An A^1-homotopy is an algebraic analogue of a homotopy in topology, where the unit interval [0,1] is replaced by the algebraic affine line A^1. As in topology, it turns out that many interesting invariants of algebraic varieties are A^1-invariant, i.e., they do not see the difference between A^1-homotopic maps. An important example is étale cohomology, which is an algebro-geometric analogue of singular cohomology.

The goal of this seminar is to learn the necessary background and study some elementary A^1-homotopical phenomena in algebraic geometry. In particular, we will discuss algebraic vector bundles and symmetric bilinear forms. The main results we will obtain are the following:

1) The A^1-homotopical classification of vector bundles: if X is a smooth affine variety, there is a bijection between isomorphism classes of vector bundles on X and A^1-homotopy classes of maps to the Grassmannian.

2) There is a bijection between the set of pointed A^1-homotopy classes of endomorphisms of the projective line and equivalence classes of nondegenerate symmetric bilinear forms.

• Organisational meeting/distribution of topics: February 5 at 16:15 in M311 or by email at
  marc.hoyois@ur.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk