The depth and LS category of a topological space
(2018) Mathematica Scandinavica — Vol. 123, p. 220-238 (2018)
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- Abstract
- The depth of an augmented ring ε:A→k is the least p, or ∞, such that \begin {equation*} \Ext _A^p(k , A)\neq 0. \end {equation*} When X is a simply connected finite type CW complex, H∗(ΩX;Q) is a Hopf algebra and the universal enveloping algebra of the Lie algebra LX of primitive elements. It is known that \depthH∗(ΩX;Q)≤\catX, the Lusternik-Schnirelmann category of X. For any connected CW complex we construct a completion Hˆ(ΩX) of H∗(ΩX;Q) as a complete Hopf algebra with primitive sub Lie algebra LX, and define \depthX to be the least p or ∞ such that \ExtpULX(Q,Hˆ(ΩX))≠0. Theorem: for any connected CW complex, \depthX≤\catX.
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Félix, Y., & Halperin, S. (2018). The depth and LS category of a topological space. Mathematica Scandinavica, 123, 220-238. https://doi.org/10.7146/math.scand.a-106920 (Original work published 2018)