Steenrod problem and some graded Stanley–Reisner rings

Takeda, Masahiro

doi:10.2140/agt.2024.24.1725

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Steenrod problem and some graded Stanley–Reisner rings

Masahiro Takeda

Algebraic & Geometric Topology 24 (2024) 1725–1738

DOI: 10.2140/agt.2024.24.1725

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 $| x | = | y | = 2^{n}$ and $x y \neq 0$ .

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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Volume 24, issue 3 (2024)