ScalingStacks

Proof. Since $M_{0}\hookrightarrow M$ is a proper embedding, a compactly supported section over $M$ can be restricted to obtain a compactly supported section over $M_{0}$, as well as over $M\smallsetminus M^{\prime}$ and over $M\smallsetminus M^{\prime\prime}$. Namely, there is a diagram among spaces of compactly supported sections

By inspection, the bottom horizontal sequence is a fiber sequence, as is the right vertical sequence, as is the diagonal sequence. Also, the inner square is pullback because $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ is a pushout. Because $M_{0}\subset M$ is equipped with a regular neighborhood, these fiber sequences are in fact Serre fibration sequences, and so the inner square is a weak homotopy pullback square. In particular, there is a right homotopy coherent action of $\Omega\Gammac(M_{0},X)$ on $\Gammac(M^{\prime},X)$, a left homotopy coherent action of $\Omega\Gammac(M_{0},X)$ on $\Gammac(M^{\prime\prime},X)$, and a continuous map of topological spaces

(7)

$$\Gammac(M^{\prime},X)\underset{\Omega\Gammac(M_{0},X)}{\times}\Gammac(M^{\prime\prime},X)\longrightarrow\Gammac(M,X)$$

from the balanced homotopy coinvariants. Because $X\to B$ is $n$-connective and $M_{0}$ is $(n-1)$-dimensional, the base $\Gammac(M_{0},X)$ is connected. It follows that the map (7) is in fact a weak homotopy equivalence. The assertion follows after the canonical identification $\Omega\Gammac(M_{0},X)\cong\Gammac(M_{0}\times\mathbb{R},X)$ as group-like $\mathcal{E}_{1}$-spaces.

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