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

Question

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 $\mathbf{B}G_+$?

Answer 1

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 $p$.

$$EG_+\wedge_GS^0:=\frac{EG\times_G\{p,q\}}{EG\times_G\{p\}}$$ $$\simeq \frac{EG\times_G\{p\}\coprod EG\times_G\{q\}}{EG\times_G\{p\}}$$ $$\simeq \frac{BG\coprod BG}{BG}$$ $$\simeq \text{pt} \coprod BG $$ $$\simeq BG_+$$