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ON THE GENERALIZED MASSEY--ROLFSEN
INVARIANT FOR LINK MAPS
A. Skopenkov
Abstract. For P = P 1 t \Delta \Delta \Delta tPs and a link map f : P ! R m let P = t i!j P i \Theta P j ,
define a map e
f : P ! S m\Gamma1 by e
f(x; y) = fx\Gammafy
jfx\Gammaf yj and a (generalized) Massey--
Rolfsen invariant ff(f) 2 ъ m\Gamma1 (P ) to be the homotopy class of e
f . We prove that
for a polyhedron P of dimension џ m \Gamma 2 under certain (weakened metastable)
dimension restrictions, ff is an onto or a 1--1 map from the set of link maps f : P !
R m up to link concordance to ъ m\Gamma1 (P ). We present calculation of ъ m\Gamma1 (P ) for
some cases. If P 1 ; : : : ; Ps are closed highly homologically connected manifolds (in
particular, homology spheres), then ъ m\Gamma1 (P ) ё = \Phi i!j ъ S
p i +p j \Gammam+1 . By the theorem
of Melikhov, 'link concordance' can be replaced by 'link homotopy' for codimension
at least 3.
1991 Mathematics Subject Classification. Primary: 57Q45, 55S15; secondary: 57Q35, 57M15,
57Q65, 57Q30, 57Q60, 55Q10, 55Q55.
Key words and phrases. Deleted product, Massey­Rolfsen invariant, link maps, link homotopy,
stable homotopy group, double suspension, codimension two, highly­connected manifolds.
Supported in part by the Russian Fundamental Research Foundation Grant No 99­01­00009
Typeset by A M S­T E X
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