Homework number 6
This time only one exercise.
Exercise 15. Show that the spaces U = Sn
× Sk
and V = Sn
∨ Sk
∨ Sn+k
have the
same homology and cohomology groups but they are not homotopy equivalent. (Hint:
Compute their structures as graded cohomology rings. To compute the cup product
on X ∨ Y use its naturality with respect to inclusions of X and Y into X ∨ Y and
retractions X ∨ Y to X and Y , respectively.)
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