The depth and LS category of a topological space

  • Yves Félix
  • Steve Halperin

Abstract

The depth of an augmented ring $\varepsilon \colon A\to 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, $H_*(\Omega X;\mathbb {Q})$ is a Hopf algebra and the universal enveloping algebra of the Lie algebra $L_X$ of primitive elements. It is known that $\depth H_*(\Omega X;\mathbb {Q}) \leq \cat X$, the Lusternik-Schnirelmann category of $X$.

For any connected CW complex we construct a completion $\widehat {H}(\Omega X)$ of $H_*(\Omega X;\mathbb {Q})$ as a complete Hopf algebra with primitive sub Lie algebra $L_X$, and define $\depth X$ to be the least $p$ or ∞ such that \[ \Ext ^p_{UL_X}(\mathbb {Q}, \widehat {H}(\Omega X))\neq 0. \] Theorem: for any connected CW complex, $\depth X\leq \cat X$.

References

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Published
2018-08-13
How to Cite
Félix, Y., & Halperin, S. (2018). The depth and LS category of a topological space. MATHEMATICA SCANDINAVICA, 123(2), 220-238. https://doi.org/10.7146/math.scand.a-106920
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Articles