By Kalantari I., Welch L.

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**Extra resources for A blend of methods of recursion theory and topology: A П 0^1 tree of shadow points**

**Example text**

3 we can diagonalize this matrix d over GLm (Op ∩ K Γ0 (p) ). But since G is irreducibel over GLm (Op ∩ K Γ0 (p) ) it follows, that d = ζσ Im , for a suitable root of unity ζσ . Now we have g σ = gζσ and at the same time g γ = gζγ for any γ ∈ Γ1 (p). Since Γ1 (p) k operates trivially on the p-th roots of unity ζ we obtain: g σ = g γ , for some integer k and therefore the two Galois automorphisms σ and γ k coincides on K Γ0 (p) (G1 ) since g is any generator of G1 . This gives the contradiction in the case, where g −1 g σ ∈ GLm (K Γ1 (p) ).

1) ι B• −−−−→ R ⊗O˜ A• commutes in D(S). We deﬁne a functor F : D(X) → T (X) setting F(C• ) = (π ∗ C• , S ⊗O C• , ι), where ι : R ⊗O C• is induced by the embedding S → R. Just as in S ⊗O C• → R ⊗O˜ (π ∗ C• ) Section 1 the following theorem holds (with almost the same proof, see [6]). 2. e. it is dense and conservative. Remark. We do not now whether it is full, though it seems to be true. 5. Configurations of type A and A˜ As it was shown in [13], even classiﬁcation of vector bundles is wild for almost all projective curves.

Note that a special word is never symmetric, a quasisymmetric word is always bispecial, and a bispecial word is always full. 6. We deﬁne a cycle as a word w such that r1 = rm−1 =∼ and xm − x1 . Such a cycle is called non-periodic if it cannot be presented in the form v − v − · · · − v for a shorter cycle v. For a cycle w we set rm = −, xqm+k = xk and rqm+k = rk for any q, k ∈ Z. 7. A (k-th) shift of a cycle w, where k is an even integer, is the cycle w[k] = xk+1 rk+1 xk+2 . . rk−1 xk . A cycle w is called symmetric if w[k] = w∗ for some k.

### A blend of methods of recursion theory and topology: A П 0^1 tree of shadow points by Kalantari I., Welch L.

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