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$\mathbb A^1$-connected components

Online, January-June 2023


The aim of this workgroup is to explore the notion of $\mathbb A^1$-connected components and the $\mathbb A^1$-homotopy sheaf of a space in degree $0$. The workgroup opens on the 2008 fundamental paper of Aravind Asok and Fabien Morel.


Date Titre Orateur Description Notes
23/02 Asok-Morel I Niels Feld $\mathbb A^1$-connectivity, $\mathbb A^1$-chain connectedness ([Asok-Morel], Section 2.1, 2.2) beamer
02/03 Asok-Morel II Samuel Lerbet Variants of rationality in birational geometry and comparison with $\mathbb A^1$-connectedness. Étale variant. ([Asok-Morel], sections 2.3 and 2.4) notes
16/03 Asok-Morel III Guillaume Bressan h-cobordism and $\mathbb A^1$-classification of rational smooth proper surfaces ([Asok-Morel] Section 3) notes
23/03 Asok-Morel IV Rakesh Pawar h-cobordisms and torsors. Classifying spaces, Eilenberg MacLane spaces, [Asok-Morel, Sections 4.1-4.3] notes
20/04 Asok-Morel V Mattia Cavicchi Algebraic groups and strong $\mathbb A^1$-invariance ([Asok-Morel, 4.4]). Non triviality of the $\mathbb A^1$-fundamental group of a smooth proper $k$-scheme [Asok-Morel, 5.1]. notes
27/04 Asok-Morel VI Samuel Lerbet Computations of $\mathbb A^1$-fundamental groups: projective line and Hirzebruch surfaces. See [Asok-Morel], sections 5.1, 5.2 and 5.3 notes
04/05 Asok-Morel VII Pierre Martinez $\mathbb A^1$-homotopical classification of rational surfaces: [Asok-Morel], sections 5.4 and 5.5 notes
11/05 Asok-Morel VIII Pierre Martinez Birational $\mathbb A^1$-invariant sheaves and the first $\mathbb A^1$-fundamental group: [Asok-Morel], section 6 notes
25/05 Contre-exemples Riccardo Pengo Présentation de la conjecture de A1-connexité de Morel et contre-exemple d'Ayoub en dimension >1. Présentation de la conjecture d'A1-invariance pour le pi_0 et contre-exemple de [Ayoub] notes, bonus1, bonus2
01/06 $\mathbb{A}^1$-connected components I François Brunault [BHS], Pages 336-348 notes
15/06 $\mathbb{A}^1$-connected components II Santiago Toro [BHS], Pages 349-354 notes
22/06 BHS $\mathbb{A}^1$-connected Santiago Toro [BHS], Fin notes
06/07 Naive homotopies Riccardo Pengo & Adrien Duobouloz [BS], Fin notes


Aravind Asok et Fabien Morel. Smooth varieties up to $\mathbb A^1$-homotopy and algebraic $h$-cobordisms. Adv. Math., 227 (2011), no. 5, 1990--2058. pdf
Joseph Ayoub. Counterexamples to F. Morel's conjecture on $\pi_0^{\mathbb A^1}$. Preprint, (2022). pdf
Chetan Balwe, Amit Hogadi, Anand Sawant. $\mathbb{A}^1$-connected components of schemes. Adv. Math., 282 (2015), 335--361. pdf
Chetan Balwe et Anand Sawant. Naive $\mathbb A^1$-Homotopies on Ruled Surfaces. Int. Math. Res. Not., 2022, no. 22, 17745--17765. pdf
Chetan Balwe, Bandna Rani, Anand Sawant. Remarks on iterations of the $\mathbb A^1$-chain connected components construction. Ann. K-Theory, 7 (2022), no. 2, 385--394. pdf