“Counterexamples to the Kneser Conjecture in Dimension Four” by Matthias Kreck, Wolfgang Lück and Peter Teichner

Abstract: We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent to M_{0}♯M_{1} unless M_{0} or M_{1} is homeomorphic to S^{4}. Let N be the nucleus of the minimal elliptic Enrique surface V_{1}(2, 2) and put M = N ∪_{∂N} N. The fundamental group of M splits as ℤ/2 ∗ ℤ/2. We prove that M♯k(S^{2} × S^{2}) is diffeomorphic to M_{0}♯M_{1} for non-simply connected closed smooth four-manifolds M_{0} and M_{1} if and only if k ≥ 8. On the hand we show that M is homeomorphic to M_{0}♯M_{1} for closed topological four-manifolds M_{0} and M_{1} with π_{1}(M_{i}) = ℤ/2.

Key words: homotopy counterexamples of Kneser’s conjecture in dimension four, examples in dimension four splitting topologically but not smoothly

AMS-classification number: 57M99

Introduction

If M is a closed connected three-manifold and α : π_{1}(M) ⟶ Γ_{0} ∗ Γ_{1} is an isomorphism then there are closed connected three-manifolds M_{0} and M_{1} with Γ_{i} = π_{1}(M_{i}) together with a diffeomorphism f : M ⟶ M_{0}♯M_{1} inducing α on the fundamental groups. This theorem is known as Kneser’s conjecture. It fails in dimension ≥ 5 by results of Cappell [1],[2]. Recently it has been shown that Kneser’s conjecture holds in dimension four stably, i.e. if one allows additional connected sums with copies of S^{2} × S^{2} [8], [11]. In this article we give counterexamples to the unstable version of Kneser’s conjecture in dimension four. The first example does not split up to homotopy, the second splits topologically but not smoothly. We prove in section 1

Theorem 0.1 For distinct prime numbers p_{0} and p_{1} there exists a connected closed smooth orientable four-manifold M such that π_{1}(M) is (ℤ/p_{0} × ℤ/p_{0}) ∗ (ℤ/p_{1} × ℤ/p_{1}) and if M is homotopy equivalent to a connected sum M_{0}♯M_{1}, then M_{0} or M_{1} is homeomorphic to S^{4}.

In section 2 we assign to a closed oriented smooth four-manifold M together with an isomorphism α : π_{1}(M) → Γ_{0} ∗ Γ_{1} an invariant σ(M, α) ∈ ℤ/16 × ℤ/16, provided that its universal covering is Spin. Namely, we split M as M_{0} ∪_{S} M_{1} according to α. Then S inherits a Spin-structure from M̃ and we define σ(M, α) = (sign(M_{0}) − R(S), sign(M_{1}) + R(S)) for sign(M_{i}) the signature and R(S) the Rohlin invariant. This invariant depends only on the stable oriented diffeomorphism type of M and we investigate its dependency on α.

Let N be the nucleus of the minimal elliptic Enriques surface V_{1}(2, 2) in the notation of Gompf [7]. Put M = N ∪_{∂N} N^{−}. The fundamental group of M is ℤ/2 ∗ ℤ/2. In section 3 we show using Freedman’s topological s-cobordism theorem in dimension four [6] and Donaldson’s result about definite intersection forms of smooth four-manifolds [4] and the invariant of section 2

Theorem 0.2 M is homeomorphic to M_{0}♯M_{1} for two closed topological four-manifolds M_{0} and M_{1} with π_{1}(M_{i}) = ℤ/2 but M ♯ k(S^{2} × S^{2}) is diffeomorphic to M_{0}♯M_{1} for non-simply connected closed smooth four-manifolds M_{0} and M_{1} if and only if k ≥ 8.

1. Examples not Splitting Homotopically

In this section we construct closed orientable four-manifolds whose fundamental group is a non-trivial free product and which are not homotopy equivalent to a non-trivial connected sum M_{0}♯M_{1} (see Theorem 1.4). As a preliminary we need the following Lemma which is taken from [9, Theorem 3 on page 162] whose proof we enclose for the reader’s convenience.

Lemma 1.3 Suppose that m_{i}, r_{i}, n_{i} and q_{i} for i = 0, 1 are integers satisfying

r_{i} > 1, r_{i}^{m_{i}} − 1 = n_{i}q_{i}, r_{i} ≡ 1 mod n_{i}, (m_{i}, n_{i}) ≠ 1 (q_{0}, q_{1}) = 1.

Then the group

π = (ℤ/m_{0} × ℤ/n_{0}) ∗ (ℤ/m_{1} × ℤ/n_{1})

has the presentation of deficiency −1

π = ⟨a_{0}, b_{0}, a_{1}, b_{1} | a_{0}^{m_{0}} = 1, [a_{0}, b_{0}] = b_{0}^{r_{0}−1}, a_{1}^{m_{1}} = 1, [a_{1}, b_{1}] = b_{1}^{r_{1}−1}, b_{0}^{n_{0}} = b_{1}^{n_{1}}⟩.

\underline{\underline{Proof}} : Obviously it suffices to show that the relation b_{0}^{n_{0}} = 1 follows from the other

relations. We start with proving inductively for k = 1, 2, … the relation a_{i}^{k}b_{i}a_{i}^{−k} = b_{i}^{−r_{1}^{k}} for i = 0, 1. The induction step follows from the calculation:

a_{i}^{k+1}b_{i}a_{i}^{−(k+1)} = a_{i}a_{i}^{k}b_{i}a_{i}^{−k}a_{i}^{−1} = a_{i}b_{i}^{r_{1}^{k}}a^{−1} = (a_{i}b_{i}a_{i}^{−1})^{r_{i}^{k}} = (b_{i}^{r_{i}})^{r_{i}^{k}} = b_{i}^{r_{i}^{k+1}}.

