Is the Trojan War based on fact?

We don't know. It came from an oral tradition of an attack by the Greeks (Achaians) on the islands and coast of Asia Minor. This tradition was used by bards who made up poems which they sang as entertainments on their circuit of towns and cities (very popular before TV and movies). They made up the poems during the performance, using stock phrases around a central theme. Versions were written down after syllabic writing was invented in the 8th Century BCE. The version we have, attributed to one Homer, probably dates about 725 BCE, with the events centred on the 12th Century BCE.

Like all legendary oral tales, there are probably facts in there somewhere. Distortions of oral stories accelerate, and after about three generations, without supportive evidence from other sources, is so corrupted that it is difficult to discover the facts from the fictions and exaggerations. As our written source came from nearly five hundred years after the supposed event, and the bards made a point of not only telling different versions to please audiences, but also varied their own versions, the distortions are much multiplied. And Homer's version itself was continually modified in written form.

So what is fact in the story - we can only guess. Ajax was dead before the reputed start of the war. The political conditions depicted are in reality those of 8th Century BCE Greece. The Boar's tooth helmet mentioned comes from 14th Century BCE.

The best guess is that the Greeks (Achaians) organised a coordinated series of pirate raids on the coast and islands of Asia Minor. After an initial failed attempt on Troy, they pillaged the rest of the area over a period of about ten years, and then came back for another go at Troy, which succeeded.