Algebraic homotopy classes of rational functions
By C. Cazanave. 2009
49 citations recorded by the source corpus.
We compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of P^1: there is an a priori monoid structure on the set of naive homotopy classes such that canonical map from this monoid to group of motivic homotopy classes is a group completion.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/755d57213ad2e633bef383c8088c89b59fe59e2f
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- A1-Algebraic Topology over a Field (2012, methodology): “Cazanave [20][18] has a different method to produce the invariant [P(1),P(1)]H•(k) → ̃ GW (k) using the approach of Barge and Lannes on Bott periodicity for orthogonal algebraic K-theory [7].”
- An arithmetic count of the lines on a smooth cubic surface (2017, background): “conjugate to the involution z7→ −α/z for some α∈ k. Proof. This is [Bea10, Theorem 4.2]. 8 Lemma 12. The A1-degree of i(z)=−α/zis hαi ∈ GW(k). Proof. This is a special case of e.g. the main result of [Caz12]. Corollary 13. If iis a nontrivial involution on P1 k and D ∈ kis the discriminant of the fix”
- La réalisation étale et les opérations de Grothendieck (2014, background): “En fait, quitte à travailler un plus, on peut vraisemblablement remplacer l’utilisation de [31, 32, 35] par les résultats très concrets et relativement élémentaires de Cazanave [10] sur le calcul de l’ensemble [P1,P1]N des classes d’homotopie naïve d’endomorphismes de la droite projective pointée.”
- Affine representability results in 1–homotopy theory, II : Principal bundles and homogeneous spaces (2015, unknown): “ive homotopy classes and thus, via Theorem4true A1–homotopy classes. Barge and Lannes [15,Chapter 4] provide such identifications inthecasewherethetarget isrelated tosymmetric bilinear forms. Cazanave [19] provides such identifications in the case where the target is Pn. In addition, Fasel [26, Theore”