By Andrey Lazarev

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Show that Hn ( i i C∗ ) ∼ = Hn (C∗i ) for all n. Now let σ : ∆n −→ X be a singular n-simplex in X. Since the image of a connected space is connected σ is actually a singular simplex in one of the connected components of X. If c = ai σi is a singular n-chain in X then grouping together the singular simplices belonging to the same connected component of X we could rewrite it as c= a1i σi1 + a2i σi2 + . . where ck := aki σik is a singular n-chain in the kth connected component of X. Thus we established a correspondence c → (c1 , c2 , .

Tm such that ti = 1 and x = ti pi . The numbers t0 , . . , tm are called the barycentric coordinates of x (relative to the ordered set p0 , . . , pm ). 11. Let p0 , . . , pm be an affine independent subset of Rn . [p0 , . . , pm ] is called the m-simplex with vertices p0 , . . , pm . 12. If p0 , . . , pm is an affine independent set then each x in the m-simplex [p0 , . . , pm ] has a unique expression of the form x = ti pi where ti = 1 and each ti ≥ 0. Proof. Indeed, any x ∈ [p0 , . .

Proof. (1) Reflexivity: f ∼ f via s∗ := 0. (2) Symmetry: if s∗ is a chain homotopy between f∗ and g∗ then −s is a chain homotopy between g∗ and f∗ . (3) Transitivity: if s∗ : f∗ ∼ g∗ and s∗ : g∗ ∼ h∗ then (s∗ + s∗ ) : f∗ ∼ h∗ . The notion of chain homotopy is analogous to the notion of homotopy for continuous maps between topological spaces. 38. If s∗ is a homotopy between f∗ , f∗ : C∗ −→ B∗ and s∗ is a homotopy between g∗ , g∗ : B∗ −→ A∗ then the chain maps g∗ ◦ f∗ and g∗ ◦ f∗ are homotopic through the chain homotopy g∗ ◦ s∗ + s∗ ◦ f∗ .