I am an undergraduate of CS and I participate in TCS.SE. I am having trouble finding the error in a proof about the polynomial hierarchy collapsing to the first level ($NP = coNP$). I believe the reasoning to be false, because it is very simple and it would be common knowledge, however it is not.
Let $M$ be any oracle TM in $NP^{NP}$ (i.e. $ \Sigma^{P}_{2} $ ) . Obviously, by ignoring its oracle, $M$ can solve any problem in NP.
On the other (and most important) direction, let $M'$ be a TM where instead of using the oracle, we encode it into a simple non-deterministic machine with a running time of $NTIME( k(n) )$. If $M$ had a running time of $NTIME( p(n) )$ , $M'$ will have a running time of $NTIME( p(n)*k(n) )$ which is polynomial. Therefore, for any oracle TM in $NP^{NP}$, there is a non-deterministic TM that can solve the same problem in polynomial time. It follows that $NP^{NP} /subset NP$.
By combining the two statements we have that $NP=NP^{NP}$ and extending this, $NP=coNP=P^{NP}=NP^{NP}$ and the polynomial hierarchy collapses to the 1st level.
Where is the error in the above proof?