Knotting of contractible two-dimensional polyhedra in $\mathbf R^4$

S. A. Popov


Abstract: In this paper the Zeeman conjecture that any piecewise linear embedding of the dunce's hat (i.e. the triangle $ABC$ with the oriented edges $AB$, $BC$, and $AC$ identified) in $\mathbf R^4$ has simply connected complement is disproven.
Indeed, the author constructs linear embeddings in $\mathbf R^4$ with non-simply-connected complements for a class of two-dimensional polyhedra. All of these, just as the dunce's hat, are contractible but not combinatorially contractible, and the author ventures to conjecture that any two-dimensional polyhedra with these properties admits a piecewise linear embedding in $\mathbf R^4$ with non-simply-connected complement.
Figures: 4.
Bibliography: 7 titles.

UDC: 513.83

MSC: Primary 55A20; Secondary 57C35

Received: 14.12.1971

 English version: Mathematics of the USSR-Sbornik, 1972, 18:2, 333–341

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