CHAPTER 1 Maps from Loop Suspensions to Loop Spaces 1.1. The James Construction In this section, we go over the James construction. Some of the results in this section may be well-known. Let X be a pointed space and let J(X) be the James construction with the James filtration {Jn(X)}. For a subspace A of X, write (X\kA)n for the subspace of Xn consisting of points (xi,£2 • • •, xn) G Xn with at least k coordinates lie in A We simply write (X\A)n for (X|iA) n . Note that Xn/(X\*)n = X^ and (X| n _i*) n = \/™=1X. Let qn: Xn — Jn(X) be the canonical quotient map. Then there is a natural commutative diagram (xi n _!*r = \j xc 3 = 1 c {x\ x *)n c xn (l.i.i) Qn Ji(x) = x c ••• c j n _i(X) c j n (x) for any (pointed) space X. In particular, there is a commutative diagram of cofibre sequences (CX\X)n — (CX)n — (EX) (n ) dn Jn-i(EX) ^ J n (EX) — ( E X ) ^ , where CX = X A [0,1] is the cone of X. The map dn: (CX\X)n- Jn_i(EX) is called the higher Whitehead product, studied in [32, 33, 34]. Since (CX)n is contractible, we have the following: PROPOSITION 1.1.1. Let X be any pointed space. There is a (functorial) cofibre sequence (CX\X)n ~ E ^ 1 ^ J ^ J n _i(EX) - Jn(EX). 77ms £/ie cofibre sequence J„_i(EX) ^ - J n (SX) — (EX)W zs principal. •

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