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Symmetric products and the stable Hurewicz homomorphism

1980; Duke University Press; Volume: 24; Issue: 4 Linguagem: Inglês

10.1215/ijm/1256047470

1945-6581

Peter J. Welcher,

Mathematics and Applications

Section 3 recalls results from various papers and studies questions of connec- tivity (at a given prime p).The techniques are classical in nature.PROPOSITION 3.1.Let X be a strongly convergent spectrum which is homo- logically (n-1)-connected.Then pr SP (X)tp) SP Xp) -KZ/ Xtp) is at least (2ff + + n-4)-connected.This improves the classical estimate by roughly a power of p.We also show" PROPOSITION 3.2.Let X be a spectrum.Then" (a) (SPPX/X)tp) -w (S/SPX)e_ (b) (SPP"X/SPp-'X)tp) -w (SPVX),p);(a) The spectrum structure map f," S 1/xSP"X SP"(S1/X) is at least (2k + 1)-connected, i.e. is an isomorphism on rti for <_ 2k.(b) S /x ZPX ZP(S / X)is at least 2k-connected.The significance of 3.3 is that it indicates that our symmetric/cyclic product functors of spectra "extend" space-level functors, in giving homotopy informa- tion in a stable range.We have traded the property that T(S/x X) _ S/x T(X), perhaps a more natural requirement of a functor which "extends" a space-level functor, for the properties of 2.1, which still give us stable information about the symmetric/cyclic products of spaces, by 3.3.This stable information about spaces was our original goal, although we will seldom spell out the space-level implications of statements about spectra (for brevity).PROPOSITION 4.1.2S ,2S S/kS /xRP [17].In general Z-PS -S/x $1/x BZp, p prime.PROPOSITION 4.2.SPPStp) -(S/xS /x BEp)tp).Putting these latter results together with 3.2, we then see that the cofibre of the inclusion Xtp) ---, SPPXtp) is S/x S/x BZp/x Xtp), up to homotopy.Thus the Puppe map from the cofibration sequence is a map (" (S / S A Bp)(p) S/k Sp).The purpose of Section 5 is to identify this map.We recall from [4, p. 49]" THEOREM (Kahn-Priddy).Let L (S/xS /x B,p)(p).(i) There is a map of spectra b'L Sp) which induces an isomorphism S onto rrep_ 3 S(p) 7 2p 2 /k S(p).