# Algebraic topology, Edition: version 5 Apr 2011 by Andrey Lazarev

By Andrey Lazarev

**Read or Download Algebraic topology, Edition: version 5 Apr 2011 PDF**

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**Extra info for Algebraic topology, Edition: version 5 Apr 2011**

**Example text**

Similarly one can introduce direct sums of arbitrary (possibly uncountable) collections of abelian groups. 30. Let {C∗i } be a collection of chain complexes. Their direct sum complex ... o dn i i Cn o dn+1 d i o n+2 i Cn+1 i i C∗ is the ... with differentials dn (a1 , a2 , . ) = (dn (a1 ), dn (a2 ), . ). 31. 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.

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 , . . , 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.

Set X2 = A and X1 = X \ U . 12 and let U X1o X2o = (X \ U )o Finally we have (X1 , X1 ¯ )o Ao ⊃ (X \ U Ao ⊃ (X \ Ao ) X2 ) = (X \ U, A \ U ) and (X1 , X2 ) = (X, A). 44 Ao = X. We’ll need the following result on long exact sequences. 14. Consider the following commutative diagram with exact rows: ... o gn ... o gn An o kn fn Dn o sn An o fn Dn o hn Cn o tn hn Cn o gn+1 An+1 o kn+1 gn+1 ... An+1 o ... in which every third map sn is an isomorphism. Then the following sequence is exact: An o ...