Eliminating higher-multiplicity intersections in the metastable dimension range

The r-fold analogues of Whitney trick were ‘in the air’ since 1960s. However, only in this century they were stated, proved and applied to obtain interesting results, most notably by Mabillard and Wagner. Here we prove and apply a version of the r-fold Whitney trick when general position r-tuple intersections have positive dimension. Theorem. Assume that D = D1 ⊔ . . . ⊔ Dr is disjoint union of k-dimensional disks, rd ≥ (r+1)k+3, and f : D → B a proper PL (smooth) map such that f∂D1∩. . .∩f∂Dr = ∅. If the map f r : ∂(D1 × . . .×Dr) → (B ) − {(x, x, . . . , x) ∈ (B) | x ∈ B} extends to D1 × . . . × Dr, then there is a proper PL (smooth) map f : D → B such that f = f on ∂D and fD1 ∩ . . . ∩ fDr = ∅.