We will show in Appendix A that this stratifold can be given the structure of a regular c-stratifold with boundary S+ D Z+S− D Z+S+ D S−. Since S+ D S− = S and our maps extend to a map from this regular c-stratifold

to U V , we have a bordism between [S+ D Z, g|S+ r]+[S− D Z, g|M− r] and [S, g].

q. e. d.

As an application we compute the homology groups of a topological sum. Let X and Y be topological spaces and X Y the topological sum (the disjoint union). Then X and Y are open subspaces of X Y and we denote them by U and V . Since the intersection U ∩ V is the empty set and the homology groups of the empty set are 0 (this is a place where it is necessary to allow the empty set as k-dimensional stratifold whose corresponding homology groups are of course 0) the Mayer-Vietoris sequence gives short exact sequences:

0 → SHk(X; Z/2) SHk(Y ; Z/2) → SHk(X Y ; Z/2) → 0,

where the zeroes on the left and right side correspond to SHn( ; Z/2) = 0 and SHn−1( ; Z/2) = 0, respectively. The map in the middle is (jX ) −(jY ) . As explained above, exactness implies that this map is an isomorphism:

(jX ) − (jY ) : SHn(X; Z/2) SHn(Y ; Z/2) → SHn(X Y ; Z/2).

Of course, this also implies that the sum (jX ) + (jY ) is an isomorphism.

2.Reduced homology groups and homology groups of spheres

For computations it is often easier to split the homology groups into the homology groups of a point and the “rest”, which will be called reduced homology. Let p : X → pt be the constant map to the space consisting of a single point. The n-th reduced homology group is SHn(X; Z/2) := ker (p : SHn(X; Z/2) → SHn(pt; Z/2)). A continuous map f : X → Y induces a homomorphism on the reduced homology groups by restriction to the kernels and we denote it again by f : SHn(X; Z/2) → SHn(Y ; Z/2).

62 5. The Mayer-Vietoris sequence and homology groups of spheres

If X is non-empty, there is a simple relation between the homology and the reduced homology of X:

The isomorphism sends a homology class a SHn(X; Z/2) to the pair (a − i p (a), p a), where i is the inclusion from pt to an arbitrary point in X. For n > 0 this means that the reduced homology is the same as the unreduced homology, but for n = 0 it di ers by a summand Z/2.

Since it is often useful to work with reduced homology, it would be nice to know if there is also a Mayer-Vietoris sequence for reduced homology. This is the case. We prepare for the argument by developing a useful algebraic result. Consider a commutative diagram of abelian groups and homomorphisms

where the horizontal sequences are exact and the map h1 is surjective. Then we consider the sequence

where the maps fi| are fi|ker hi . The statement is that the sequence

is again exact. This is proved by a general method called diagram chasing, which we introduce in proving this statement. We chase in the commutative diagram given by Ai and Bj above. The first step is to show that im f2| ker f3| or equivalently (f3|)(f2|) = 0. This follows since f3f2 = 0. To show that ker f3| im f2|, we start the chasing by considering x ker h3 with f3(x) = 0. By exactness of the sequence given by the Ai, there is y A2 with f2(y) = x. Since h3(x) = 0 and h3f2(y) = g2h2(y), we have g2h2(y) = 0 and thus by the exactness of the lower sequence and the surjectivity of h1, there is z A1 with g1h1(z) = h2(y). Since g1h1(z) = h2f1(z), we conclude h2(y − f1(z)) = 0 or y − f1(z) ker h2. Since f2f1(z) = 0, we are done since we have found y − f1(z) ker h2 with f2(y − f1(z)) = f2(y) = x.

With this algebraic information, we can compare the Mayer-Vietoris sequences for X = U V with that of the space pt given by U := pt =: V :

· · · → SHn(U ∩ V ; Z/2) → SHn(U; Z/2) SHn(V ; Z/2) →

↓ ↓

→SHn(U ∩ V ; Z/2) → SHn(U ; Z/2) SHn(V ; Z/2) →

→SHn(U V ; Z/2) → SHn−1(U ∩ V ; Z/2) → · · · .

Since U ∩ V = U = V = U V = pt, all vertical maps are surjective if U ∩ V is non-empty (and thus U and V as well), and therefore by the argument above, the reduced Mayer-Vietoris sequence

· · · → SHn(U ∩ V ; Z/2) → SHn(U; Z/2) SHn(V ; Z/2)

→ SHn(X; Z/2) → SHn−1(U ∩ V ; Z/2) → · · ·

is exact if U ∩ V is non-empty.

Now we use the homotopy axiom (Proposition 4.8) and the reduced Mayer-Vietoris sequence to express the homology of the sphere Sm := {x

| ||x|| = 1} in terms of the homology of a point. For this we decompose Sm into the complement of the north pole N = (0, ..., 0, 1) and the south pole S = (0, ..., 0, −1), and define S+m : Sm − {S} and S−m := Sm − {N}. The inclusion Sm−1 → S+m ∩ S−m mapping y −→ (y, 0) is a homotopy equiva-

lence with homotopy inverse r : (x1, . . . , xm+1) −→ (x1, . . . , xm)/||(x1,...,xm)|| (why?). Both S+m and S−m are homotopy equivalent to a point, or equiva-

lently the identity map on these spaces is homotopic to the constant map (why?). Since S+m S−m is Sm, the reduced Mayer-Vietoris sequence gives an exact sequence

· · · → SHn(S+m ∩ S−m; Z/2) → SHn(S+m; Z/2) SHn(S−m; Z/2)

If we use the isomorphisms induced by the homotopy equivalences above this becomes

· · · → SHn(Sm−1; Z/2) → SHn(pt; Z/2) SHn(pt; Z/2)

64 5. The Mayer-Vietoris sequence and homology groups of spheres

and so by induction:

=

m 0

SHn(S ; Z/2) −→ SHn−m(S ; Z/2).

