On Embeddings of Polyhedra and Compacta
by
S. Spiez
University of Washington

The well known Menger-Nöbeling Theorem says that any n-dimensional compact metric space X can be embedded in Euclidean space \R²ⁿ⁺¹, moreover any continuous mapping X --> \R²ⁿ⁺¹ can be approximated arbitrarily closely by embeddings. We characterize n-dimensional compact metric spaces for which the dimension of the Euclidean space in the later statement can be decreased; we also survey some related results. In 1933, E. R. van Kampen constructed n-dimensional polyhedra which are not embeddable in \R²ⁿ. He also gave a rough description of a certain \Z/2\Z-equivariant 2n-dimensional cohomology class of the deleted product K^*={(x, y) in K×K:x =/= y} of an n-dimensional polyhedron K, which vanishes if and only if K is embeddable in \R²ⁿ, provided n >= 3. In 1967, C. Weber generalized van Kampen result, by proving that if there exists an equivariant map K^* --> S^m-1 then the n-dimensional polyhedron K is embeddable in \R^m, provided the dimensions are in A. Haefliger result on embedding of compact differentiable manifolds into Euclidean spaces. We discuss some results, due to Mardesi\'c-Segal, Segal-Spie\.z, Freedman-Krushkal-Teichner and Segal-Skopenkov-Spie\.z, showing that the Weber's theorem can not be extended beyond the metastable range for m >= 4. The cases m=3 and n=2 or 3 are open. If m=1 or 2, the existence of an equivariant map K^* --> S^m-1, implies the existence of an embedding K in \R^m.

Date received: February 15, 1998