4. Exercises

3) Similarly, Xj := CP[j/2] gives a CW -decomposition of CPn.

Here is a first instance showing that it is useful to consider CW -decompo- sitions.

Theorem 9.5. A finite CW -complex X is homologically and Z/2-homo- logically finite. Denote the number of j-cells of a finite CW -complex X by βj. Then:

Proof: We prove the statement inductively over the cells. Suppose that Y

U = Dk and V = Z − {0}, where 0 Dk. The space U ∩ V is homotopy equivalent to Sk−1, U is homotopy equivalent to a point, and V is homotopy equivalent to Y .

The Mayer-Vietoris sequence implies that Z is homologically and Z/2- homologically finite, thus, by Theorem 7.4,

e(Z) = e(Y ) + e(pt) − e(Sk−1) = e(Y ) + 1 − (1 + (−1)k−1) = e(Y ) + (−1)k,

which implies the statement. q.e.d.

Remark: In this case as well as in many other instances, it is enough to require that X is homotopy equivalent to a finite CW -complex.

4.Exercises

(1)Let h be a homology theory. Prove the following:

a)If f : A → B is a homotopy equivalence then f is an isomorphism.

b)For every two topological spaces there is a natural map hn(A) hn(B) → hn(A B) and it is an isomorphism.

c)hn( ) = 0 for all n.

(2)Let h be a homology theory.

a)If hn(pt) = 0 for all n, what can you say about h?

1009. A comparison theorem for homology theories and CW -complexes

b)If there exists a non-empty space with hn(X) = 0 for all n, what can you say about h?

(3)Which of the following are homology theories? Prove or disprove:

a)Given a topological space A define for every topological space

X the homology groups hn(X) = SHn(X × A) and for a map f : X → Y the homomorphism SHn(X × A) → SHn(Y × A) induced by the map f × id : X × A → Y × A.

b) Given a topological space A define for every topological space X the homology groups hn(X) = SHn(X A) and for a map f : X → Y the homomorphism SHn(X A) → SHn(Y A) induced by the map f id : X A → Y A.

c) Define for every topological space X the homology groups hn(X) = SHn(X × X) and for a map f : X → Y the homomorphism SHn(X ×X) → SHn(Y ×Y ) induced by the map f × f : X ×X →

Y × Y .

(4)Define for every topological space a series of functors hn(X) = SHn(X) Z/2 with the boundary operator d Z/2.

a) Show that there is a natural transformation

η: hn(X) → SHn(X; Z/2).

b)Is η a natural isomorphism?

c)Is h a homology theory?

(5)Let h and h be two homology theories. Show that h h is a homology theory.

(6)Let h be a homology theory. Show that h defined by hn(X) = hn+k(X) for a given k is a homology theory.

(7)Let X be a topological space and f1 : M1 → X, f2 : M2 → X be two continuous maps from closed manifolds of dimension n. We say that the two maps are bordant if there is a compact manifold M

with boundary equal to M1 M2 and a map f : M → X such that f|Mi = fi. Define Nn(X) to be the set of bordism classes of maps f : Mn → X where Mn is a closed manifold of dimension n and f is a continuous map. Show that N is a homology theory with the boundary operator d defined in a similar way to the one we defined

for SHn. Show that N1(pt) is trivial and N2(pt) is generated by

RP2.

(8)a) Define SHnp(X) in a similar way to the way we defined SHn(X), but instead of stratifolds we use p-stratifolds and the same for stratifolds with boundary. Show that this is a homology theory. What can you say about its connection to SHn(X)?

b) Show that every class in SHnp(X) for n ≤ 2 can be represented by a map from a stratifold which is actually a manifold.