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Localisation of unstable \(A_p\)-algebras and Smith theory. (English) Zbl 0896.55015
From the author’s introduction: “In [W. Dwyer and C. W. Wilkerson, Ann. Math., II. Ser. 127, No. 1, 191-198 (1988; Zbl 0675.55011)] the Borel-Quillen localization theorem is reformulated to produce an expression for the \(\text{mod }p\) cohomology of the fixed point set of a finite \(\mathbb{Z}_p\)-complex \(X\) in terms of the \(\text{mod }p\) equivariant cohomology of \(X\). Here \(\mathbb{Z}_p\) denotes the cyclic group of prime order \(p\). This result contrasts with previous work where the localized cohomology of the fixed point set is obtained. The key to this extension is to localize in the category of unstable \(A_p\)-algebras, where \(A_p\) is the \(\text{mod }p\) Steenrod algebra. Many of the classical results concerning the cohomology of fixed point sets then become statements concerning this localization process.
“In this paper we reproduce some of these results giving \(p\) roots in the category of unstable \(A_p\)-algebras. Thus we only require \(X\) to have finite \(\text{mod }p\) cohomology.” The main sections of the paper treat only the case when \(p=2\).
MSC:
55S10 Steenrod algebra
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
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