zbMATH — the first resource for mathematics

Homotopy groups of certain highly connected manifolds via loop space homology. (English) Zbl 1422.55017
Let \(n\geq 2\) and \(d\) be positive integers such that \(d\leq 3n-2\). In this paper the authors study the homotopy types of \((n-1)\)-connected \(d\)-dimensional closed manifolds. In particular, when \(\dim H^*(M;\mathbb{Q})>4\) they show that the \(p\)-local homotopy groups of \(M\) can be determined by the dimension of the space of indecomposable elements in the cohomology ring \(H^*(M;\mathbb{Q})\) and prove that these \(p\)-local homotopy groups can be expressed as the finite \(p\)-local homotopy groups of spheres. Their proof is based on a careful analysis of the loop homology \(H_*(\Omega M;\mathbb{Q})\).
55P15 Classification of homotopy type
55P35 Loop spaces
55Q52 Homotopy groups of special spaces
16S37 Quadratic and Koszul algebras
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
Full Text: Euclid arXiv
[1] S. Basu and S. Basu: Homotopy groups of highly connected manifolds, Adv. Math. 337 (2018), 363-416. · Zbl 1433.55005
[2] S. Basu and S. Basu: Homotopy groups and periodic geodesics of closed 4-manifolds, Internat. J. Math. 26 (2015), 1550059, 34pp. · Zbl 1331.53064
[3] A. Berglund and K. Börjeson: Free loop space homology of highly connected manifolds, Forum Math. 29 (2017), 201-228. · Zbl 1422.55023
[4] G.M. Bergman: The diamond lemma for ring theory, Adv. in Math. 29 (1978), 178-218. · Zbl 0326.16019
[5] P. Cartier: Remarques sur le théorème de Birkhoff-Witt, Ann. Scuola Norm. Sup. Pisa (3), 12 (1958), 1-4.
[6] Y. Félix, S. Halperin and J.-C. Thomas: Rational homotopy theory, Graduate Texts in Mathematics 205, Springer-Verlag, New York, 2001.
[7] B. Gray: On the sphere of origin of infinite families in the homotopy groups of spheres, Topology 8 (1969), 219-232. · Zbl 0159.24701
[8] P.J. Hilton: On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154-172. · Zbl 0064.17301
[9] P. Lalonde and A. Ram: Standard Lyndon bases of Lie algebras and enveloping algebras, Trans. Amer. Math. Soc. 347 (1995), 1821-1830. · Zbl 0833.17003
[10] M. Lazard: Sur les algèbres enveloppantes universelles de certaines algèbres de Lie, Publ. Sci. Univ. Alger. Sér. A. 1 (1954), 281-294 (1955).
[11] J.-L. Loday and B. Vallette: Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 346, Springer, Heidelberg, 2012.
[12] J. Milnor and D. Husemoller: Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. · Zbl 0292.10016
[13] J. Neisendorfer: Algebraic methods in unstable homotopy theory, New Mathematical Monographs 12, Cambridge University Press, Cambridge, 2010. · Zbl 1190.55001
[14] A. Polishchuk and L. Positselski: Quadratic algebras, University Lecture Series 37, American Mathematical Society, Providence, RI, 2005. · Zbl 1145.16009
[15] D.C. Ravenel: Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press, Inc., Orlando, FL, 1986.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.