I have looked around for hours and although I have seen many definitions of bordism and cobordism (for some authors these two coincide and for some other not (without mentioning explicitly what's the difference)), I wasn't able to understand how they defined and what's the difference (if exists).
I know one definition, of iso. classes of $n$-manifolds over a manifold, say $X$ modulo co(bordism), but I have seen this definition to mention both bordism and cobordism too. So I don't know to what really corresponds.
Moreover, when looking of cobordism (co)-homology theory thinks are getting more complicated. For instance, I am aware of the Thom's theorem which associates the cobordism ring with $\pi_{*}(MU)$, therefore I understand the terminology. However, the complex cobordism theory many authors say that is represented by the Thom complex, but the latter is not $\Omega$-spetrum, thus it cannot represent a cohomology theory (corresponds to such a theory but not represents). What struggles me is the usage of terminology, and how the authors use them interchangeably. Could you please explain me what's the difference, and how the Thom spectrum represents the coborism theory?
P.S.1 By Brown's representability theorem, even though the Thom spectrum is not representing as I said, but constructs a cohomology theory, we know that this theory is represented by some $\Omega$-spectrum. So an additional (natural) question is, what's this representing $\Omega$-spectrum that represents the cobordism theory?
P.S.2 I have read many other related question MSE but no answer replies really to this question. So hope to finally someone write out an answer to sort out th above!
Thank you!