By L. F. McAuley
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Additional resources for Algebraic and Geometrical Methods in Topology: Conference on Topological Methods in Algebraic Topology SUNY Binghamton, October 3–7, 1973
If Z is P-local nilpotent, then we may find a principal refinement of its Postnikov system. Moreover this principal refinement will be such that the fibre at each stage is a space K(A,n), where is P-local abelian. A = ri. 6, for some i, and it is easy to see [ 3 ] that FiB is n P-local if B is P-local. Given g: X ~ Z, the obstructions to the existence and uniqueness of a counterimage to lie in the groups H*(f;A) g under f* will thus and, as in the corresponding argument in the 1-connected case (note that we have trivial coefficients here, too), these groups will vanish if f induces P-localization in homology.
That if X is a (pointed) CW-complex and then the function space of a CW-complex. XW a finite (pointed) CW-complex, of pointed maps W ~ X has the homotopy type However its components will, of course, fall to be 1-connected even if components of XW W We know, following Milnor, X is l-connected. are nilpotent if X However, it turns out that the is nilpotent. Moreover, the category of nilpotent CW-complexes is suitable for homotopy theory (as first pointed out by E. Dror), and for localization techniques [ii].
3) Hp(~IX;HqX) ~ Hp(~IY;HqY). 5 ~i X nilpotently on if operates nilpotently H Y. 3) is localization unless p = q = 0. Passing through the spectral sequences and the appropriate filtrations of infer that H f localizes if HnX , HnY , we n ~ i. n Now let (t) be th~ st~temen~ f*: [~,Z]--~'~X,Z] fvr ~Zl P-local Z Note that thi~ statement differs from (i) only in not requiring that P-local. We prove that (iii) = (i'). Y This will, of course, imply that (ii) = (i). If Z is P-local nilpotent, then we may find a principal refinement of its Postnikov system.
Algebraic and Geometrical Methods in Topology: Conference on Topological Methods in Algebraic Topology SUNY Binghamton, October 3–7, 1973 by L. F. McAuley