By Christoph Schweigert

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Additional info for Algebraic Topology [Lecture notes]

Example text

Xn ] ∈ KP n . We define subsets for 0 i n Xi := {[x0 : . . : xn ]|xi = 0, xi+1 = . . = xn = 0} ⊂ KP n and consider the map ξ i : Xi → K i , ξi [x0 : . . , ). xi xi The map ξi is a homeomorphism; thus Xi is a cell of KP n of real dimension idimR (K) = im. We can write KP n as X0 . . Xn and we have characteristic maps Φ i : Dmi → KP n as Φi (y) = Φi (y0 , . . , yi−1 ) = [y0 : . . : yi−1 : 1 − ||y|| : 0 : . . : 0] with Xi = Φi (˚ Dmi ). This defines a structure of a CW complex on KP n . 1.

Then W 2 /W 1 ∼ = 6 2 i=1 S . 12 Cellular homology In the following, X will always be a CW complex. 1. For the relative homology of the skeleta, we have Hq (X n , X n−1 ) = 0 for all q = n 1. Proof. Using the identification of relative homology and reduced homology of the quotient gives ˜ q (X n /X n−1 ) ∼ Hq (X n , X n−1 ) ∼ =H = ˜ q (Sn ). 8. 7. 2. Consider the inclusion in : X n → X of the n-skeleton X n into X. 1. The induced map Hn (in ) : Hn (X n ) → Hn (X) is surjective. 2. On the (n + 1)-skeleton we get an isomorphism Hn (in+1 ) : Hn (X n+1 ) ∼ = Hn (X).

Proof. Using the identification of relative homology and reduced homology of the quotient gives ˜ q (X n /X n−1 ) ∼ Hq (X n , X n−1 ) ∼ =H = ˜ q (Sn ). 8. 7. 2. Consider the inclusion in : X n → X of the n-skeleton X n into X. 1. The induced map Hn (in ) : Hn (X n ) → Hn (X) is surjective. 2. On the (n + 1)-skeleton we get an isomorphism Hn (in+1 ) : Hn (X n+1 ) ∼ = Hn (X). Proof. 54 • Using the inclusion of skeleta, we can factor in : X n → X as in Xn GQ gkgkgkgkxgkxSY X g g g g k in+1 ggggg kkk xx α1 gggggkkkkkk xxxxi g g g g n+3 g k  gggggG n+2 kk iG n+2n+3x G ...