ScalingStacks

Proof. We first consider the case that $\mathcal{V}=\bigl({\sf Spectra},\wedge\bigr)$ and $V=\Sigma^{\infty}X$ is the suspension spectrum of a pointed connected space $X$. There is a natural equivalence of spaces

$$\int_{S^{m}\times\mathbb{R}^{n-m}}\Omega^{n}\Sigma^{n}X~\underset{\rm Cor~\ref{non-abel}}{\simeq}~(\Omega^{n-m}\Sigma^{n}X)^{S^{m}}~\simeq~\Omega^{n}\Sigma^{n}X\times\Omega^{n-m}\Sigma^{n}X$$

where the first equivalence is by nonabelian Poincaré duality and the second is by the standard trivialization of each fiber sequence $\Map_{*}(K,G)\rightarrow G^{K}\rightarrow G$ whose base is equipped with the structure of a group-like $\mathcal{E}_{1}$-space. (It is this second step that requires the strict inequality $m<n$.) Passing to suspension spectra, Proposition 5.5 begets the further equivalence

$$\bigvee_{k\geq 0}\conf_{k}(S^{m}\times\mathbb{R}^{n-m})\underset{\Sigma_{k}}{\otimes}\Sigma^{\oo}X^{\otimes k}\simeq\Bigl(\bigvee_{i\geq 0}\conf_{i}(\mathbb{R}^{n})\underset{\Sigma_{i}}{\otimes}\Sigma^{\oo}X^{\otimes i}\Bigr)\otimes\Bigl(\bigvee_{j\geq 0}\conf_{j}(\mathbb{R}^{n-m})\underset{\Sigma_{j}}{\otimes}\Sigma^{\oo}(\Sigma^{m}X)^{\otimes j}\Bigr)~.$$

Collecting coefficients of terms which are homogeneous in $X$ determines a $\Sigma_{k}$-equivariant stable homotopy equivalence

$$\Sigma^{\infty}_{\ast}\conf_{k}(S^{m}\times\mathbb{R}^{n-m})\simeq\coprod_{i+j=k}\Bigl(\Sigma_{k}\underset{\Sigma_{i}}{\times}\Sigma^{\infty}_{\ast}\conf_{i}(\mathbb{R}^{n})\Bigr)\otimes\Bigr(\Sigma_{k}\underset{\Sigma_{j}}{\times}\Sigma^{\infty}_{\ast}\bigl(\Sigma^{mj}_{+}\conf_{j}(\mathbb{R}^{n-m})\bigr)\Bigl)$$

where $\Sigma_{k}\underset{\Sigma_{l}}{\times}-$ is induction from $\Sigma_{l}$-spectra to $\Sigma_{k}$-spectra.

Now consider the general case for $\mathcal{V}$, according to the hypothesis. After Proposition 5.5, both sides of the equivalence split as coproducts in homogeneous terms $V^{\otimes k}$, so it suffices to show that the coefficients of these terms are degreewise equivalent. Inspecting, we thus seek an equivalence in $\mathcal{V}$:

$$\conf_{k}(S^{m}\times\mathbb{R}^{n-m})\underset{\Sigma_{k}}{\otimes}V^{\otimes k}~\simeq~\bigoplus_{i+j=k}\Bigl(\conf_{i}(\mathbb{R}^{n})\underset{\Sigma_{i}}{\otimes}V^{\otimes i}\Bigr)\otimes\Bigr(\conf_{j}(\mathbb{R}^{n-m})\underset{\Sigma_{j}}{\otimes}(\Sigma^{m}V)^{\otimes j}\Bigl)~.$$

This equivalence follows from the conclusion of the previous paragraph upon tensoring with $V^{\otimes k}$ and taking balanced $\Sigma_{k}$-coinvariants.

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