Derived functors of nonadditive functors and homotopy theory. (English) Zbl 1214.18012

Breen, Lawrence; Mikhailov, Roman

Algebr. Geom. Topol. 11, No. 1, 327-415 (2011).

Summary: The main purpose of this paper is to extend our knowledge of the derived functors of certain basic nonadditive functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie functors, as well as that of the main cubical functors. We also present a functorial approach to the study of the homotopy groups of spheres and of Moore spaces \(M(A,n)\), based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. As an illustration, we retrieve in a purely algebraic manner the 3-torsion components of the homotopy groups of the 2-sphere in low degrees, and give a unified presentation of the homotopy groups \(\pi_i(M(A,n))\) for small values of both \(i\) and \(n\).

MSC:

18G10	Resolutions; derived functors (category-theoretic aspects)
18G55	Nonabelian homotopical algebra (MSC2010)
54E30	Moore spaces
55Q40	Homotopy groups of spheres

Keywords:

nonadditive derived functor; Moore space

Software:

Kenzo

References:

[1]	K Akin, D A Buchsbaum, J Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982) 207 · Zbl 0497.15020
[2]	H J Baues, Homotopy type and homology, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford Univ. Press (1996) · Zbl 0857.55001
[3]	H J Baues, J Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996) 573 · Zbl 0881.55008
[4]	H J Baues, P Goerss, A homotopy operation spectral sequence for the computation of homotopy groups, Topology 39 (2000) 161 · Zbl 0958.55013
[5]	H J Baues, T Pirashvili, A universal coefficient theorem for quadratic functors, J. Pure Appl. Algebra 148 (2000) 1 · Zbl 0972.18008
[6]	D Blanc, C Stover, A generalized Grothendieck spectral sequence (editors N Ray, G Walker), London Math. Soc. Lecture Note Ser. 175, Cambridge Univ. Press (1992) 145 · Zbl 0813.18006
[7]	A K Bousfield, Homogeneous functors and their derived functors, Unpublished manuscript, Brandeis University (1967)
[8]	A K Bousfield, Operations on derived functors of nonadditive functors, Unpublished manuscript, Brandeis University (1967)
[9]	A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, J W Schlesinger, The mod-\(p\) lower central series and the Adams spectral sequence, Topology 5 (1966) 331 · Zbl 0158.20502
[10]	L Breen, On the functorial homology of abelian groups, J. Pure Appl. Algebra 142 (1999) 199 · Zbl 0942.20038
[11]	H Cartan, J C Moore, R Thom, J P Serre, Algèbres d’Eilenberg-Mac Lane et homotopie, Séminaire Henri Cartan de l’Ecole Normale Supérieure 1954/1955, Secrétariat math. (1955)
[12]	F R Cohen, J C Moore, J A Neisendorfer, Torsion in homotopy groups, Ann. of Math. \((2)\) 109 (1979) 121 · Zbl 0405.55018
[13]	F R Cohen, J Wu, A remark on the homotopy groups of \(\Sigma^n\mathbfR\mathrmP^2\) (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 65 · Zbl 0841.55007
[14]	E B Curtis, Lower central series of semi-simplicial complexes, Topology 2 (1963) 159 · Zbl 0119.18205
[15]	E B Curtis, Some relations between homotopy and homology, Ann. of Math. \((2)\) 82 (1965) 386 · Zbl 0139.16503
[16]	E B Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971) 107 · Zbl 0225.55002
[17]	G J Decker, The integral homology algebra of an Eilenberg-Mac Lane space, PhD thesis, University of Chicago (1974)
[18]	A Dold, Homology of symmetric products and other functors of complexes, Ann. of Math. \((2)\) 68 (1958) 54 · Zbl 0082.37701
[19]	A Dold, Lectures on algebraic topology, Grund. der Math. Wissenschaften 200, Springer (1980) · Zbl 0434.55001
