5-manifolds. Since the linking form and the second Stiefel-Whitney class are homotopy invariants, we get:
Corollary 7.21. If two closed, simply-connected 5-manifoldsX, Y are homotopy equivalent, then they are diffeomorphic.
Proof. Ifh:X →Y is a homotopy equivalence, thenθ=h∗preserves linking numbers (Proposition 7.3) andw2(Y)◦θ=w2(X)(Proposition 7.12); hence there exists a diffeomorphismf:X →Y such thatf∗=h∗.
VII.5 Construction of building blocks
Recall the following definition:
Definition 7.22. A smooth manifoldXnof dimensionnis called irreducible if in any connected sum decompositionX =Y1#Y2one of the summands is diffeomorphic toSn.
There is a different definition, used for example in Section III.1.2, where a smoothn-manifold is called irreducible if and only if in any connected sum decomposition one of the summands is homeo-morphic toSn. In the 5-dimensional case this difference is inessential by Corollary 7.21.
Note that a connected sum of two manifolds is simply-connected if and only if both summands are simply-connected. It is possible to give a complete list of all simply-connected, closed, irreducible 5-manifolds. They are constructed in [6] (see Table VII.2). There are three special manifolds (W, S2×S3,S2טS3) and several families: a familyXk wherek ∈ N= {1,2, . . .}and for every prime numberpa familyMpk,k∈N. The manifoldX1 is exceptional in this list because it is diffeomorphic toW#W, cf. Proposition 7.28. All other manifolds in Table VII.2 are irreducible.
ManifoldX H2(X;Z) w2(X) b(X) i(X) W3(X)
(1) Xk,k∈N Z2k⊕Z2k 6= 0 C2k k 6= 0
(2) Wu-manifoldW Z2 6= 0 A 1 6= 0
(3) Mpk,pprime,k∈N Zpk⊕Zpk 0 Bpk 0 0
(4) S2×S3 Z 0 – 0 0
(5) S2טS3 Z 6= 0 – ∞ 0
Table VII.2: Building blocks of simply-connected 5-manifolds.
HereS2טS3 denotes the non-trivialS3-bundle overS2 (which is unique up to isomorphism, be-cause π1(SO(5)) = Z2) and W3(X) ∈ H3(X;Z) denotes the third integral Stiefel-Whitney class, given by the image ofw2(X)under the Bockstein homomorphismβ,
. . .−→H2(X;Z)−→p∗ H2(X;Z2)−→β H3(X;Z)−→. . .
associated to the short exact sequence of coefficients0 → Z →·2 Z →p Z2 → 0. The manifolds in the table above are pairwise not homotopy equivalent, distinguished by their invariants.
We want to give an explicit construction of the manifolds in Table VII.2. The following theorem is a generalization of the Heegaard decomposition of3-manifolds to manifolds of higher dimension (see [76, Chapter VIII, Cor. 6.3]).
Theorem 7.23. A (k−1)-connected closed(2k+ 1)-dimensional manifold, k ≥ 1, is obtained by identifying the boundaries of two manifolds, each of which is a connected sum along the boundary of a number of(k+ 1)-disc bundles overSk.
In particular fork= 2, all simply-connected closed 5-manifolds can be obtained in this way from D3-bundles overS2. We explicitly describe this decomposition for the manifolds in Table VII.2 and then prove that all simply-connected 5-manifolds can be obtained by connected sums of these building blocks.
Up to isomorphism, there are twoD3-bundles over S2 (because π1(SO(3)) = Z2): the trivial bundleA =S2×D3 and a non-trivial bundleB =S2טD3. The boundaries are∂A=S2×S2and
∂B = CP2#CP2, since∂B = S2טS2 is the non-trivialS2-bundle over S2. LetA0, B0 denote the boundary connected sums
A0 =A#bA, B0 =B#bB.
ThenA0 andB0are simply-connected compact 5-manifolds with boundary
∂A0 =∂A#∂A, ∂B0 =∂B#∂B.
We want to show that all building blocks in Table VII.2 are constructed by taking two copies of a manifold of the same typeA, A0, BorB0 and gluing them together along their boundaries via certain orientation reversing diffeomorphisms.
