Algebraic homotopy classes of rational functions
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 4, p. 511-534

Let k be a field. We compute the set 𝐏 1 ,𝐏 1 N of naive homotopy classes of pointed k-scheme endomorphisms of the projective line 𝐏 1 . Our result compares well with Morel’s computation in [11] of the group 𝐏 1 ,𝐏 1 𝐀 1 of 𝐀 1 -homotopy classes of pointed endomorphisms of 𝐏 1 : the set 𝐏 1 ,𝐏 1 N admits an a priori monoid structure such that the canonical map 𝐏 1 ,𝐏 1 N 𝐏 1 ,𝐏 1 𝐀 1 is a group completion.

Soit k un corps. Nous déterminons l’ensemble 𝐏 1 ,𝐏 1 N des classes d’homotopie naïve d’endomorphismes pointés de k-schémas de la droite projective 𝐏 1 . Notre résultat se compare bien avec le calcul de Morel [11] du groupe 𝐏 1 ,𝐏 1 𝐀 1 des classes d’𝐀 1 -homotopie d’endomorphismes pointés de 𝐏 1  : l’ensemble 𝐏 1 ,𝐏 1 N admet a priori une structure de monoïde pour laquelle l’application canonique 𝐏 1 ,𝐏 1 N 𝐏 1 ,𝐏 1 𝐀 1 est une complétion en groupe.

DOI : https://doi.org/10.24033/asens.2172
Classification:  14F42,  55Q40,  12Y05
Keywords: naive homotopy classes, rational functions, projective line, group completion
@article{ASENS_2012_4_45_4_511_0,
     author = {Cazanave, Christophe},
     title = {Algebraic homotopy classes of rational functions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {4},
     year = {2012},
     pages = {511-534},
     doi = {10.24033/asens.2172},
     mrnumber = {3059240},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2012_4_45_4_511_0}
}
Cazanave, Christophe. Algebraic homotopy classes of rational functions. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 4, pp. 511-534. doi : 10.24033/asens.2172. http://www.numdam.org/item/ASENS_2012_4_45_4_511_0/

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