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Let $\Sigma^{\infty}$ denote the suspension spectrum functor from pointed topological spaces (=CGWH spaces) to orthogonal spectra. As usual, a weak equivalence of spaces is a continuous map inducing a bijection on $\pi_0$ and an iso on all $\pi_i$, $i\ge 1$, for all choices of basepoint. For orthogonal spectra, I refer to the usual notion of stable equivalence, see e.g. Mandell-May-Shipley-Schwede.

Question: If $f\colon X\longrightarrow Y$ is a weak equivalence, is $\Sigma^\infty f\colon \Sigma^\infty X\longrightarrow \Sigma^\infty Y$ a stable equivalence?

Remarks: 1. This is true if $X$ and $Y$ are well-pointed, see this MO discussion.

  1. In general, if $X$ and $Y$ are not well-pointed, then the reduced suspension does not preserve weak equivalences. For example, take the set $\{0\} \cup\{\frac 1n\}$ which receives a weak equivalence from a countable discrete set. The reduced suspension is a map from a wedge of spheres to the Hawaiian earrings which is not a weak equivalence.
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  • $\begingroup$ Doesn't your counterexample in $2$ answer this in the negative if $X,Y$ are not well-pointed? $\endgroup$ – Tyrone Jul 14 at 19:47
  • $\begingroup$ @Tyrone: it is not clear to me whether the map becomes a weak equivalence after suspending further $\endgroup$ – nikola karabatic Jul 15 at 9:45
  • $\begingroup$ Enough of the homology of the Hawaiian earing is known and is not the homology of the wedge. In fact $\pi_n$ of the $n$-dimensional Hawaiian earing is known (and it is not even free abelian if I recall). $\endgroup$ – Tyrone Jul 15 at 9:52
  • $\begingroup$ Hmm you are right about the homology, but I think this argument uses the suspension iso for reduced suspension and reduced homology implicitely and this does NOT hold for not-well-pointed spaces (since the reduced cone is not acyclic, at least I don't see why it should be). $\endgroup$ – nikola karabatic Jul 15 at 9:57
  • $\begingroup$ Or did you intend to argue in another way? $\endgroup$ – nikola karabatic Jul 15 at 9:57

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