Let $E$ be a spectrum. For any CW complex $X$, define $h\_\*=\pi\_i(E\wedge X)$. Then we know that $h\_\*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a cofiber sequence of spaces to a long exact sequence of abelian groups, also satisfy the wedge axiom in the definition of a homology theory. I want to know the converse case. Is every homology theory given by a spectrum in such way?


Thanks for all your comments. This is not really a problem. Anybody knows how to close it?



Is every homology theory given by a spectrum?
For homology theories on CW-complexes or homology theories that map weak equivalences to isomorphisms, that's Brown's representability theorem, which you can find in any textbook on stable homotopy theory. You forgot the important axiom of excision, by the way. The short answer is yes.


