000			01978nam a22002657a 4500
001			th617
003			ISI Library, Kolkata
005			20250220151923.0
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040			_aISI Library _bEnglish
082	0	4	_223rd _a514.24 _bR888
100	1		_aRoy, Biman _eauthor
245	1	0	_aA1-homotopy types of A2 and A2 \ {(0, 0)}/ _cBiman Roy
260			_aKolkata: _bIndian Statistical Institute, _c2024
300			_avii, 114 pages,
502			_aThesis (Ph.D) - Indian Statistical Institute, 2024
504			_aIncludes bibliography
505	0		_aIntroduction -- 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 --
508			_aGuided by Prof. Utsav Choudhury
520			_aMorel-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)}.
650		4	_aMathematics
650		4	_aHomotopy
856			_uhttp://hdl.handle.net/10263/7485 _yFull text
942			_2ddc _cTH
999			_c436753 _d436753