Abstract
We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered.
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Baues, H.-J.: Commutator calculus and group of homotopy classes. In: Lecture Notes, no. 50. Cambridge University Press, Cambridge (1981)
Baues, H.-J.: Combinatorial Homotopy and 4-Dimensional Complexes. Walter de Gruyter, Berlin (1991)
Baues, H.-J.: The algebra of secondary cohomology operations. In: Progress in Mathematics, vol. 247. Birkhäuser, Cambridge, MA (2006)
Barcus, W.D., Barratt, M.G.: On the homotopy classification of the extensions of a fixed map. Trans. Amer. Math. Soc. 88, 57–74 (1958)
Baues, H.-J., Jibladze, M.: Suspension and loop objects and representability of track. Georgian Math. J. 8(4), 683–696 (2001)
Barratt, M.G., Jones, J.D.S., Mahowald, M.E.: The Kervaire invariant problem. In: Proceedings of the Northwestern Homotopy Theory Conference, Evanston, IL, 1982. Contemporary Mathematics, vol. 19, pp. 9–22. American Mathematical Society, Providence, RI (1983)
Baues, H.-J., Jibladze, M., Pirashvili, T.: Quadratic algebra of square groups. Preprint of the Max-Planck-Institut für Mathematik MPIM2006-9, http://arxiv.org/abs/math.CT/0601777 (2005)
Baues, H.-J., Muro, F.: Secondary algebras associated to ring spectra. Preprint of the Max-Planck-Institut für Mathematik MPIM2006-137, http://arxiv.org/abs/math.AT/0610523 (2006)
Baues, H.-J., Muro, F.: Secondary homotopy groups. Preprint of the Max-Planck-Institut für Mathematik MPIM2006-36. http://arxiv.org/abs/math.AT/0604029. Forum Mathematicum (2006)
Baues, H.-J., Muro, F.: The symmetric action on secondary homotopy groups. Preprint of the Max-Planck-Institut für Mathematik MPIM2006-37. http://arxiv.org/abs/math.AT/0604030 (2006)
Baez, J.C., Neuchl, M.: Higher-dimensional algebra. I. Braided monoidal 2-categories. Adv. Math. 121(2), 196–244 (1996)
Borceux, F.: Handbook of categorical algebra 1. In: Encyclopedia of Mathematics and Its Applications, no. 50. Cambridge University Press, Cambridge (1994)
Baues, H.-J., Pirashvili, T.: Quadratic endofunctors of the category of groups. Adv. Math. 141, 167–206 (1999)
Bröcker, T., Dieck, tom T.: Representations of compact Lie groups. In: Graduate Texts in Mathematics, vol. 98. Springer, New York (1985)
Conduché, D.: Modules croisés généralisés de longueur 2. J. Pure Appl. Algebra 34(2–3), 155–178 (1984)
Crans, S.E.: Generalized centers of braided and sylleptic monoidal 2-categories. Adv. Math. 136(2), 183–223 (1998)
Ellis, G.: Crossed squares and combinatorial homotopy. Math. Z. 214(1), 93–110 (1993)
Hardie, K.A., Mahowald, M.E.: On cup-one and Toda bracket. Quaestiones Math. 16(2), 193–207 (1993)
Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Amer. Math. Soc. 13(1), 149–208 (2000)
Marcum, H.J.: Obstructions for a map to be cyclic. In: Algebraic Topology, Oaxtepec, 1991. Contemporary Mathematics, vol. 146, pp. 277–295. American Mathematical Society, Providence, RI (1993)
Rutter, J.W.: A homotopy classification of maps into an induced fibre space. Topology 6, 379–403 (1967)
Toda, H.: Composition methods in homotopy groups of spheres. In: Annals of Mathematics Studies, no. 49. Princeton University Press, Princeton, NJ (1962)
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The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract.
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Baues, HJ., Muro, F. Smash Products for Secondary Homotopy Groups. Appl Categor Struct 16, 551–616 (2008). https://doi.org/10.1007/s10485-007-9071-x
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DOI: https://doi.org/10.1007/s10485-007-9071-x
Keywords
- Secondary homotopy group
- Square group
- Crossed module
- Smash product
- Lax symmetric monoidal functor
- Whitehead product
- Cup-one product
- Toda bracket