[20]	A Dold, D Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier Grenoble 11 (1961) 201 · Zbl 0098.36005
[21]	W Dreckmann, Distributivgesetze in der Homotopietheorie, PhD Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (1992); Bonner Math. Schriften 249 (1993) · Zbl 0838.55012
[22]	S Eilenberg, S Mac Lane, On the groups \(H(\Pi,n)\). II. Methods of computation, Ann. of Math. \((2)\) 60 (1954) 49 · Zbl 0055.41704
[23]	V Franjou, Cohomologie de de Rham entière
[24]	V Franjou, J Lannes, L Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994) 513 · Zbl 0798.18009
[25]	W Fulton, Young tableaux. With applications to representation theory and geometry, London Math. Society Student Texts 35, Cambridge Univ. Press (1997) · Zbl 0878.14034
[26]	W Fulton, J Harris, Representation theory. A first course, Graduate Texts in Math. 129, Springer (1991) · Zbl 0744.22001
[27]	P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Math. und ihrer Grenzgebiete 35, Springer (1967) · Zbl 0186.56802
[28]	R M Hamsher, Eilenberg-Mac Lane algebras and their computation. An invariant description of \(H(\Pi,1)\), PhD thesis, University of Chicago (1973)
[29]	L Illusie, Complexe cotangent et déformations. I, Lecture Notes in Math. 239, Springer (1971) · Zbl 0224.13014
[30]	F Jean, Foncteurs dérivés de l’algébre symétrique: Application au calcul de certains groupes d’homologie fonctorielle des espaces \(K(B,n)\), PhD thesis, University of Paris 13 (2002)
[31]	B Köck, Computing the homology of Koszul complexes, Trans. Amer. Math. Soc. 353 (2001) 3115 · Zbl 0980.13009
[32]	A Lascoux, Syzygies des variétés déterminantales, Adv. in Math. 30 (1978) 202 · Zbl 0394.14022
[33]	D Leibowitz, The \(E^1\) term of the lower central series spectral sequence for the homotopy of spaces, PhD thesis, Brandeis University (1972)
[34]	S Mac Lane, Triple torsion products and multiple Künneth formulas, Math. Ann. 140 (1960) 51 · Zbl 0093.02101
[35]	S Mac Lane, Decker’s sharper Künneth formula (editor F Borceux), Lecture Notes in Math. 1348, Springer (1988) 242 · Zbl 0655.18010
[36]	W Magnus,
[37]	J P May, Simplicial objects in algebraic topology, Van Nostrand Math. Studies 11, Van Nostrand (1967) · Zbl 0165.26004
[38]	R Mikhailov, Homotopy theory of Lie functors, in preparation
[39]	R Mikhailov, On the homology of the dual de Rham complex
[40]	R Mikhailov, I B S Passi, Lower central and dimension series of groups, Lecture Notes in Math. 1952, Springer (2009) · Zbl 1157.20002
[41]	R Mikhailov, J Wu, On homotopy groups of the suspended classifying spaces, Algebr. Geom. Topol. 10 (2010) 565 · Zbl 1196.55017
[42]	J A Neisendorfer, \(3\)-primary exponents, Math. Proc. Cambridge Philos. Soc. 90 (1981) 63 · Zbl 0483.55007
[43]	D Quillen, On the (co-) homology of commutative rings (editor A Heller), Proc. Sympos. Pure Math. 17, (New York, 1968) Amer. Math. Soc. (1970) 65 · Zbl 0234.18010
[44]	N Roby, Lois polynomes et lois formelles en théorie des modules, Ann. Sci. École Norm. Sup. \((3)\) 80 (1963) 213 · Zbl 0117.02302
[45]	J W Schlesinger, The semi-simplicial free Lie ring, Trans. Amer. Math. Soc. 122 (1966) 436 · Zbl 0163.17901
[46]	H Toda, Composition methods in homotopy groups of spheres, Annals of Math. Studies 49, Princeton Univ. Press (1962) · Zbl 0101.40703
[47]	C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press (1994) · Zbl 0797.18001
[48]	J H C Whitehead, A certain exact sequence, Ann. of Math. \((2)\) 52 (1950) 51 · Zbl 0037.26101
[49]	E Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937) 152 · Zbl 0016.24401
[50]	J Wu, Homotopy theory of the suspensions of the projective plane, Mem. Amer. Math. Soc. 162 no. 769, Amer. Math. Soc. (2003) · Zbl 1022.55009