There is no DNA theory, there are some inherently flawed studies by people who set out holding the conclusion as one of their assumptions (that Neandertal went extinct, iow, has no living descendants).
There was no fraud involved with "Nebraska Man". It was a mistake.
BTW, the primary researchers never even referred to it as a "man". That was an over enthusiastic English scientist, who made this suggestion in the popular press there. The serious scientific advocates of this short lived fossil phenom were only claiming that it was an anthropoid ape of some sort. That was extraordinary enough, as no apes whatever are known from the Americas, and not even monkeys from North America.
It was a single badly worn molar, actually from a pig as it turned out. Unfortunately, not only are pig teeth and ape/human teeth grossly similar (ironically the primary researcher studying and describing Hesperopithecus had made just this point in a previous paper!) but this particular tooth had been rotated in the jaw in life, in such a way that it acquired a human-like wear pattern.
Anyway, within a few years they found more of the material. The claim was corrected and retracted in 1927. (The tooth was first described in 1922.)
See this article for more.