By Botvinnik B.

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**Example text**

First we need some new “standard spaces” which live happily inside the Euclidian space Rn . Let q ≤ n + 1, and v1 , . . , vq+1 be vectors those endpoints do not belong to any (q − 1)dimensional subspace. We call the set ∆q (v1 , . . , vq+1 ) = { t1 v1 + . . + tq vq+1 | t1 + . . + tq+1 = 1, t1 ≥ 0, . . , tq+1 ≥ 0 } 38 BORIS BOTVINNIK a q -dimensional simplex. 5. ∆(v1 , . . , vq+1 ) is homeomorphic (moreover, by means of a linear map) to the standard simplex ∆q = (x1 , . . , xq+1 ) ∈ Rq+1 | x1 ≥ 0, .

Such maps ϕ : I −→ X that ϕ(0) = ϕ(1) = x0 . The loops ϕ, ϕ′ are homotopic if there is a homotopy ϕt : I −→ X , (0 ≤ t ≤ 1) such that ϕ0 = ϕ, ϕ1 = ϕ′ . A “product” of the loops ϕ, ψ is the loop ω , difined by the formula: ω(t) = ϕ(2t), for 0 ≤ t ≤ 1/2, ψ(2t − 1), for 1/2 ≤ t ≤ 1. This product operation induces a group structure of π1 (X). It is easy to check that a group operation is well-defined. 4. Write an explicit formula givinig a null-homotopy for the composition ϕ¯ · ϕ. 3. Dependence of the fundamental group on the base point.

Consider the case when q > p. Here we need the following lemma. 6. (Free-point-Lemma) Let U be an open subset of Rp , and ϕ : U → D be a ◦q continuous map such that the set V = ϕ−1 (dq ) ⊂ U is compact for some closed disk dq ⊂ D . ◦q If q > p there exists a continuous map ψ : U −→ D such that 1. ψ|U \V = ϕ|U \V ; 2. e. there exists a point y0 ∈ dq \ ψ(U ). We postpone a proof of this Lemma for a while. Remark. 6 are homotopic relatively to U \ V : it is ◦q enough to make a linear homotopy: ht (x) = (1 − t)ϕ(x) + tψ(x) since the disk D is a convex set.