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Realising unstable modules as the cohomology of spaces and mapping spaces. (English) Zbl 1259.55006

This is a survey article on the problem of realizing unstable modules over the Steenrod algebra as the cohomology of spaces. Despite its short length it ranges from classical results such as J. F. Adams’s solution of the Hopf invariant one problem [Ann. Math. (2) 72, 20–104 (1960; Zbl 0096.17404)], through L. Schwartz’s own proof [Invent. Math. 134, No.1, 211–227 (1998); erratum ibid. 182, No. 2, 449–450 (2010; Zbl 0919.55007)] of N. J. Kuhn’s conjectures [Ann. Math., II. Ser. 141, No. 2, 321–347 (1995; Zbl 0849.55022)] (with details of recent corrections to that work [L. Schwartz, Invent. Math. 182, No. 2, 449–450 (2010; Zbl 1215.55007)], and of an alternative proof by N. J. Kuhn himself [Algebr. Geom. Topol. 8, No. 4, 2109–2129 (2008); correction ibid. 10, No. 1, 531–533 (2010; Zbl 1169.55011)]), to the more recent work on the Kervaire invariant problem by M. A. Hill, M. J. Hopkins and D. C. Ravenel [In: Jerison, David (ed.) et al., Current developments in mathematics, 2010. Somerville, MA: International Press. 1–43 (2011; Zbl 1249.55005)].

MSC:

55S10 Steenrod algebra
55S35 Obstruction theory in algebraic topology
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