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A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map.

Not every ghost map $f$ is the zero object of the abelian group $\operatorname{Hom}_{SH}(X,Y)$ (more precisely, its image $\bar f\in \operatorname{Hom}_{SH}(X,Y)$ in the stable homotopy category $SH$).

Suppose $X=\Sigma^\infty A$ and $Y=\Sigma^\infty B$ are the suspension spectra of finite CW-complexes, let $f=\Sigma^\infty f'$ for a map $f'\colon A'\to B'$ and suppose that $f$ is a ghost map. Is $f$ necessarily zero in this case?

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    $\begingroup$ This is Freyd's Generating Hypothesis. It has been an open question for nearly 50 years. Many interesting consequences would follow if it were true; Mark Hovey's paper "On Freyd's Generating Hypothesis" is a nice survey. $\endgroup$ – Neil Strickland Jun 25 '14 at 14:04
  • $\begingroup$ Neil: why don't you put that as an answer to the question? $\endgroup$ – André Henriques Jun 25 '14 at 21:09
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Following André's suggestion I will promote this comment to an answer.

The OP's question is Freyd's Generating Hypothesis, which has been an open question for nearly 50 years. It is very hard to test, because there is no example of a nontrivial finite spectrum $X$ where all the groups $\pi_k(X)$ are known. Devinatz had a potentially interesting approach using chromatic theory and Gross-Hopkins duality, but that seems to have petered out. There is also some work on analogous questions in other triangulated categories.

Many interesting and surprising consequences would follow if the Generating Hypothesis were true; Mark Hovey's paper "On Freyd's Generating Hypothesis" is a nice survey. In particular, the $p$-completion of the stable homotopy groups of spheres would be a non-Noetherian, totally incoherent ring that is injective as a module over itself. Recently Leigh Shepperson and I wrote a paper investigating the algebraic theory of such rings, in the hope of shedding some indirect light on the Generating Hypothesis.

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