-CONNECTEDNESS OF MODULI OF VECTOR BUNDLES ON A CURVE

In this note, we prove that the moduli stack of vector bundles on a curve with a fixed determinant is ${\mathbb A}^1$ -connected. We obtain this result by classifying vector bundles on a curve up to ${\mathbb A}^1$ -concordance. Consequently, we classify ${\mathbb P}^n$ -bundles on a curve up to ${\mathbb A}^1$ -weak equivalence, extending a result in [3] of Asok-Morel. We also give an explicit example of a variety which is ${\mathbb A}^1$ -h-cobordant to a projective bundle over ${\mathbb P}^2$ but does not have the structure of a projective bundle over ${\mathbb P}^2$ , thus answering a question of Asok-Kebekus-Wendt [2].