On embedding polyhedra and manifolds

Horvatić, Krešo

doi:10.1090/S0002-9947-1971-0278314-4

On embedding polyhedra and manifolds
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Trans. Amer. Math. Soc. 157 (1971), 417-436 Request permission

Abstract:

It is well known that every $n$-polyhedron PL embeds in a Euclidean $(2n + 1)$-space, and that for PL manifolds the result can be improved upon by one dimension. In the paper are given some sufficient conditions under which the dimension of the ambient space can be decreased. The main theorem asserts that, for there to exist an embedding of the $n$-polyhedron $X$ into $2n$-space, it suffices that the integral cohomology group ${H^n}(X - \operatorname {Int} A) = 0$ for some $n$-simplex $A$ of a triangulation of $X$. A number of interesting corollaries follow from this theorem. Along the line of manifolds the known embedding results for PL manifolds are extended over a larger class containing various kinds of generalized manifolds, such as triangulated manifolds, polyhedral homology manifolds, pseudomanifolds and manifolds with singular boundary. Finally, a notion of strong embeddability is introduced which allows us to prove that some class of $n$-manifolds can be embedded into a $(2n - 1)$-dimensional ambient space.

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Seminar on combinatorial topology