# Reconstructing a group from deformed classifying space

Suppose that we have a finite group $G$. Its classifying space $BG$ is not naturally pointed, so if we would like to consider it as a spectrum there are two possible approaches. The first is to take any point in the classifying space and let it be the basepoint when constructing the suspension spectrum; the second is to add a disjoint basepoint.

My question is: is it possible to reconstruct $G$ from these spectra? If so, how? What information do we know about $G$ if we are given one of these spectra?

• 1) Yes. 2) Taking the fundamental group (of the nontrivial component in the second case. 3) I don't understand this question. – Fernando Muro Jul 1 '13 at 20:35
• Sorry, am I just being daft? I always thought $\mathrm B G$ was pointed for any group $G$. Take the canonical map of groups $1 \to G$ (where $1$ is the one-element group) and apply $\mathrm B$ to it. But $\mathrm B 1 = \mathrm{pt}$. – Theo Johnson-Freyd Jul 1 '13 at 20:44
• @TheoJohnson-Freyd, it is pointed if you point it! One can decide not to :-) – Mariano Suárez-Álvarez Jul 1 '13 at 21:11
• @FernandoMuro: if G is not abelian, then $\pi_1^s BG \neq G$. And for question 3, I want to know how much, if not all, information can we reconstruct from the space? (This assumes that the answer to the first question is "no.") – Inna Jul 1 '13 at 21:21
• @MarianoSuárez-Alvarez, sure, you may always forget the pointing of a pointed space, or re-point it some other way. But OP had said "not naturally pointed", and I think that the functor $B: \mathrm{Groups} \to \mathrm{Spaces}$ does receive a canonical natural transformation from the constant functor $pt$. In any case, trying to turn it into a spectrum will, at least a priori, lose information. – Theo Johnson-Freyd Jul 1 '13 at 21:44

Let $X_+$ be a space $X$ together with a disjoint base-point. Since $\Sigma^\infty(BG_+)=\Sigma^\infty(BG)\vee S$, where $S$ is the sphere spectrum, there's not much difference between the two spectra you propose. In Example 5.2 of Martino–Priddy's 'Stable homotopy classification of $BG^{\hat{}}_p$' you can find non-isomorphic finite groups whose classifying spaces have the same suspension spectrum, so the answer to your first question in its current form is no. In that paper you can also find very interesting results concerning your last question.
The same problem arises with disjoint basepoints of course. The homology of $BG_{+}$ is the same as that of $B(e)_{+} \simeq S^0$.