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The homotopy classification of self-maps of infinite quaternionic projective space. (English) Zbl 0625.55013

The paper contains the proof of the fact that self-maps of \(BS^ 3\) are classified up to homotopy by their degree \((= usual\) degree of the induced map on \(\Omega BS^ 3=S^ 3)\). This is the end of a long story. S. Feder and S. Gitler [Bol. Soc. Mat. Mex., II. Ser. 18, 33- 37 (1973; Zbl 0314.55025)] have noticed that the nonzero possible degrees must be odd squares. D. Sullivan [Ann. Math., II. Ser. 100, 1-79 (1974; Zbl 0355.57007)] proved that all odd square degrees actually occur. Zabrodsky has recently succeeded in showing that deg f\(=0\Rightarrow f\simeq 0.\)
The author’s impetus for closing this problem was provided by his recent joint work with E. M. Friedlander on locally finite approximation of Lie groups [Invent. Math. 83, 425-436 (1986; Zbl 0566.55011)] and by a nice general result of Dwyer, which establishes that the components of map(BP,BG) \((P= finite\) p-group, \(G= compact\) connected Lie group) are parameterized by the conjugacy classes of homomorphisms \(r: P\to G\) and computes the \({\mathbb{Z}}/p {\mathbb{Z}}\) homology of the component indexed by r as \(H_*(BC(r); {\mathbb{Z}}/p {\mathbb{Z}})\) \((C(r)= centralizer\) of r in G).
The author’s proof of the fact that “deg f\(=\deg g\Rightarrow f\simeq g''\) is quite elaborate: many previously used techniques are present; the final step consists in careful computation with (co-) homology spectral sequences for homotopy limits.
Reviewer: St.Papadima

MSC:

55S37 Classification of mappings in algebraic topology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55P60 Localization and completion in homotopy theory
55T99 Spectral sequences in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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