Van Kampen’s Embedding Obstruction Is Incomplete for 2-complexes in R Michael H. Freedman, Vyacheslav Krushkal and Peter Teichner

In 1933, anticipating formal cohomology theory, van Kampen [5] gave a slightly rough description of an obstruction o(K) ∈ H Z/2(K ,Z) which vanishes if and only if an n-dimensional simplicial complex K admits a piecewise-linear embedding into R, n ≥ 3. Here K is the deleted product of a complex K. The cohomology in question is the Z/2-equivariant cohomology where Z/2 acts on the space by exchanging the factors of K and acts on the coefficients by multiplication with (−1). Many details were clarified by Shapiro [3] and Wu [6] in the 1950’s. In 1991 Sarkaria [2] showed that for n = 1 this obstruction provides a necessary and sufficient condition in that dimension as well (and is thus identical to Kuratowski’s subgraph condition). Sarkaria recently asked the first named author if it were possible that the vanishing of o(K) might also imply embeddability for n = 2. It is the purpose of this paper to exhibit a simplicial 2-complex K with 14 vertices, 43 1-cells and 69 2-cells for which o(K) is trivial but which does not admit an embedding into R. If one considers relative settings (K, L) ⊂ (D, S) there is an analogous obstruction and a high dimensional theorem analogous to van Kampen’s. But setting n = 2, K = D ⊔D ⊔D and L ⊂ S the Borromean rings gives an elementary (and well known) example where although the obstruction vanishes there is no relative embedding. Our task was to ”unrelativize” this simple example. In section 2 we recall van Kampen’s obstruction (generalized to a relative setting) and give a modern proof that its vanishing implies the existence of a P.L. embedding into R, n ≥ 3. Section 3 describes the example, proves the absence of a P.L. embedding and the vanishing of the obstruction. In section 4 we show that K does not embed, even topologically, into R.