SinceA andB are homotopy equivalent to S2 they have homology only in degree 0 and2. We denote the generator ofH2(A;Z)byuand the generator ofH2(B;Z)byv. Letx, ydenote the stan-dard generators of H2(∂A;Z), corresponding to theS2-factors, and p, q the standard generators of H2(∂B;Z). Ifidenotes the inclusion of the boundary into the manifold, we have
i∗(x) =u, i∗(y) = 0, and i∗(p) =v=i∗(q).
The claim forBfollows becausepandqare the fundamental classes of the image of sections inS2טS2. Similarly,H2(A0;Z)has generatorsu1, u2andH2(∂A0;Z)has generatorsx1, y1, x2, y2such that
i∗(xj) =uj, i∗(yj) = 0,
whereasH2(B0;Z)has generatorsv1, v2andH2(∂B0;Z)has generatorsp1, q1, p2, q2 such that i∗(pj) =vj =i∗(qj).
LetA(k)andB(n)for1≤k, n <∞denote the matrices
A(k) =
1 0 0 −k
0 1 0 0
0 k 1 0
0 0 0 1
, B(n) =
1 n −n 0
n 1 0 n
n 0 1 n
0 −n n 1
.
We writeφ∗(e1, e2, e3, e4)as a shorthand notation for(φ∗e1, φ∗e2, φ∗e3, φ∗e4).
The 4-manifolds∂Aand∂Bhave natural orientation reversing self-diffeomorphisms, given by an orientation reversing self-diffeomorphism on oneS2-factor and the identity on the otherS2-factor for
∂Aand by interchanging the summands in∂B. They induce orientation reversing self-diffeomorphisms on the connected sums ∂A0 and∂B0 (see Lemma 2 in [144]). Ifφis an orientation preserving self-diffeomorphism of one of the manifolds ∂A, ∂B, ∂A0, ∂B0, we can compose it with this orientation reversing self-diffeomorphism to get an orientation reversing self-diffeomorphism, which we denote byφ.
We construct the following manifolds:
VII.5 Construction of building blocks 149
• S2טS3=B∪g∞B, where
g∞:∂B −→∂B
is an orientation preserving self-diffeomorphism realizing on second homology(g∞)∗(p, q) = (p, q).
• W =B∪g−1B, where
g−1:∂B−→∂B
is a orientation preserving self-diffeomorphism realizing on second homology (g−1)∗(p, q) = (p,−q).
• M[A(k)] =A0∪fkA0, where
fk:∂A0 −→∂A0
is a orientation preserving self-diffeomorphism realizing on second homology (fk)∗(x1, y1, x2, y2) = (x1, y1, x2, y2)A(k).
• X[B(n)] =B0∪gnB0, where
gn:∂B0 −→∂B0
is a orientation preserving self-diffeomorphism realizing on second homology (gn)∗(p1, q1, p2, q2) = (p1, q1, p2, q2)B(n).
Since the maps on homology above always preserve the intersection form, the existence of the corresponding diffeomorphisms follows from a theorem of Wall (see [144]):
Theorem 7.24. LetM be a closed, simply-connected 4-manifold which is a connected sum of copies of CP2,CP2 andS2 ×S2. For b2(M) > 10 exclude the case that b+2(M) = 1 orb−2(M) = 1.
Then any automorphism of the intersection form QM can be realized by an orientation preserving self-diffeomorphism ofM.
The corresponding building blocks in Table VII.2 are defined as Mpk = M[A(pk)] and Xk = X[B(2k−1)]. Since the manifolds A, A0, B, B0 are simply-connected, the manifolds we have con-structed are simply-connected closed oriented 5-manifolds. We now compute their homology, which can be reduced to computingH2(X;Z)by Section VII.3.1.
Proposition 7.25. The second homology groups are given by:
(1.) H2(S2טS3;Z) =Z (2.) H2(W;Z) =Z2
(3.) H2(M[A(k)];Z) =Zk⊕Zk (4.) H2(X[B(n)];Z) =Z2n⊕Z2n
Proof. We need the following form of the Mayer-Vietoris sequence: SupposeU, V are manifolds with boundary andX=U ∪φV with a diffeomorphismφ:∂U −→∂V. LetiU, iV denote the inclusion of the boundary in the manifolds. Then there is the exact sequence, cf. Section V.1.1:
. . .−→Hn+1(X)−→Hn(∂U)−→Ψ Hn(U)⊕Hn(V)−→Hn(X)−→. . . withΨ(x) = (iU∗(x), iV∗φ∗(x)). In our situation we have
0−→H3(X)−→H2(∂U)−→Ψ H2(U)⊕H2(V)−→H2(X)−→0.
