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The rigidity of L(n). (English) Zbl 0632.55004

Algebraic topology, Proc. Workshop, Seattle/Wash. 1985, Lect. Notes Math. 1286, 286-292 (1987).
[For the entire collection see Zbl 0621.00017.]
Let \(SP^ k(S)\) be the localization at the prime p of the kth symmetric product of the sphere spectrum S. There are inclusions \(SP^ k(S)\to SP^{k+1}(S)\). Let \(L(n)=\Sigma^{-n}SP^{p^ n}(S)/SP^{p^{n- 1}}(S)\). The author proves the following rigidity property of \(H^*(L(n)):\) \(End_ A(H^*(L(n)))={\mathbb{Z}}/p\) where A denotes the mod p Steenrod algebra. As a consequence of this theorem the author obtains the Corollary: Any map f: L(n)\(\to L(n)\) with \(H_{2p^ n-n-2}(f)\neq 0\) is a homotopy equivalence. The rigidity of \(H^*(L(n))\) is proved by combining classical linear algebra with the formula \({\mathbb{Z}}/p[M_ n({\mathbb{Z}}/p)]\cong End_ A(H^*(({\mathbb{Z}}/p)^ n))\) of Adams, Gunawardena and Miller.
Reviewer: R.Kultze

MSC:

55P42 Stable homotopy theory, spectra

Citations:

Zbl 0621.00017