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## Cocycle invariants of codimension 2 embeddings of manifolds

### Volume 103 / 2014

Banach Center Publications 103 (2014), 251-289 MSC: 57Q45, 57M25. DOI: 10.4064/bc103-0-11

#### Abstract

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 a 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).

#### Authors

• Józef H. PrzytyckiDepartment of Mathematics
The George Washington University
Washington, DC 20052, U.S.A.
University of Maryland CP
and
University of Gdańsk
Poland
e-mail
• Witold RosickiInstitute of Mathematics
University of Gdańsk
Poland
e-mail

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