We consider the classical problem of a position of \(n\)-dimensional
manifold \(M^n\) in \(\mathbb{R}^{n+2}\). We show that we can define
the fundamental \((n+1)\)-cycle and the shadow fundamental
\((n+2)\)-cycle for a fundamental quandle of knotting \(M^n \to
\mathbb{R}^{n+2}\). In particular, we show that for any fixed
quandle, quandle coloring, and shadow quandle coloring of a diagram
of \(M^n\) embedded in \(\mathbb{R}^{n+2}\) we have \((n+1)\)- and
\((n+2)\)-(co)cycle invariants (i.e., invariant under Roseman moves).
The case \(n=2\) is well-known, and the case \(n=3\) we can explane in a
geometric way. The general case we described in arXiv:1310.3030v1.
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