Formality of spaces with Lusternik-Schnirelmann category 1
Torgeir Aamb{\o}
TL;DR
This paper proves that spaces with Lusternik-Schnirelmann category 1 are formal, by establishing a converse to a known fact about dg-algebras with trivial induced product on cohomology.
Contribution
It demonstrates that spaces of Lusternik-Schnirelmann category 1 are formal, extending the understanding of formality in algebraic topology.
Findings
Spaces of Lusternik-Schnirelmann category 1 are formal.
Dg-algebras with trivial induced product on cohomology are formal.
The converse of the known fact about Massey products is established.
Abstract
It is a well known fact that formal dg-algebras admit no non-trivial Massey products, while the converse fails. We prove that by restricting to dg-algebras whose induced product on cohomology is trivial, we do in fact get this converse. This allows us to prove that spaces of Lusternik-Schnirelmann category 1 are formal spaces.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra