The depth and LS category of a topological space
DOI:
https://doi.org/10.7146/math.scand.a-106920Abstract
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 ˆH(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,ˆH(ΩX))≠0. Theorem: for any connected CW complex, \depthX≤\catX.
References
Félix, Y. and Halperin, S., Malcev completions, LS category, and depth, Bol. Soc. Mat. Mex. (3) 23 (2017), no. 1, 267–288. https://doi.org/10.1007/s40590-016-0097-7
Félix, Y., Halperin, S., Jacobsson, C., Löfwall, C., and Thomas, J.-C., The radical of the homotopy Lie algebra, Amer. J. Math. 110 (1988), no. 2, 301–322. https://doi.org/10.2307/2374504
Félix, Y., Halperin, S., and Thomas, J.-C., Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0105-9
Félix, Y., Halperin, S., and Thomas, J.-C., Rational homotopy theory. II, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. https://doi.org/10.1142/9473
Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers, New York-London-Sydney, 1966.
Milnor, J. W. and Moore, J. C., On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. https://doi.org/10.2307/1970615
Quillen, D., Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. https://doi.org/10.2307/1970725
Sullivan, D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331.