A1-homotopy types of A2 and A2 \ {(0, 0)}/ Biman Roy
Material type:
TextPublication details: Kolkata: Indian Statistical Institute, 2024Description: vii, 114 pagesSubject(s): DDC classification: - 23rd 514.24 R888
- Guided by Prof. Utsav Choudhury
| Item type | Current library | Call number | Status | Notes | Date due | Barcode | Item holds | |
|---|---|---|---|---|---|---|---|---|
| THESIS | ISI Library, Kolkata | 514.24 R888 (Browse shelf(Opens below)) | Available | E-Thesis. Guided by Prof. Utsav Choudhury | TH617 |
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| 514.24 P588 Lectures on homotopy theory | 514.24 R253 Complex cobordism and stable homotopy groups of spheres | 514.24 R253 Nilpotence and periodicity in stable homotopy theory | 514.24 R888 A1-homotopy types of A2 and A2 \ {(0, 0)}/ | 514.24 R982 Spaces of homotopy self-equivalences | 514.24 R989 Homotopy index and partial differential equations | 514.24 S635 Statistical theory of shape |
Thesis (Ph.D) - Indian Statistical Institute, 2024
Includes bibliography
Introduction -- A1-homotopy theory: An Introduction -- A1-invariance of πA1/ 0 (−) -- Birational Connected Components -- Existence of A1 and A1-Connectedness of a Surface -- A1-homotopy theory and log-uniruledness -- Kan Fibrant Property of Sing∗(X)(−) -- Characterisation of the Affine Space -- A1-homotopy type of A2 \ {(0, 0)} -- Regular Functions on S(X) -- Naive 0-th A1-homology --
Guided by Prof. Utsav Choudhury
Morel-Voevodsky developed A^1-homotopy theory which is a bridge between algebraic geometry and algebraic topology. In this thesis we study the A^1-connected component of a smooth variety in great detail. We have shown that the A^1-connected component of a smooth variety contains the information about the existence of affine lines in the variety. Using this and Miyanishi-Sugie's algebraic characterisation, we determine that the affine plane is the only A^1-contractible smooth affine surface over the field of characteristic zero. In the other part of the thesis, we studied the A^1-homotopy type of A^2-{(0,0)}. We showed that over the field of characteristic zero, if an open subvariety of a smooth affine surface is A^1-weakly equivalent to A^2-{(0,0)}, then it is isomorphic to A^2-{(0,0)}.
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