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1-minimal models for $C_{\infty}$-algebras and flat connections. (arXiv:1712.04789v1 [math.AT])
来源于:arXiv
Given a smooth manifold $M$ equipped with a properly and discontinuous smooth
action of a discrete group $G$, the nerve $M_{\bullet}G$ is a simplicial
manifold and its vector space of differential forms
$\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ carry a
$C_{\infty}$-algebra structure $m_{\bullet}$. We show that each
$C_{\infty}$-algebra $1$-minimal model $g_{\bullet}\: : \: \left(W,
{m'}_{\bullet} \right) \to \left(
\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right) $
gives a flat connection $\nabla$ on a smooth trivial bundle $E$ on $M$ where
the fiber is the Malcev Lie algebra of $\pi_{1}(M/G)$ and its monodromy
representation is the Malcev completion of $\pi_{1}(M/G)$. This connection is
unique in the sense that different $1$-models give isomorphic connections. In
particular, the resulting connections are isomorphic to Chen's flat connection
on $M/G$. If the action is holomorphic and $g_{\bullet}$ has holomorphic image
(with logarithmic singula 查看全文>>