Abstract
A map f:KRd of a simplicial complex is an almost embedding if f(sigma)f()=empty set whenever sigma, are disjoint simplices of K.TheoremFix integersd,k2such thatd=3k/2+1. Assume that PNP.
Then there exists a finitek-dimensional complexKthat does not admit an almost embedding inRdbut for which there exists an equivariant map from the deleted productK -1.The algorithmic problem of recognition of almost embeddability of finitek-dimensional complexes inRdis NP-hard. The proof is based on the technique from the Matouek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma.
The new part of our argument is a stronger almost embeddings' version of the generalized van Kampen-Flores theorem.