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Steenrod problem and some graded Stanley–Reisner rings

Masahiro Takeda

Algebraic & Geometric Topology 24 (2024) 1725–1738
Abstract

“What kind of ring can be represented as the singular cohomology ring of a space?” is a classic problem in algebraic topology, posed by Steenrod. We consider this problem when rings are the graded Stanley–Reisner rings, in other words the polynomial rings divided by an ideal generated by square-free monomials. We give a necessary and sufficient condition that a graded Stanley–Reisner ring is realizable when there is no pair of generators x,y such that |x| = |y| = 2n and xy0.

Keywords
Steenrod problem, Stanley–Reisner ring, homotopy colimit, Steenrod algebra
Mathematical Subject Classification
Primary: 55N10
Secondary: 55R35, 13F55
References
Publication
Received: 24 July 2022
Revised: 9 February 2023
Accepted: 9 March 2023
Published: 28 June 2024
Authors
Masahiro Takeda
Faculty of Mathematics
Kyushu University
Fukuoka
Japan

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