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**The Sullivan conjecture on maps from classifying spaces.**
*(English)*
Zbl 0552.55014

From the author’s introduction: ”We prove the following theorem, resolving in the affirmative a conjecture of D. Sullivan [Geometric topology. Localization, periodicity and Galois symmetry. M.I.T. press 1970, p. 5.118. For a review of the Russian translation see Zbl 0366.57003]. Theorem A. Leg G be a discrete group which is locally finite (i.e. every finitely generated subgroup is finite), and let X be a connected finite dimensional CW complex. then the space of pointed maps to X from the classifying space of G has the weak homotopy type of a point: \(\pi_*map_*(BG,X)=0.\)

”This theorem presents a curious feature of loop spaces of finite dimensional complexes X: For any \(n\geq 0\) and any locally finite group G, every pointed map \(BG\to \Omega^ nX\) is null-homotopic through pointed maps. Thus, for example, no essential map from \({\mathbb{R}}P^ m\) to \(\Omega^ nX\) can be extended over \({\mathbb{R}}P^{m+s}\) for all s.

A. Zabrodsky has pointed out that the following extension is a corollary. Theorem A’. Let W be a connected CW complex such that each homotopy group \(\pi_ i(W)\) is locally finite and such that \(\pi_ i(W)\) is nonzero for only finitely many i. Let X be a connected finite dimensional CW complex. Then \(\pi_*map_*(W,X)=0.-\) We give the simple deduction of this from Theorem A in Section 9. A. Zabrodsky [On phantom maps and theorems of H. Miller (to appear)] has carried this further, to obtain information even when \(\pi_*(W)\) is not torsion. We remark also that C. McGibbon and J. Neisendorfer [Comment. Math. Helv. 59, 253-257 (1984; Zbl 0538.55010)] have used Theorem A (or rather, Theorem C below) to prove the conjecture of J-P. Serre [Comment. Math. Helv. 27, 198-232 (1953; Zbl 0052.195)]: Let X be a simply connected space such that \(H_*(X;{\mathbb{F}}_ p)\) is nonzero for only finitely many degrees. Then either \(\pi_*(X)\) contains no elements of order p at all, or it contains them in arbitrarily large dimensions.

”One may view Theorem A as an unstable analogue of the Burnside ring conjecture of G. B-Segal [cf. F. J. Adams, Bull. Am. Math. Soc., New Ser. 6, 201-210 (1982; Zbl 0483.55009)]. A proof of this conjecture has recently been completed by G. Carlsson. For a finite group G, it computes the stable cohomotopy of BG, and asserts, among other things, that \(\{BG,S^ q\}=[BG,\lim_{\to_{n}}\Omega^ nS^{n+q}]\) is trivial for \(q>0\). On the other hand if \(q\leq 0\) it predicts that \(\{BG,S^ q\}\) will be nonzero - indeed, when \(q=0\), it will generally be uncountable. This is in contrast to the consequence of Theorem A: For any \(q\in {\mathbb{Z}}\), \(\lim_{\to_{n}}[BG,\Omega^ nS^{n+q}]=0.\) The moral one draws is that none of the essential maps to \(\lim_{\to}\Omega^ nS^{n+q}\) can be compressed to \(\Omega^ nS^{n+q}\), for any n. This may be regarded as a dramatic instance of J. F. Adams’ dictum [Stable homotopy and generalised homology (1974; Zbl 0309.55016)], ”Cells now, maps later.”

”This theorem presents a curious feature of loop spaces of finite dimensional complexes X: For any \(n\geq 0\) and any locally finite group G, every pointed map \(BG\to \Omega^ nX\) is null-homotopic through pointed maps. Thus, for example, no essential map from \({\mathbb{R}}P^ m\) to \(\Omega^ nX\) can be extended over \({\mathbb{R}}P^{m+s}\) for all s.

A. Zabrodsky has pointed out that the following extension is a corollary. Theorem A’. Let W be a connected CW complex such that each homotopy group \(\pi_ i(W)\) is locally finite and such that \(\pi_ i(W)\) is nonzero for only finitely many i. Let X be a connected finite dimensional CW complex. Then \(\pi_*map_*(W,X)=0.-\) We give the simple deduction of this from Theorem A in Section 9. A. Zabrodsky [On phantom maps and theorems of H. Miller (to appear)] has carried this further, to obtain information even when \(\pi_*(W)\) is not torsion. We remark also that C. McGibbon and J. Neisendorfer [Comment. Math. Helv. 59, 253-257 (1984; Zbl 0538.55010)] have used Theorem A (or rather, Theorem C below) to prove the conjecture of J-P. Serre [Comment. Math. Helv. 27, 198-232 (1953; Zbl 0052.195)]: Let X be a simply connected space such that \(H_*(X;{\mathbb{F}}_ p)\) is nonzero for only finitely many degrees. Then either \(\pi_*(X)\) contains no elements of order p at all, or it contains them in arbitrarily large dimensions.

”One may view Theorem A as an unstable analogue of the Burnside ring conjecture of G. B-Segal [cf. F. J. Adams, Bull. Am. Math. Soc., New Ser. 6, 201-210 (1982; Zbl 0483.55009)]. A proof of this conjecture has recently been completed by G. Carlsson. For a finite group G, it computes the stable cohomotopy of BG, and asserts, among other things, that \(\{BG,S^ q\}=[BG,\lim_{\to_{n}}\Omega^ nS^{n+q}]\) is trivial for \(q>0\). On the other hand if \(q\leq 0\) it predicts that \(\{BG,S^ q\}\) will be nonzero - indeed, when \(q=0\), it will generally be uncountable. This is in contrast to the consequence of Theorem A: For any \(q\in {\mathbb{Z}}\), \(\lim_{\to_{n}}[BG,\Omega^ nS^{n+q}]=0.\) The moral one draws is that none of the essential maps to \(\lim_{\to}\Omega^ nS^{n+q}\) can be compressed to \(\Omega^ nS^{n+q}\), for any n. This may be regarded as a dramatic instance of J. F. Adams’ dictum [Stable homotopy and generalised homology (1974; Zbl 0309.55016)], ”Cells now, maps later.”

Reviewer: M.Mimura