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Summary: HOUSTON JOURNAL OF MATHEMATICS, Volume 3. No. 2, 1977.
THOROUGHLY KNOTTED HOMOLOGY SPHERES
Martin Scharlemann *
For Hn a homologyn-sphere,considerthe problemof classifyinglocallyflat
imbeddingsHn·'-·Sn+2uptoisotopy.Sinceanyimbeddingmaybealteredbyadding
knotsSn· Sn+2,the classificationproblemis at leastascomplexastheisotopy
classificationof knots. Elsewhere[8] we showthat thereisa naturalcorrespondence
betweenknottheoryandtheclassificationofthoseimbeddingsHn· Sn+2which
satisfya certainconditionon fundamentalgroup.Thoseimbeddingswhich do not
satisfythe fundamentalgroupconditionwill be calledthoroughlyknotted.Thegoal
ofthepresentpaperistheconstruction,foralln·>3,ofhornologyspheresHnandof
PLlocallyfiatimbeddingsHn%Sn+2whicharethoroughlyknotted.Acorollarywill
be that, for thesenomologyspheres,the codimensiontwo classificationproblemis
measurablymore complex than knot theory.
The outline is as follows: In õ1 we define thoroughly knotted imbeddings and
present,for anyhomologyn-sphereHn, necessaryconditionsforrrl(H)to bethe
fundamental group of a thoroughly knotted homology n-sphere.Theseconditionsare
shown to be sufficient if n ·> 5. Also, for n ·> 3, the conditions are shown to be
sufficienttoproduceathoroughlyknottedimbeddingsH# H· Sn+2.
In õ2 we review enough of Milnor's K-theory and of Steinberg'sresultson
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