Endomorphisms of ℙ¹ and 𝔸ⁿ: Motivic homotopy classes and open images
Abstract
This dissertation is about geometric objects called affine space and the projective line, which you may picture as a flat plane and as a circle. I have studied the algebraic functions from these objects to themselves. Are there functions from affine space to affine space that hit everything except for some specified point or curve? Requiring only a weak condition on the dimensions, we show that the answer is yes. There are infinitely many functions from a circle to itself, and they can be sorted into homotopy classes. Interestingly, these homotopy classes can be added and subtracted. In this thesis we construct a geometric model which makes the addition and subtraction of motivic homotopy classes of endomorphisms of the projective line more geometrically explicit.List of papers
Paper I. Viktor Balch Barth, William Hornslien, Gereon Quick, Glen Matthew Wilson, “Making the motivic group structure on the endomorphisms of the projective line explicit”. To be published. The paper is not available in DUO awaiting publishing. Preprint available in arXiv: 2306.00628. DOI: 10.48550/arXiv.2306.00628 |
Paper II. Viktor Balch Barth “Surjective morphisms from affine space to its Zariski open subsets”. In: International Journal of Mathematics. Vol. 34, no. 12 (2023), Paper No. 2350075. DOI: 10.1142/S0129167X23500751. The paper is included in the thesis. Also available at: https://doi.org/10.1142/S0129167X23500751 |
Paper III. Viktor Balch Barth, Tuyen Trung Truong “Images of dominant endomorphisms of affine space”. To be published. The paper is not available in DUO awaiting publishing. Preprint available in arXiv: 2311.08238. DOI: 10.48550/arXiv.2311.08238 |