Math 527/427
Homework 4
23 November 2015
1. Read the following sections in Hatcher:
(a) The section on Split Exact Sequences in section 2.2 on Page 147. Reconstruct the proof of the
Splitting lemma yourself (not to be turned in)
(b) Lemma 3.1 and its proof
2. Do the following problems from Hatcher: 3.1: 6, 9, 11. Referring to the statement of Theorem 3.12,
also do 6 and 7 from 3.2.
Regarding question 3.2.6, the following may be useful:
The space CP n has a cellular structure laid out in Example 0.6 on page 7. In particular there are
only even-dimensional cells in this structure, and the 2k-skeleton of CP n (where k ≤ n) is a subspace
homeomorphic to CP k .
CP n consists of equivalence classes of n + 1-tuples of complex numbers, not all of which are 0, under
the relation (z 0 , . . . , z n ) ∼ (λz 0 , . . . , λz n ) where λ ∈ C∗ . If you follow through what the cell structure on on
page 7 is asserted to be, you discover that the 2k-skeleton (where k ≤ n) of CP n is none other than the
image of those n-tuples of complex numbers (z 0 , z 1 , . . . , z k , 0, . . . , 0).
It is customary to write the class of (z 0 , . . . , z n ) in CP n in projective coordinates, as [z 0 ; z 1 ; . . . ; z n ]. Then
the k-skeleton consists of those elements of the form [z 0 ; . . . ; z k ; 0; . . . ; 0].
Note that the standard incusion CP 1 ,→ CP n , the inclusion of the 2 skeleton, is compatible with the
m-th power map in the question.
The cohomology of CP n (with Z coefficients) is given in Proposition 3.22. It is the truncated polynomial ring Z[α]/(αn+1 ) where |α| = 2.
It is well known that CP 1 ≈ S 2 , the Riemann Sphere, that is also the 1-point compactification of C
(the point at infinity being [1; 0], the point z ∈ C corresponding to [z; 1]). You may cite the result of Exercise 2.2.8 without proof if it helps you answer this question, although if you find an alternative solution
avoiding this, or if you solve 2.2.8 along the way, I’ll happily read it.
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