Desuspensions of S1 ∧ (ℙℚ1 −{0,1,∞})

Abstract

We use the Galois action on πé t 1 (P 1 Q − {0, 1, ∞}) to show that the homotopy equivalence

𝔸1-contractibility of affine modifications

“…but there is no A 1 -weak equivalence between G m ∨ G m and P 1 {0, 1, ∞} by [52]. Theorem 1.13 does not allows us to conclude that X (s, R) is A 1 -contractible.…”

“…Remark 1.14. In general, A 1 -weak equivalences do not desuspend, e.g., by homotopy purity we have [52]. Theorem 1.13 does not allows us to conclude that X (s, R) is A 1 -contractible.…”

The simplicial EHP sequence in 𝔸1–algebraic topology

“…In a different direction, it can be shown that there is an A 1 weak equivalence Σ(P 1 −{0, 1, ∞}) ∼ = Σ(G m ∨G m ) between the S 1 suspensions of P 1 − {0, 1, ∞} and G m ∨ G m . By comparing the actions of the absolute Galois group on geometric étale fundamental groups, it can be shown that this weak equivalence does not desuspend [Wic16]. Because the action of the absolute Galois group on π ét 1 (P 1 Q − {0, 1, ∞}) is both tied to interesting mathematics [Iha91] and obstructs desuspension, it is potentially also of interest to have systematic tools like those provided by the EHP sequence to study the obstructions to desuspension.…”

Families of \mathbb{A}^1-contractible affine threefolds

“…Let us note however that examples of non isomorphic spaces in H(k) that become isomorphic after a single smash product with P 1 are abundant in nature. We refer the interested reader to [AM11,Proposition 5.22] or to [Wic15,Theorem 4.2] for such examples. The first theorem of the present paper shows that the Koras-Russell threefolds are indeed contractible.…”