On the metastable Mabillard-Wagner conjecture

The purpose of this note is to attract attention to the following conjecture (metastable r-fold Whitney trick) by clarifying its status as not having a complete proof, in the sense described in the paper. Assume that D = D1 ⊔ . . . ⊔ Dr is disjoint union of r disks of dimension s, f : D → B a proper PL map such that f∂D1∩. . .∩f∂Dr = ∅, rd ≥ (r+1)s+3 and d ≥ s+ 3. If the map f r : ∂(D1 × . . .×Dr) → (B ) − {(x, x, . . . , x) ∈ (B) | x ∈ B} extends to D1 × . . .×Dr, then there is a PL map f : D → B such that f = f on Dr ∪ ∂D and fD1 ∩ . . . ∩ fDr = ∅. The purpose of this note is to attract attention to the following conjectures by clarifying their status as not having complete proofs, in the sense described below. Let B := [0, 1] denote the standard PL (piecewise linear) ball and Sd−1 = ∂B the standard PL sphere. We need to speak about PL balls of different dimensions and we will use the word ‘disk’ for lower-dimensional objects and ‘ball’ for higher-dimensional ones in order to clarify the distinction (even though, formally, the disk D is the same as the ball B). A map f : M → B from a manifold with boundary to a ball is called proper, if f−1Sd−1 = ∂M . Conjecture 1 (Metastable Local Disjunction). Assume that • D = D1 ⊔ . . . ⊔Dr is disjoint union of r disks; • f : D → B a proper PL map such that f∂D1 ∩ . . . ∩ f∂Dr = ∅; • rd ≥ (s1 + s2 + . . .+ sr) + si + 3 and d ≥ si + 3 for each i. If the map f r : ∂(D1 × . . .×Dr) → (B) − {(x, x, . . . , x) ∈ (B) | x ∈ B} extends to a continuous map of D1 × . . .×Dr , then there is a PL map f : D → B such that f = f on Dr ∪ ∂D and fD1 ∩ . . . ∩ fDr = ∅. A continuous (or PL) map f : K → R of a finite simplicial complex is an almost r-embedding if f(σ1) ∩ . . . ∩ f(σr) = ∅ whenever σ1, . . . , σr are pairwise disjoint simplices of K. Denote by Σr the permutation group of r elements. The group Σr acts on the set of real d× r-matrices by permuting the columns. Denote by S d(r−1)−1 Σr the set of the set of real d× rmatrices such that the sum in each row is zero, and the sum of squares of the matrix elements is 1. This set is homeomorphic to the sphere of dimension d(r − 1)− 1. ∗Research supported by the Russian Foundation for Basic Research Grant No. 15-01-06302, by Simons-IUM Fellowship and by the D. Zimin’s Dynasty Foundation Grant. I would like to thank I. Mabillard U. and Wagner for helpful discussions. Moscow Institute of Physics and Technology, and Independent University of Moscow. Email: