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By Christoph Schweigert

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Example text

For every continuous map f : X → Y induces a chain map S∗ (f ) : S∗ (X) → S∗ (Y ); for the evaluation, we have εY ◦ S0 (f ) = εX . 3. e. for a continuous map f : X → Y we get an induced map H∗ (S∗ (f )) : H∗ (S∗ (X)) → H∗ (S∗ (Y )) such that the identity on X induces the identity and composition of maps is respected. As a consequence, H∗ (−) is a functor. 4 For ∅ = A ⊂ X we define Hn (X, A) := Hn (X, A). 3 reduced homology groups with relative homology groups we obtain a reduced version of the Mayer-Vietoris sequence.

For the cellular complex, we find ∙2 0 0 ... → Z → Z → Z → 0 . Thus, depending on n we get   k=0 Z n Hk (RP ) = Z/2Z k n, k odd   0 otherwise. for n even. For odd dimensions n we get   k = 0, n Z n Hk (RP ) = Z/2Z 0 < k < n, k odd   0 otherwise. Note that RP 1 ∼ = S1 and RP 3 ∼ = SO(3). 13 Homology with coefficients Let G be an arbitrary abelian group. 1 The singular chain complex of a topological space X with coefficients in G, S∗ (X; G), has as n elements in Sn (X; G) finite sums of the form N i=1 gi αi with gi in G and αi : Δ → X a singular n-simplex.

Axiom (b) is not satisfied, since the closure of the two-cell has a non-trivial intersection with infinitely many 0-cells. But axioms (a) and (c) hold. 5. The unit interval [0, 1] has a CW structure with two zero cells and one 1-cell. But for 1 instance the decomposition σ00 = {0}, σk0 = { k1 }, k > 0 and σk1 = ( k+1 , k1 ) does not give a CW structure on [0, 1]. Consider the following subset A ⊂ [0, 1] A := 1 2 1 1 + k k+1 |k ∈ N . 1 Then A ∩ σ ˉk1 is precisely the point 12 ( k1 + k+1 ). This is closed, but A is not closed in [0, 1], since it does not contain the limit point 0 of A.

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