ScalingStacks

Remark 3.29. One could replace the $\oo$-category of topological $n$-manifolds and embeddings with that of smooth $n$-manifolds and smooth embeddings, $\mfld^{\sm}_{n}$, or piecewise linear $n$-manifolds and piecewise linear embeddings, $\mfld^{\sf PL}_{n}$, and Theorem 3.24 is still valid. The proof, in fact, is even simpler, as the existence of handlebody structures is much easier than in the case of topological manifolds. However, a consequence of smoothing theory [KS] is an equivalence

$$\mfld^{\sm}_{n}\simeq\mfld_{n}^{{\sf BO}(n)}$$

between smooth $n$-manifolds and $\BO(n)$-framed topological $n$-manifolds, so long as $n$ is not equal $4$, so nothing new is obtained by considering smooth or piecewise linear manifolds rather than $B$-framed topological manifolds. In the case of smooth 4-manifolds, there is still an equivalence $\disk_{4}^{\sm}\simeq\disk_{4}^{{\sf BO}(4)}$, and so combining the smooth and topological versions of Theorem 3.24 gives an equivalence

$$\mathbf{H}(\mfld_{n}^{\sm},\mathcal{V})\simeq\mathbf{H}(\mfld_{4}^{{\sf BO}(4)},\mathcal{V})$$

between homology theories for smooth 4-manifolds and homology theories for ${\sf BO}(4)$-framed topological 4-manifolds. Since the ${\sf BO}(4)$-framing on the tangent microbundle is only a very weak measure of a smooth structure in dimension 4 (e.g., there is a single ${\sf BO}(4)$-framing of $\mathbb{R}^{4}$, in contrast to the uncountably many smooth structures), this form of factorization homology is not a refined invariant of smooth 4-manifolds.