ALGEBRAIC TOPOLOGY II, EXERCISE SHEET 5, 20.03.2014
Exercise 1. (Tensor product of chain complexes)
Let R be a commutative ring and let C, D ∈ Ch≥0 (R). In the lecture we defined homomorphisms
∂ : (C ⊗R D)k → (C ⊗R D)k−1 , k ≥ 1.
(1) Check that these define a chain complex C ⊗R D ∈ Ch≥0 (R), i.e., verify the relation ∂ 2 = 0.
(2) Show that we obtain an additive functor ⊗R : Ch≥0 (R) × Ch≥0 (R) → Ch≥0 (R).
Exercise 2. (Homology of tori)
Let T n = S 1 × . . . × S 1 be the n-dimensional torus, i.e., the n-fold product of S 1 with itself. For
an arbitrary abelian group A there is a natural isomorphism
n
H (T n ; A) ∼
= A(k ) , 0 ≤ k ≤ n.
k
In the case of A = Z and n = 2, use the isomorphism between cellular and singular homology to
draw a picture of the generators. Calculate also the cohomology groups H k (T n ; A), 0 ≤ k ≤ n.
Exercise 3. (Products of real projective planes)
Let RP 2 denote the real projective plane.
(1) Calculate the homology groups Hk (RP 2 × RP 2 ) and Hk (RP 2 × RP 2 ; Z/2Z).
(2) Calculate the cohomology groups H k (RP 2 × RP 2 ).
Exercise 4. (Symmetry constraint of tensor product of chain complexes)
Let R be a commutative ring and let C, D ∈ Ch≥0 (R).
(1) Prove that the R-linear maps Cp ⊗R Dq → Dq ⊗R Cq : c ⊗ d 7→ (−1)pq d ⊗ c, p, q ≥ 0, together
assemble into a natural isomorphism of chain complexes τ : C ⊗R D → D ⊗R C.
(2) Check that the R-linear maps Cp ⊗R Dq → Dq ⊗R Cq : c⊗d 7→ d⊗c, p, q ≥ 0 do not assemble
to a chain map. Thus the signs are really essential in this business!
Exercise 5. (Applications of method of acyclic models)
Let X, X 0 be topological spaces.
(1) Let α, β : C∗ (X) ⊗ C∗ (X 0 ) → C∗ (X) ⊗ C∗ (X 0 ) be natural transformations such that in
degree zero α0 = β0 = id : C0 (X) ⊗ C0 (X 0 ) → C0 (X) ⊗ C0 (X 0 ). Then α and β are naturally
chain homotopic.
(2) Let α, β : C∗ (X × X 0 ) → C∗ (X × X 0 ) be natural transformations such that in degree zero
α0 = β0 = id : C0 (X × X 0 ) → C0 (X × X 0 ). Then α and β are naturally chain homotopic.
By means of the tensor product of chain complexes, we obtain a different description of the
cylinder and cone construction for chain complexes. Given a space X, let X+ = X t {∗} be the
pointed space obtained from X by adding a disjoint base point. There is a unique pointed map
∗ → X+ . If K is a CW complex, then K+ is naturally again a CW complex with one additional
0-cell, given by the base point. Let us say that a CW complex K is pointed if it comes with a
base point which is a 0-cell. This is the same as asking for a pointed, cellular map ∗ → K. The
reduced cellular chain complex of a pointed CW complex K is defined as the cokernel
e cell (K) = cok(C cell (∗) → C cell (K)).
C
∗
∗
1
∗
2
ALGEBRAIC TOPOLOGY II, EXERCISE SHEET 5, 20.03.2014
This applies in particular to pointed CW complexes of the form K+ .
Exercise 6.
(1) Let S 0 be the 0-sphere, i.e., the space ∗ t ∗, considered as a CW complex with
0
two 0-cells only. Let S+
be the induced pointed CW complex and let us abuse notation by
0
cell
0
0
e
). There is a natural isomorphism S+
⊗C ∼
also writing S+ for C∗ (S+
= C ⊕ C of functors
Ch≥0 (Z) → Ch≥0 (Z).
(2) Let I = [0, 1] denote the interval with the usual CW structure consisting of two 0-cells and
one 1-cell. Let I+ be the induced pointed CW complex and let us abuse notation by also
e∗cell (I+ ). There is a natural isomorphism of functors
writing I+ for C
I+ ⊗ − ∼
= cyl : Ch≥0 (Z) → Ch≥0 (Z),
where cyl is the cylinder construction from exercise sheet 2. The boundary inclusion S 0 → I
0
→ I+ . The induced natural transformation
is a cellular map and it induces a chain map S+
0
S+ ⊗ (−) → I+ ⊗ (−) of functors Ch≥0 (Z) → Ch≥0 (Z) is naturally isomorphic to the
inclusion functor (i0 , i1 ) : C ⊕ C → cyl(C) from exercise sheet 2.
(3) Let S 0 again be the 0-sphere considered as a CW complex with two 0-cells only. We consider
e∗cell (S 0 ).
S 0 = (S 0 , 0) as a pointed CW complex and abuse notation by also writing S 0 for C
There is a natural isomorphism of functors
S0 ⊗ − ∼
= id : Ch≥0 (Z) → Ch≥0 (Z).
(4) Let I = [0, 1] denote the interval with the usual CW structure as above. We consider
e∗cell (I).
I = (I, 0) as a pointed CW complex and abuse notation by also writing I for C
There is a natural isomorphism of functors
I ⊗−∼
= cone : Ch≥0 (Z) → Ch≥0 (Z),
where cone is the cone construction from exercise sheet 3. The boundary inclusion S 0 → I
is a cellular map and it induces a chain map S 0 → I. The induced natural transformation
S 0 ⊗ (−) → I ⊗ (−) of functors Ch≥0 (Z) → Ch≥0 (Z) is naturally isomorphic to the inclusion
functor i1 : C → cone(C) from exercise sheet 3.