  This article is devoted to the study of a higher-dimensional generalisation
of de Rham epsilon lines. To a holonomic $D$-module $M$ on a smooth variety $X$
and a generic tuple of $1$-form $(\nu_1,\dots,\nu_n)$, we associate a point of
the $K$-theory space $K(X,Z)$. If $X$ is proper this $K$-theory class is
related to the de Rham cohomology $R\Gamma_{dR}(X,M)$. The novel feature of our
construction is that $Z$ is allowed to be of dimension $0$. Furthermore, we
allow the tuple of $1$-forms to vary in families, and observe that this leads
naturally to a crystal akin to the epsilon connection for curves. Our approach
is based on combining a construction of Patel with a homotopy invariance
property of algebraic $K$-theory with respect to $(\mathbb{P}^1,\infty)$. This
homotopical viewpoint leads us naturally to the definition of an epsilon
connection in higher dimensions. Along the way we prove the compatibility of
Patel's epsilon factors with the graded lines defined by Deligne and
Beilinson--Bloch--Esnault in the case of curves.
