# Does the suspension spectrum functor preserve weak equivalences?

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.
• Doesn't your counterexample in $2$ answer this in the negative if $X,Y$ are not well-pointed? – Tyrone Jul 14 at 19:47
• @Tyrone: it is not clear to me whether the map becomes a weak equivalence after suspending further – nikola karabatic Jul 15 at 9:45
• 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). – Tyrone Jul 15 at 9:52
• 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). – nikola karabatic Jul 15 at 9:57
• Or did you intend to argue in another way? – nikola karabatic Jul 15 at 9:57