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NON-PLIMBEDDINGS OF 3-MANIFOLDS. 1. Introduction. All imbeddings will be locally flat and all isotopies will
 

Summary: NON-PLIMBEDDINGS OF 3-MANIFOLDS.

1. Introduction. All imbeddings will be locally flat and all isotopies will
be ambient isotopies. Let M",N"+~ be closed PL manifolds, m >3. For any
imbedding f :M"+N"+~ there is an obstruction in H ~ ( M ;2,) to isotopingf to
a PL imbedding [6]. However, it follows from the twisted product structure
theorem of [9] that for m >5 there is a PL manifold M' and a homeomorphism
g :M'+M such that f 0 g :M'+N is isotopic to a PL imbedding. Thus for m >5
there is a natural way of replacing any imbedding of M in N with a PL
imbedding of a homeomorphic manifold M'.
Here we study the analogous situation for m=3. Since PL structures on
3-manifolds are unique, the above theorem does not extend directly. However,
we show there is a biunique correspondence between isotopy classes of non-PL
imbeddings of M in N and isotopy classes of PL imbeddings (with an ap-
propriate condition on fundamental group) of a manifold M' homology equiv-
alent to M.
In particular, we will show in Section 4 that the cobordism classificationof
non-PL 3-knots in s5[4,2] is equivalent to the cobordism classification of PL
imbeddings of certain homology 3-spheres in s5. The correspondence is natural
in the sense that a 3-knot and the corresponding imbedded homology sphere

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics