Algebraic topology by Andrey Lazarev

By Andrey Lazarev

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Indeed, any x ∈ [p0 , . . , pm ] is such a convex combination. If this expression had not been unique the barycentric coordinates would also have not been unique. Example. For i = 0, 2, . . , n let ei denote the point in Rn+1 whose coordinates are all zeros except for 1 in the (i + 1)st place. Clearly {e0 , . . en } is affine independent. The set [e0 , . . , en ] is called the standard n-simplex in Rn+1 and denoted by ∆n . Thus, ∆n consists of all convex combinations x = ti ei . In this case, barycentric and cartesian coordinates of a point x ∈ ∆n coincide and we see that ∆n is a collection of points (t0 , .

N ∆n ∆n n It follows that dn (t∆ ∗ − b∗ − sn−1 dn )(δ) is a singular n-cycle in ∆ × I. 34 implies that all cycles in ∆ × I are boundaries and therefore there exists βn+1 ∈ Cn+1 (∆n × I) for which n n n ∆ ∆ dn+1 βn+1 = dn (t∆ ∗ − b∗ − sn−1 dn )(δ). Define sX n : Cn (X) −→ Cn+1 (X × I) by sX n (σ) = (σ × id)∗ (βn+1 ) where σ is an n-simplex in X and extend by linearity. 6)) =tX σ − bX σ − sX n−1 dn σ∗ (δ) =(tX − bX − sX n−1 dn )(σ). 6); here τ : ∆n −→ ∆n is a singular n-simplex in ∆n and σ : ∆n −→ X is a singular n-simplex in X: n (σ × id)∗ s∆ n (τ ) = (σ × id)∗ (τ × id)∗ (βn+1 ) = (στ × id)(βn+1 ) X = sX n (στ ) = sn σ∗ (τ ).

Fk ] such that every loop fi is a loop in some Aα . We will call this a factorization of f . It is, thus, a word in the free product of π1 (Aα )s that is mapped to [f ] via Φ. We showed above that each homotopy class of loop in X has a factorization. To describe the kernel of Φ is tantamount to describing possible factorizations of a given loop of X. We will call two factorizations equivalent if they are related by two sorts of moves or their inverses: • Combine adjacent terms [fi ][fi+1 ] into a single term [fi fi+1 ] if fi and fi+1 lie in the same space Aα .

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