New PDF release: Algebraic topology

By Andrey Lazarev

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Proof. For a convex combination ti pi define f˜( ti pi ) = ti f (pi ). Uniqueness is obvious. 15. Let ∆n be the standard n-simplex. Its ith face map i = ni : ∆n−1 −→ ∆n is the affine map from the standard n − 1-simplex ∆n−1 to ∆n given in the barycentric coordinates by the formula n i (t0 , . . , tn−1 ) = (t0 , . . , ti−1 , 0, ti , . . , tn−1 ). 16. If k < j the face maps satisfy n+1 n k j = n+1 n j−1 k : ∆n−1 −→ ∆n+1 . Proof. Just evaluate these affine maps on every vertex ei for 0 ≤ i ≤ n − 1.

This show that H0 (X) = Z and the generator corresponds to the 0-chain x. Now suppose that X is not connected and denote by Xα its path components. Pick a point xα ∈ Xα . Arguing as before we see that any 0-chain in X is homologous to the unique chain of the form aα xα (where the sum is, of course, finite). Moreover the chain aα xα is homologous to zero iff all aα = 0. Therefore H0 (X) is the free abelian group on the set of generators xα . Suppose that {Xi } is the collection of connected components of a space X.

Also note that the intersection of any number of affine (convex) sets is affine (convex). Thus, it makes sense to speak about the affine (convex) set in Rn spanned by a subset X ⊂ Rn , namely, the intersection of all affine (convex) sets in Rn containing X. We will denote by [X] the convex set spanned by X and by [X]a the affine set spanned by X. 3. An affine combination of points p0 , . . , pn ∈ Rn is a point x := t0 p0 +. +tm pm . where m i=0 ti = 1. A convex combination is an affine combination for which ti ≥ 0 for all i.

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Algebraic topology by Andrey Lazarev

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