Affine algebraic geometry techniques in A^1-homotopy theory

For a space X, the property that Sing∗(X) satisfies affine Nisnevich exci-
sion, is very useful in unstable A

1
-homotopy theory and it has been studied
in the works of Morel, Asok-Hoyois-Wendt. Using a result of Dubouloz, we
will see if an affine surface X admits an A
1
-fibration over an A
1
-rigid curve
C, in the sense of Gurjar-Miyanishi, then X has such property. We will see
that if X is an affine surface, then π
A1
0
(X) is A
1
-invariant, which answers
a question of Morel for affine surfaces. If time permits, using a result of
Dubouloz-Fasel, we will see that if X is the Koras-Russell threefold of the
first kind, then Sing∗(X) satisfies the affine Nisnevich excision property.
This is based on a joint work with Utsav Choudhury and Aritra Mandal.

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