On maps with unstable singularities. (English) Zbl 1013.57016

Melikhov, Sergey A.

Topology Appl. 120, No. 1-2, 105-156 (2002).

Let \(X\) be a compact \(n\)-polyhedron and \(Q\) a PL \(m\)-manifold. A map \(f: X\rightarrow Q\) is called realizable or discretely realizable if it can be \(\epsilon\)-approximated by embeddings \(f_\epsilon :X \rightarrow Q\) for each \(\epsilon > 0\). \(f\) is isotopically realizable if \(f\) and \(f_\epsilon\) can be related by pseudo-isotopy [W. Holsztyński, Proc. Am. Math. Soc. 27, 598-602 (1971; Zbl 0211.26501)]. Furthermore, \(f:X \rightarrow Q\) is called continuously realizable if for each \(\epsilon > 0\) there exists a \(\delta >0\) such that each embedding \(g_\delta :X\rightarrow Q\) which is \(\delta\)-close to \(f\) can be taken onto \(f\) by an \(\epsilon\)-pseudo-isotopy. Clearly, if \(f:X\rightarrow Q\) is continuously realizable, then it is isotopically realizable. The paper also treats the fourth notion of realizability called concordant realizability. The author treats the question when does discrete realizability imply isotopic realizability? The proof of the obtained result is done via continuous realizability. The main results on this line are the following:
If \(f:X \rightarrow Q\) is discretely realizable, then it is isotopicaly realizable if
(1) \(m > 2n, (m,n)\) different from \((1,3)\);
(2) \(m > 3(n+1)/2\) and \(\Delta(f) = \{(x,y) \mid f(x) = f(y)\}\) has a \(\mathbb{Z}_2\)-equivariant mapping cylinder neighbourhood in \(X\times X\);
(3) \(m > n + 2\) and \(f\) is the composition of a PL map and a topological embedding.
For \((m,n) = (3,6)\) a map discretely realizable may not be isotopically realizable. The paper contains a series of examples illustrating different relations between the treated notions of realizability. Furthermore, there are results in the smooth category when \(X,Q\), and \(f\) belong to that category. Also, one can find a series of results related to this topic.

Reviewer: Ivan Ivanšić (Zagreb)

MSC:

57N37	Isotopy and pseudo-isotopy
57R45	Singularities of differentiable mappings in differential topology
57N35	Embeddings and immersions in topological manifolds
57Q35	Embeddings and immersions in PL-topology
57N45	Flatness and tameness of topological manifolds
57N75	General position and transversality
57Q37	Isotopy in PL-topology
57Q55	Approximations in PL-topology
57Q60	Cobordism and concordance in PL-topology

Keywords:

approximations by embeddings; pseudo-isotopy; discretely relizable map; isotopically realizable map; continuously realizable map; concordantly realizable map; equivariant mapping cylinder neighbourhood

Citations:

Zbl 0211.26501

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