't want to start.

If you find no vacuum leaks, and the vacuum line to the petcock is attached, not kinked or leaking - then you might consider a carb cleaning / rebuild. But before that you should do a complete tune up - check plugs, wires, valve clearances, etc. But it's a good possibility your carburetors are in need of attention.

Kilgoretrout is correct, too. If yours is a California bike with all the emission control equipment stripped off by someone who didn't know what they were doing you could have trouble making the bike run.

Good luck. Let us know how things turn out.

 

The more I got to thinking about it, I don't know if his year came with PAIR.
Mines a '95, and I don't know if the older one's had it??
Maybe not...

Battery is a good place to start, though. You could hook up jumpers to another 12V battery to see if it helps.

 

My Opinion.........think you have a lot of little items adding up

Get a new battery, check your regulator, new plugs anyway, add your new airbox and filer then have the carbs cleaned and sync'd and idle set proper.

They take time to warm up no matter what.

 

I ended up taking the bike to a shop in SF and they confirmed the worse bad compression (i though it felt a little sluggish) all cylinders have bad compression all around 100, 110, one is even 90. Then I had them do a leakdown test and its the rings and an intake or 2. They are going to try a valve adjustment and see if that makes it startable.

Well it was fun to ride for a couple days, I can't afford an overhual nor have the skills time or resources to do it myself. I guess i'll cut my losses and sell it to a salvage yard or something. Maybe part it out to try and make my $$ back when I'm feeling less depressed.

thanks for everyones help.