Is every homology theory given by a spectrum? - MathOverflow

Is every homology theory given by a spectrum?

2011-05-05T06:07:23Z

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?

Answer by Tilman for Is every homology theory given by a spectrum?

2011-05-05T06:57:53Z

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.

Answer by Mark Grant for Is every homology theory given by a spectrum?

2011-05-05T07:03:45Z

The answer is yes, if you replace the wedge axiom with the stronger direct limit axiom $h_{i}(X) = \mathrm{lim}\ h_{i}(X_{\alpha})$, where $X$ is the direct limit of subcomplexes $X_{\alpha}$.

As well as Switzer, this is discussed in Chapter 4.F of Hatcher's "Algebraic Topology", Adams' little blue book "Stable homotopy and generalised homology", and Adams' paper "A variant of E. H. Brown's representability theorem".