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Homotopy groups of highly connected manifolds. (English) Zbl 1433.55005
The authors prove the following result (Theorem A). Let \(M\) be a closed \((n-1)\)-connected \(2n\)-manifold with \(n\ge 2\) and \(n\)th Betti number \(r\ge 2\). (a) The homotopy groups of \(M\) can be expressed as a direct sum of homotopy groups of spheres. (b) The homotopy groups of \(M\) are determined by \(r\), i.e., the number of factors \(\pi_jS^\ell\) occurring in the homotopy group \(\pi_jM\) is a function of \(\ell,r\) and \(n\) only. The case when the \(n\)th Betti number is 1 turns out to be different. As a consequence of Theorem A, a Moore conjecture, regarding the relationship between the rational and torsion homotopy groups of finite simply connected CW-complexes, is established in this particular case. Similar results were obtained by P. Beben and S. Theriault [Adv. Math. 262, 213–238 (2014; Zbl 1296.55012)] by different methods, \(n=4\) and \(n=8\) excepted. The primary techniques used in the proof of this theorem are quadratic algebras, Koszul duality of associative algebras and quadratic Lie algebras. The homology of the loop space \(\Omega M\) which is a tensor algebra modulo a relation corresponding to the attaching map of the top cell, is calculated via a quadratic Lie algebra. These techniques are also used to thoroughly analyze homotopy groups of a CW complex \(X=(\bigvee_rS^n) \cup e^{2n}\) obtained by attaching a \(2n\)-dimensional cell to a finite wedge of \(n\)-dimensional spheres (Theorem C) when the intersection form of \(X\) has rank at least 2 in Theorem A.

MSC:
55P35 Loop spaces
55Q52 Homotopy groups of special spaces
57N65 Algebraic topology of manifolds
16S37 Quadratic and Koszul algebras
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[1] Adams, J. F., On the cobar construction, Proc. Natl. Acad. Sci. USA, 42, 409-412, (1956) · Zbl 0071.16404
[2] Adams, J. F., Algebraic topology—A Student’s guide, London Mathematical Society Lecture Note Series, vol. 4, (1972), Cambridge University Press London-New York · Zbl 0234.55002
[3] Adams, J. F.; Hilton, P. J., On the chain algebra of a loop space, Comment. Math. Helv., 30, 305-330, (1956) · Zbl 0071.16403
[4] Ballmann, W.; Ziller, W., On the number of closed geodesics on a compact Riemannian manifold, Duke Math. J., 49, 629-632, (1982) · Zbl 0495.53042
[5] Basu, S.; Basu, S., Homotopy groups and periodic geodesics of closed 4-manifolds, Internat. J. Math., 26, (2015) · Zbl 1331.53064
[6] Beben, P.; Theriault, S., The loop space homotopy type of simply-connected four-manifolds and their generalizations, Adv. Math., 262, 213-238, (2014) · Zbl 1296.55012
[7] Berglund, A.; Börjeson, K., Free loop space homology of highly connected manifolds, Forum Math., 29, 201-228, (2017)
[8] Berglund, A.; Madsen, I., Homological stability of diffeomorphism groups, Pure Appl. Math. Q., 9, 1-48, (2013) · Zbl 1295.57038
[9] Bergman, G. M., The diamond lemma for ring theory, Adv. Math., 29, 178-218, (1978) · Zbl 0326.16019
[10] Cartier, P., Remarques sur le théorème de Birkhoff-Witt, Ann. Sc. Norm. Super. Pisa (3), 12, 1-4, (1958) · Zbl 0081.26302
[11] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A., Torsion in homotopy groups, Ann. of Math. (2), 109, 121-168, (1979) · Zbl 0405.55018
[12] Cohn, P. M., A remark on the Birkhoff-Witt theorem, J. Lond. Math. Soc., 38, 197-203, (1963) · Zbl 0109.26202
[13] Duan, H.; Liang, C., Circle bundles over 4-manifolds, Arch. Math. (Basel), 85, 278-282, (2005) · Zbl 1078.57020
[14] Félix, Y.; Halperin, S.; Thomas, J.-C., Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, (2001), Springer-Verlag New York
[15] Gromov, M., Homotopical effects of dilatation, J. Differential Geom., 13, 303-310, (1978) · Zbl 0427.58010
[16] Hilton, P. J., On the homotopy groups of the union of spheres, J. Lond. Math. Soc., 30, 154-172, (1955) · Zbl 0064.17301
[17] James, I. M., On the homotopy groups of certain pairs and triads, Quart. J. Math., Oxf. Ser. (2), 5, 260-270, (1954) · Zbl 0058.38901
