He first declines a drink, saying he's tired, but then decides he'll have some whisky.
Fowler hands him a glass, showing him his calm nerves, and asks why Vigot continues to suspect him. Is it because he wanted Phuong back?
Vigot says no, that's not it. That motive wouldn't account for Fowler's taking Pyle's book by York Harding.
Speaking of Harding, Fowler tells the police officer that the author killed Pyle at long range.
This confuses Vigot, as you might expect.
So Fowler explains that Pyle came out here indoctrinated on York Harding's idea about the Third Way, and his meddling got him killed.
Vigot says that he doesn't think Fowler is a criminal, but he knows he's more involved than he's admitted, and he lied about not seeing Pyle the night he died.
Why does he say this? Because Fowler's chronology of the events of that evening don't add up. Plus Pyle's dog had cement on his paws that he had likely gotten from the wet cement outside Fowler's apartment.
Vigot also finds it suspicious that Fowler is not at all angry with him for the continued questioning.
After Vigot leaves, Fowler tells us that he did see Pyle the night he died and wishes he had the courage to call Vigot back and tell him.