HenceH2(X)is isomorphic to the cokernel ofΨ. SinceU =V in our case, we denote the homology generators of the manifoldV with a bar, likev.
(1.) Ψis given by
p7→v+v q 7→v+v.
Hence ImΨ =Z(v+v). Sincev, v+vis a basis forH2(B)⊕H2(B)we get CokerΨ∼=Z. (2.) Ψis given by
p7→v+v q 7→v−v.
Hence ImΨ = 2Zv⊕Z(v−v). With the same basis as in (a) this implies CokerΨ∼=Z2. (3.) Ψis given by
x17→u1−u1 y17→ku2
x27→u2−u2
y27→ −ku1.
We take the basisu1, u2, u1−u1, u2−u2 forH2(A0)⊕H2(A0). Then CokerΨ∼=Zk⊕Zk. (4.) Ψis given by
p1 7→v1+ (n+ 1)v1+nv2
q1 7→v1+ (n+ 1)v1−nv2
p2 7→v2−nv1+ (n+ 1)v2 q2 7→v2+nv1+ (n+ 1)v2.
Hence a basis for the image ofΨis2nv1,2nv2, v1+ (n+ 1)v1+nv2, v2+nv1+ (n+ 1)v2. ForH2(B0)⊕H2(B0)we take as basis the last two elements of the basis of ImΨtogether with v1, v2. Then CokerΨ∼=Z2n⊕Z2n.
VII.5 Construction of building blocks 151 We want to determine the i-invariant of the closed, simply-connected 5-manifolds constructed above. Because of Theorems 7.8 and 7.9 this will determine their linking forms. It is clear that
i(S2×S3) = 0, and i(W) = 1,
sinceS2×S3 is spin and the only possible linking form onH2(W;Z) ∼=Z2 is of typeA, which has i-invariant1.
Lemma 7.26. A connected sumX =Y1#Y2ofn-dimensional oriented manifolds is spin if and only if bothY1andY2are spin. A similar result holds for boundary connected sums.
Proof. To define the connected sum ofY1 andY2 one chooses embedded disksD1nandDn2 inY1, Y2
and an orientation reversing diffeomorphismφ: D1n → D2n. SupposeY1 andY2 are spin and choose spin structures. Since there is only one spin structure onDnup to homotopy, the image underφof the induced spin structure onDn1 and the induced spin structure onD2nare homotopic. This is also true for the induced spin structures on∂(Y1\intD1n)and∂(Y2\intDn2). Hence the image underφof the induced spin structure on∂(Y1\intDn1)extends overY2\Dn2 to give a spin structure onX.
Conversely, suppose thatXis spin. We only prove the casen≥3. A spin structure onX induces spin structures onY1\intDn1 andY2\intDn2. SinceH1(Sn−1;Z2) = 0ifn≥3, there is only one spin structure onSn−1. It extends overDn. Hence the spin structures on∂(Y1\intD1n)and∂(Y2\intD1n) extend overDn1 andDn2 to give spin structures onY1 andY2.
Lemma 7.27. The manifoldsM[A(k)]are spin for allk≥1and the manifoldsX[B(n)]are non-spin for alln≥1.
Proof. Suppose a manifold of type X[B(n)] is spin. A spin structure on X[B(n)] induces a spin structure on B0, which induces a spin structure on∂B0 = 2CP2#2CP2. This is impossible, since 2CP2#2CP2has odd intersection form.
We denoteM[A(k)]byA01∪fkA02. The manifoldAis spin, since it is homotopy equivalent toS2. By Lemma 7.26,A0is spin. SinceH1(S2×S2#S2×S2;Z2) = 0, there exists a unique spin structure on∂A0, up to homotopy. Choose spin structures onA01, A02. The image underfkof the induced spin structure on∂A01 is homotopic to the induced spin structure on∂A02, hence extends overA02 to give a spin structure onM[A(k)].
In particular,Xk is not spin fork ≥ 1andMpk is spin for all primespand integersk ≥1. This implies that
i(Mpk) = 0 for all primespand integersk≥1.
On the other hand,i(Xk)6= 0for allk≥1. Since
H2(Xk;Z) =Z2k⊕Z2k, it follows by Theorem 7.9 thatb(Xk)∼=C2k. In particular,
i(Xk) =k for all integersk≥1.