On Rudyak's "On Thom spectra, orientability, and Cobordism", the following axioms are given:

Let $\mathscr{S}$ be the category of spectra, and let $\Sigma\colon \mathscr{S}\to \mathscr{S}$ be the translation functor defined as follow: for a spectrum $X$, $(\Sigma X)_n:=X_{n+1}$

An homology theory on $\mathscr{S}$ is a family $\{h_n,\hat{\mathfrak{s}}_n\}$, $n\in \mathbb{Z}$ of covariant functors $h_n \colon \mathscr{S}\to \textbf{Ab}$ and natural transformations $\hat{\mathfrak{s}}_n\colon h_n \to h_{n+1}\Sigma$ satisfying the following axioms:

  • The homotopy axiom. If the morphisms $f,g\colon X \to Y$ are homotopic, then the induced homomorphism $h_n(f),h_n(g)\colon h_n(X)\to h_n(Y)$ coincide for every $n$.
  • The exactness axiom. For every cofiber sequence $X\xrightarrow{f} Y \xrightarrow{g} Z$ of spectra, the sequence $$h_n(X)\xrightarrow{h_n(f)} h_n(Y) \xrightarrow{h_n(g)} h_n(Z)$$ is exact.

The problem is the very next Proposition 3.11 (3.11 page 62)

We are given a cofiber sequence and the associated long cofiber sequences $$ \cdots \to \Sigma^{-1}Z\to X \to Y \to Z \to \Sigma X \to \Sigma Y \to \Sigma Z \to \cdots$$ and by the second axiom we have a l.e.s.

$$ \cdots \to h_n(\Sigma^{-1}Z)\to h_n(X) \to h_n(Y) \to h_n(Z) \to h_n(\Sigma X) \to h_n(\Sigma Y) \to h_n(\Sigma Z) \to \cdots$$

Then, the author states that the $\hat{\mathfrak{s}}_n$ are isomorphisms and concludes the proof. I don't think there are particular hypothesis here, so I don't know why it is true, since by axioms $\hat{\mathfrak{s}}_n$ is only a natural transformation.

I'm unable to find another reference for a definition of an homology theory on spectra in order to check whether there is something missing.

Can someone clarify this passage?

  • 1
    $\begingroup$ Yes you are right, $\hat{s}_n$ is probably meant to be an isomorphism. Otherwise you could take $h_{n+1}=0$ for every spectrum, in that case, you won't have long exact sequences. $\endgroup$ – Roland Mar 31 '16 at 14:25
  • $\begingroup$ @Roland yeah I suspected that as well, since then he build the suspension isomorphism for the induced reduced homology theory on pointed CW-complexes, and you need that guy to be an iso there. you can turn your comment in an answer so we can close this question! $\endgroup$ – Luigi M Mar 31 '16 at 15:49

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