HOMEWORK 7 – 2017
Exercise 1. Let f : X → Y be a constant map. Prove that f∗ : Hn(X) → Hn(Y )
and f∗
: Hn
(Y ) → H(
X) are zero maps for n ≥ 1. (Hint: One can do it from the
definition, but much easier is to factor f as a composition of suitable two maps and
use the fact that H∗ and H∗
are a functor and a cofunctor, respectively.)
Exercise 2. Let the cohomology rings of the spaces X and Y are the following
H∗
(X) ∼= Z[x]/ xn
, H∗
(Y ) ∼= Z[y]/ ym
where x ∈ H1
(X) and y ∈ H1
(Y ). Prove that
H∗
(X ∨ Y ) ∼= Z[u, v]/ un
, vm
, uv .
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