Corollary 4.6 (0N4Z) · ScalingStacks

0N4Z

Corollary 4.6 (Nonabelian Poincaré duality). For any $X$ in $\Space_{B}^{\geq n}$ , with associated $\disk^{B}_{n}$ -algebra $\Omega_{B}^{n}X\in\Alg_{\disk^{B}_{n}}(\Space)$ , there is a natural equivalence

\int_{M}\Omega_{B}^{n}X~\simeq~\Gammac(M,X)

between the factorization homology of a $B$ -framed $n$ -manifold $M$ with coefficients in $\Omega^{n}_{B}X$ and the space of compactly supported sections of $X$ over $M$ .

0N50

Proof. We apply Theorem 3.24: Since $\Gammac(-,X)$ is a homology theory, it is equivalent to factorization homology with coefficients in $\Gammac(\mathbb{R}^{n},X)$ , which is identified as the $n$ -fold loop space of the fiber of the map $X\rightarrow B$ , $\Gammac(\mathbb{R}^{n},X)\simeq\Omega^{n}_{B}X$ . ∎

David Ayala, John Francis

Original source: arXiv:1206.5522v6