Starting Problem

I have an 94 acura integra Ls 5 speed manual. Recently, it failed to start after I got out of class. The battery was still good, all the lights came on as usual.I had not idea why the engine didn't turn on.I tried to jump start it. Cable and manually pushing it up and down the parking lot. No luck.I ended up towing it the next day to my in laws. Next day I tore the alternator apart to check if that was the problem. Seems like the alternator was still working because I tested it out on the car battery with wires. It spun as usual. Checked the spark plugs. Looks ok. Checked all the wires from the alternator to the ignition with a guage. They all have continuity and nonthing seems to have came off or broke. Checked all the fuses. Nothing wrong. Checked the relay switch took it off. Connected a wire instead( almost like hot wiring it) and turned the key. And voila. The cAr came to life again. The car will only start with the wire in place.I dont even have to press on the clutch to turn it on. Which is weird.I have no clue as why that is happening.I am stumped.I bought a new relay switch but it still did not solve the problem.I have asked around and no one seems to know what is wrong. Some say wire problems, or clutch solenoid malfunction.I have no idea. Once the wire is in place and the engine comes on. It runs like normal. Even when I take the wire off. But. Without the wire in the relay switch's place, the car wont start.