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A characterization of groups of closed orientable surfaces in 4-space. (English) Zbl 0820.57017
A group is the fundamental group of the complement of an embedding in \(\mathbb{R}^ 4\) of a closed orientable surface (not necessarily connected) if and only if it has a finite Wirtinger presentation, one in which each relator equates one of the generators with a conjugate of another [J. Simon, Pac. J. Math. 90, 177-190 (1980; Zbl 0461.57008)]. The main result of this paper is a similar characterization of the groups corresponding to surfaces with a given number of components and total genus. In particular, this gives a characterization of 2-knot groups in terms of presentations which is analogous to Artin’s characterization of classical knot groups. (However the problem of characterizing knot groups in these dimensions intrinsically remains open.) The key ideas used are a normal form for surfaces in 4-space (the “motion picture method”) and an adaption of Alexander’s argument for representing links as closed braids.

MSC:
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F05 Generators, relations, and presentations of groups
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