[18] James, I. M., The suspension triad of a sphere, Ann. of Math. (2), 63, 407-429, (1956) · Zbl 0071.17101
[19] James, I. M., Multiplication on spheres. II, Trans. Amer. Math. Soc., 84, 545-558, (1957) · Zbl 0085.17202
[20] Lalonde, P.; Ram, A., Standard Lyndon bases of Lie algebras and enveloping algebras, Trans. Amer. Math. Soc., 347, 1821-1830, (1995) · Zbl 0833.17003
[21] Lambrechts, P., The Betti numbers of the free loop space of a connected sum, J. Lond. Math. Soc. (2), 64, 205-228, (2001) · Zbl 1018.55006
[22] Lambrechts, P., On the Betti numbers of the free loop space of a coformal space, J. Pure Appl. Algebra, 161, 177-192, (2001) · Zbl 0981.55004
[23] Lazard, M., Sur LES algèbres enveloppantes universelles de certaines algèbres de Lie, Publ. Sci. Univ. Alger., Sér. A, 1, 281-294, (1954), (1955)
[24] Loday, J.-L.; Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, vol. 346, (2012), Springer Heidelberg · Zbl 1260.18001
[25] Lothaire, M., Combinatorics on words, (1997), Cambridge Mathematical Library, Cambridge University Press Cambridge, With a foreword by Roger Lyndon and a preface by Dominique Perrin, Corrected reprint of the 1983 original, with a new preface by Perrin · Zbl 0874.20040
[26] McGibbon, C. A.; Neisendorfer, J. A., Various applications of haynes Miller’s theorem, (Conference on Algebraic Topology in Honor of Peter Hilton, Saint John’s, Nfld., 1983, Contemp. Math., vol. 37, (1985), Amer. Math. Soc. Providence, RI), 91-98 · Zbl 0565.55016
[27] McGibbon, C. A.; Wilkerson, C. W., Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc., 96, 698-702, (1986) · Zbl 0594.55006
[28] Miller, H., The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2), 120, 39-87, (1984) · Zbl 0552.55014
[29] Milnor, J., On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2), 64, 399-405, (1956) · Zbl 0072.18402
[30] Milnor, J.; Husemoller, D., Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, (1973), Springer-Verlag New York-Heidelberg · Zbl 0292.10016
[31] Milnor, J. W.; Moore, J. C., On the structure of Hopf algebras, Ann. of Math. (2), 81, 211-264, (1965) · Zbl 0163.28202
[32] Mimura, M., The homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ., 6, 131-176, (1967) · Zbl 0171.44101
[33] Neisendorfer, J., Algebraic methods in unstable homotopy theory, New Mathematical Monographs, vol. 12, (2010), Cambridge University Press Cambridge · Zbl 1190.55001
[34] Neisendorfer, J.; Miller, T., Formal and coformal spaces, Illinois J. Math., 22, 565-580, (1978) · Zbl 0396.55011
[35] Neisendorfer, J. A.; Selick, P. S., Some examples of spaces with or without exponents, (Current Trends in Algebraic Topology, Part 1, London, Ont., 1981, CMS Conf. Proc., vol. 2, (1982), Amer. Math. Soc. Providence, R.I.), 343-357
[36] Polishchuk, A.; Positselski, L., Quadratic algebras, University Lecture Series, vol. 37, (2005), American Mathematical Society Providence, RI · Zbl 1145.16009
[37] Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, (1986), Academic Press, Inc. Orlando, FL · Zbl 0608.55001
[38] Samelson, H., Classifying spaces and spectral sequences, Amer. J. Math., 75, 744-752, (1953) · Zbl 0051.13904
[39] Serre, J.-P., Homologie singulière des espaces fibrés. applications, Ann. of Math. (2), 54, 425-505, (1951) · Zbl 0045.26003
[40] Smale, S., On the structure of 5-manifolds, Ann. of Math. (2), 75, 38-46, (1962) · Zbl 0101.16103
[41] Stasheff, J., H-spaces from a homotopy point of view, Lecture Notes in Mathematics, vol. 161, (1970), Springer-Verlag Berlin-New York · Zbl 0205.27701
[42] Stasheff, J. D., Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc., 108, 293-312, (1963), ibid. · Zbl 0114.39402
[43] Tamura, I., 8-manifolds admitting no differentiable structure, J. Math. Soc. Japan, 13, 377-382, (1961) · Zbl 0109.16302
[44] Toda, H., On the double suspension \(E^2\), J. Inst. Polytech. Osaka City Univ., Ser. A, 7, 103-145, (1956)
[45] Wall, C. T.C., On simply-connected 4-manifolds, J. Lond. Math. Soc., 39, 141-149, (1964) · Zbl 0131.20701
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