On maps with unstable singularities

Abstract

If a continuous map f : X → Q is approximable arbitrary closely by embeddings X ֒→ Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised byŠčepin and Akhmet'ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the answer is 'generally no' for (n, m) = (3, 6); (1, 3), and 'yes' when:and ∆(f ) = {(x, y) | f (x) = f (y)} has an equivariant (with respect to the factor exchanging involution) mapping cylinder neighborhood in X × X;3) m > n + 2 and f is the composition of a PL map and a TOP embedding. In doing this, we answer affirmatively (with a minor preservation) a question of Kirby: does small smooth isotopy imply small smooth ambient isotopy in the metastable range, verify a conjecture of Kearton-Lickorish: small PL concordance implies small PL ambient isotopy in codimension ≥ 3, and a conjecture set of Repovs-Skopenkov.

Projected and near-projected embeddings

“…• By recent work of the second author [31], f is a k-prem if and only if ∆ f := {(x, y) ∈ N × N \ ∆ N | f (x) = f (y)} admits a Z/2-equivariant map to S k−1 , assuming that either f is a fold map or 3n − 2m ≤ k in the smooth case. • Building on some previous work [35], [4], [27], we show that f is k-realizable if and only if ∆ f admits a stable Z/2-map to S k−1 , i.e. for some x there exists a Z/2-map ∆ f * S x → S k−1 * S x = S k+x .…”

“…Alternatively, Lemma 2.7(a) can be viewed as a special case (with P = M = pt and L = ∅) of Lemma 2.7(b), which we now prove by adapting the geometric proof of the Freudenthal suspension theorem in [12] (see also [27;Lemma 7.7…”