The space S0 consists of two points {+1} and {−1} which are open subsets and by the formula above for the homology of a topological sum we have

class of a compact Z/2-oriented regular stratifold. Let S be a n-dimensional Z/2-oriented compact regular stratifold. We define its fundamental class as [S]Z/2 := [S, id] SHn(S, Z/2). As the name indicates this class is important. We will see that it is always non-zero. In particular, we obtain for each compact smooth manifold the fundamental class [M]Z/2 = [M, id]. In the case of the spheres the non-vanishing is clear since by the inductive computation one sees that the non-trivial element of SHm(Sm; Z/2) is given by the fundamental class [Sm]Z/2.

As an immediate consequence of Theorem 5.3 the spheres Sn and Sm are not homotopy equivalent for m = n, for otherwise their homology groups would all be isomorphic. In particular, for n = m the spheres are not homeomorphic. In the next chapter, we will show for arbitrary manifolds that the dimension is a homeomorphism invariant.

3. Exercises

fg

(1)Let 0 → A −→ B −→ C → 0 be an exact sequence of abelian groups.

a)Show the the following are equivalent:

s

i) There is a map C →− B such that g ◦ s = idC .

p

ii) There is a map B →− A such that p ◦ f = idA.

h

iii) There is an isomorphism B −→ A C such that h ◦ f(a) = (a, 0)

and g ◦ h−1(a, c) = c.

In this case we say that this is a split exact sequence.

b)Show that if all groups are vector spaces and all maps are linear then the sequence splits.

c)Show that if C is free then the sequence splits.

(2)Prove the five lemma: Assume that the following diagram is commutative with exact rows:

a)Show that if f2, f4 are injective and f1 is surjective then f3 is injective.

b)Show that if f2, f4 are surjective and f5 is injective then f3 is surjective.

Conclude that if f2, f4 are isomorphisms, f1 is surjective and f5 is injective then f3 is an isomorphism.

Remark: This lemma is usually used when f1, f2, f4, f5 are isomorphisms.

(4)Let 0 → Z/2 → A → Z/2 → 0 be an exact sequence of abelian groups. What are the possibilities for the group A? What can you say if you know that the maps are linear maps of Z/2 vector spaces?

(5)Compute SHn(X; Z/2) for the following spaces using the MayerVietoris sequence. If possible represent each class by a map from a stratifold.

a)The wedge of two circles S1 S1 or more generally the wedge of two pointed spaces (X, x0) and (Y, y0). Assume that both x0 and y0 have a contractible neighborhood.

Remark: A pointed space (X, x0) is a topological space X together with a distinguished point x0 X. The wedge of two pointed spaces (X, x0) and (Y, y0) is the pointed space (X Y/x0 y0, [x0]) and denoted by X Y .

b)The two dimensional torus T 2 = S1 × S1 or more generally the n-torus which is the product of n copies of S1.

c)The M¨obius band which is the space obtained from I × I after

identifying the points (0, x) and (1, 1 − x) for every x I, where I = [0, 1].

d) Let M1 and M2 be two smooth n-dimensional manifolds. Compute the homology of M1#M2 as defined in the exercises in ch. 1.

66 5. The Mayer-Vietoris sequence and homology groups of spheres

e) Any compact surface (using that an orientable surface is either homeomorphic to S2 or a connected sum of copies of S1 × S1, and a non-orientable surface is homeomorphic to a connected sum of copies of RP2).

f)Any compact orientable surface with one point removed.

g)R2 with n points removed. What about if we remove an infinite discrete set?

(6)Let X be a topological space.

a)Compute SHn(X ×S1; Z/2) or more generally SHn(X ×Sk; Z/2) in terms of SHm(X; Z/2).

b)Compute SHn(ΣX; Z/2) (the suspension of X) or more generally SHn(ΣkX; Z/2) (the kth iterated suspension of X) in terms of SHn(X; Z/2), where the suspension is defined in the exercises in chapter 2.

c)Compute SHn(T X; Z/2) (the suspension of X modulo both end

points) or more generally SHn(T kX; Z/2) (where we define T kX by induction, T kX = T (T k−1X)) in terms of SHm(X; Z/2).

(7)Let M be a non-empty connected closed n-dimensional manifold. Construct a map f : M → Sn with

f : SHn(M; Z/2) → SHn(Sn; Z/2)

non-trivial. (Hint: Use the fact that Sn is homeomorphic to Dn with the boundary collapsed to a point.)

(8)Determine the map SHn(Sn; Z/2) → SHn(RPn; Z/2) induced by the quotient map Sn → RPn.