# What is the pointed Borel construction of the $0$-sphere?

From what I understand, the Borel construction takes a $$G$$-space $$X$$ and produces a topological space $$X\times_{G}\mathbf{E}G$$―the homotopy quotient $$X/\!\!/G$$ of $$X$$ by $$G$$ in the $$\infty$$-category of spaces $$\mathcal{S}$$―satisfying $$\mathrm{H}^*(X\times_G\mathbf{E}G;R)\cong\mathrm{H}^*_G(X;R).$$ One partiularly important example is given by $$\mathbf{B}G$$, the Borel construction / homotopy quotient of the point: $$\mathbf{B}G\simeq\mathbf{E}G\times_G*$$.

Moving to the pointed setting, one has the pointed Borel construction, which takes a pointed $$G$$-space $$X$$ and returns $$\mathbf{E}G_+\wedge_{G}X$$, the homotopy quotient of $$X$$ by $$G$$ in the $$\infty$$-category of pointed spaces $$\mathcal{S}_*$$. Concretely, it is given by \begin{align*} \mathbf{E}G_+\wedge_{G}X &\overset{\mathrm{def}}{=} \frac{\mathbf{E}G\times_G X}{\mathbf{E}G\times_G*},\\ &\cong \frac{\mathbf{E}G\times_G X}{\mathbf{B}G}. \end{align*} Now, $$\mathbf{E}G_+\wedge_G*\cong*$$, rather than $$\mathbf{B}G$$. But while $$*$$ is the monoidal unit of $$\mathcal{S}$$, it is not that of $$\mathcal{S}_*$$, which is $$S^{0}$$. Hence it would be interesting to know:

Question. What is the pointed Borel construction $$\mathbf{E}G_+\wedge_G S^0$$ of the $$0$$-sphere? Is it related to $$\mathbf{B}G_+$$?

• Your definition is wrong. It's the same as EG x_G X modulo EG x_G pt; the basepoint you have in the numerator gets collapsed when you take the smash product. Beyond that, this is a definition push, so I'm not sure where you run into trouble. Yes, EG_+ ^_G S^0 = BG_+. But this is not really interesting: (X_+ ^ Y_+) = (X ^ Y)_+ in general. Quotienting by G changes nothing...
– mme
Aug 23, 2021 at 9:28
• This is a sidenote: for any $\infty$-category $\mathcal C$ and a $G$-equivariant object $X\colon BG\to\mathcal C$, the homotopy orbit (i.e. "homotopy quotient" in your post) $X_{hG}$ is simply the colimit taken over $BG$.
– Z. M
Aug 23, 2021 at 16:48
• @mme Thanks! I edited the question and corrected the definition.
– Théo
Aug 23, 2021 at 21:12

Let's apply your definition (which I think has typos on the RHS - the two "+" subscripts on the EG should not be there I think). Let's model everything as topological spaces and do the calculation there. Let's model $$S^0$$ as the discrete set $$\{p,q\}$$ with basepoint $$p$$.
$$EG_+\wedge_GS^0:=\frac{EG\times_G\{p,q\}}{EG\times_G\{p\}}$$ $$\simeq \frac{EG\times_G\{p\}\coprod EG\times_G\{q\}}{EG\times_G\{p\}}$$ $$\simeq \frac{BG\coprod BG}{BG}$$ $$\simeq \text{pt} \coprod BG$$ $$\simeq BG_+$$