Mod \(p\) loop space homology. (English) Zbl 0667.55007

Félix, Yves; Halperin, Stephen; Lemaire, Jean-Michel; Thomas, Jean-Claude

Invent. Math. 95, No. 2, 247-262 (1989).

It was shown by Y. Felix, S. Halperin, C. Jacobsson, C. Löfwall, and J.-C. Thomas [Am. J. Math. 110, 301-322 (1988; Zbl 0654.55011)] that the rational Lusternik-Schnirelmann category of a space \(X\) forms an upper bound for the depth of the rational homology algebra of the loop space \(\Omega\) X. The present work extends this result to the case of an arbitrary coefficient field \(\Bbbk\). As in the rational context, the proof relies on techniques of differential algebra. However, unlike the rational situation, characteristic \(p>0\) necessitates modeling the non-commutative DGA of singular \(\Bbbk\)-cochains and proving a non-commutative version of Theorem B of the paper cited above (Theorem A’). The \(\Bbbk\)-depth theorem above is applied, together with structure results on radicals of Hopf algebras and facts about Gorenstein algebras and spaces [see Y. Felix, S. Halperin and J.-C. Thomas, Adv. Math. 71, No.1, 92-112 (1988; Zbl 0659.57011)], to prove the following interesting generalization of Serre’s theorem on the existence of infinitely many nontrivial homotopy groups: Let \(X\) be simply connected with finite category and suppose that for some prime \(p>0\) each \(H_ i(X;{\mathbb{Z}}/p)\) is finite dimensional. If \(X\) admits an \(n\)-stage Postnikov decomposition at \(p\), then \(H_ i(X:{\mathbb{Z}}/p)=0\) for all \(i\geq 1\). This result says (for \(p\)-local \(X\) say) that noncontractibility not only implies the existence of infinitely many homotopy groups, but that these groups in fact require an infinite number of principal fibrations to assemble \(X\).

Reviewer: J.Oprea

MSC:

55P35	Loop spaces
55S45	Postnikov systems, \(k\)-invariants
55P62	Rational homotopy theory

Keywords:

rational Lusternik-Schnirelmann category; depth of the rational homology algebra of the loop space; non-commutative DGA; radicals of Hopf algebras; Gorenstein algebras; existence of infinitely many nontrivial homotopy groups; \(n\)-stage Postnikov decomposition

Citations:

Zbl 0654.55011; Zbl 0659.57011

References:

[1]	Anick, D.: Non-commutative graded algebras and their Hilbert series. J. Algebra78, 120-140 (1982) · Zbl 0502.16002 · doi:10.1016/0021-8693(82)90104-1
[2]	Avramov, L., Halperin, S.: Through the looking glass: a dictionary between rational homotopy theory and local algebra. In: Algebra, algebraic topology and their interactions, Roos, J.E., (ed.) (Lecture Notes in Math., Vol. 1183, pp. 1-27). Berlin-Heidelberg-New York: Springer 1986 · Zbl 0588.13010
[3]	Borel, A.: Sur la cohomologie des espaces fibr?s principaux et des espaces homog?nes de groupes de Lie compacts. Ann. Math.57, 115-207 (1953) · Zbl 0052.40001 · doi:10.2307/1969728
[4]	Felix, Y., Halperin, S., Jacobsson, C., L?fwall, C., Thomas, J.C.: The radical of the homotopy Lie algebra. Am. J. Math.110, 301-322 (1988) · Zbl 0654.55011 · doi:10.2307/2374504
[5]	Felix, Y., Halperin, S., Thomas, J.C.: Gorenstein spaces. Adv. Math.71, 92-112 (1988) · Zbl 0659.57011 · doi:10.1016/0001-8708(88)90067-9
[6]	Halperin, S., Lemaire, J.M.: Notions of category in differential algebra. In: Algebraic topology (Rational homotopy), Proceedings, Louvain-la-Neuve Felix, Y., (ed.) (Lecture Notes in Math., Vol. 1318, pp. 138-154. Berlin-Heidelberg-New York: Springer 1988 · Zbl 0656.55003
[7]	McGibbon, C.A., Neisendorfer, J.: On the homotopy groups of a finite dimensional space. Comment. Math. Helv.59, 253-257 (1984) · Zbl 0538.55010 · doi:10.1007/BF02566349
[8]	Milnor, J.W., Moore, J.C.: The structure of Hopf algebras. Ann. Math.81, 211-264 (1965) · Zbl 0163.28202 · doi:10.2307/1970615
[9]	Moore, J.C., Smith, L.: Hopf algebras and multiplicative fibrations, I. Am. J. Math.90, 752-780 (1968) · Zbl 0194.24501 · doi:10.2307/2373482
[10]	Serre, J.-P.: Cohomologie modulo 2 des complexes d’Eilenberg-MacLane. Comment. Math. Helv.27, 198-232 (1953) · Zbl 0052.19501 · doi:10.1007/BF02564562