Homework number 7
Do both exercises.
Exercise 16. With all details prove the exactness of the long exact sequence of the
homotopy groups for a couple (X, A) in one of the terms πn(A, xo) and πn(X, A, xo).
Exercise 17. For 1 ≤ k < n we have a canonical inclusion RPk
֒→ RPn
. Using the
cup product in cohomology of real projective spaces with Z2 coefficients prove that
RPk
is not a retract of RPn
.
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