Endomorphisms of ℙ¹ and 𝔸�ⁿ: 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.