This implies for k = m_{i} and i = 0, 1

(b_{i}^{n_{i}})^{q_{i}} = b_{i}^{r_{i}^{m_{i}}−1} = 1.

Since b_{0}^{n_{0}} = b_{1}^{n_{1}} holds we conclude

(b_{0}^{n_{0}})^{q_{0}} = (b_{0}^{n_{0}})^{q_{1}} = 1.

Since q_{0} and q_{1} are prime, we get b_{0}^{n_{0}} = 1.

We mention that for distinct primes p_{0} and p_{1} one can find the numbers m_{i}, r_{i}, n_{i} and q_{i} as required in Theorem 1.4 so that it applies to π = (ℤ/p_{0} × ℤ/p_{0}) ∗ (ℤ/p_{1} × ℤ/p_{1}) [9, page 163].

Theorem 1.4 Let M be the boundary of a regular neighborhood of an embedding into ℝ^{5} of a 2-dimensional CW -complex X which realizes a presentation of

π = (ℤ/m_{0} × ℤ/n_{0}) ∗ (ℤ/m_{1} × ℤ/n_{1})

of deficiency −1. Then M is not homotopy equivalent to a connected sum M_{0}♯M_{1} unless M_{0} or M_{1} is homeomorphic to S^{4}.

For the proof we need the following well-known Lemma.

Lemma 1.5 Let M be a connected closed orientable four-manifold with fundamental group π and classifying map f : M ⟶ Bπ. Denote by b_{p}(π; F) the p-th Betti number of Bπ with coefficients in the field F. If f_{∗}([M]) = 0 in H_{4}(Bπ; F), then

2 · (b_{2}(π; F) − b_{1}(π; F) + b_{0}(π; F)) ≤ χ(M).

\underline{\underline{Proof}} : Since the classifying map is 2-connected, the map f^{p} : H^{p}(Bπ; F) ⟶ H^{p}(M; F) is bijective for p = 0, 1 and injective for p = 2. Because of f_{∗}([M]) = 0 its image for p = 2 is a totally isotropic subspace of H^{2}(M; F) with respect to the intersection form. If we write the intersection form as an isomorphism b : H^{2}(M; F) ⟶ H^{2}(M; F)^{∗}, this is equivalent to the fact that the composition i^{∗} ∘ b ∘ i for the inclusion i : im(f^{2}) ⟶ H^{2}(M; F) is zero. Hence H^{2}(M; F) contains a subspace which is isomorphic to the direct sum of two copies of H^{2}(Bπ; F). This shows b_{p}(M; F) = b_{p}(π; F) for p = 0, 1 and b_{2}(M; F) ≥ 2 · b_{2}(π; F). From Poincaré duality, χ(M) = b_{2}(M; F) − 2 · b_{1}(M; F) + 2 · b_{0}(M; F) and the claim follows.

Now we are ready to prove Theorem 1.4. We first explain the construction of M which depends on the presentation of π given in Lemma 1.3. Let X be a 2-dimensional CW - complex given by this presentation. Embed X into ℝ^{5} and let M be the boundary of a regular neighboorhood N of X. The resulting manifold M comes with a reference map f to Bπ which induces an isomorphism on the fundamental groups. Obviously we have [M, f] = 0 in Ω_{4}(Bπ), a nullbordism is given by the regular neighborhood N. This implies f_{∗}([M]) = 0 in H_{4}(π; ℤ). One easily checks χ(M) = 2 · χ(N) = 2 · χ(X) = 4.

Choose for i = 0, 1 a prime number p_{i} dividing both m_{i} and n_{i}. Let 𝔽_{p_{i}} be the field of p_{i} elements. One easily checks b_{k}(ℤ/p_{i}^{l}; 𝔽_{p_{i}}) = 1 for k ≥ 0 and l ≥ 1 and computes using the Künneth fromula

b_{2}(ℤ/m_{i} × ℤ/n_{i}; 𝔽_{p_{i}}) − b_{1}(ℤ/m_{i} × ℤ/n_{i}; 𝔽_{p_{i}}) + b_{0}(ℤ/m_{i} × ℤ/n_{i}; 𝔽_{p_{i}}) = 2.

Assume that M is homotopy equivalent to M_{0}♯M_{1}. By Kurosh subgroup theorem [12, Theorem 1.10 on page 178]) (and possible renumbering M_{0} and M_{1}) it suffices to treat the two cases where π_{1}(M_{i}) = ℤ/m_{i} × ℤ/n_{i} for i = 0, 1 or where M_{0} is simply connected. In the first case we get χ(M_{i}) ≥ 4 from Lemma 1.5 and hence χ(M_{0}♯M_{1}) ≥ 6. This contradicts the assumption χ(M) = 4. In the second case we have π_{1}(M_{1}) = π_{1}(M) and again by Lemma 1.5 and the additivity of k-th Betti number b_{k}(π; F) for k ≥ 1 under free products we conclude χ(M_{1}) ≥ 4. This implies χ(M_{0}) ≤ 2. Hence M_{0} is a homotopy sphere and by Freedman’s result [5, Theorem 1.6, page 280] homeomorphic to S^{4}. This finishes the proof of Theorem 1.4.

2. A stable diffeomorphism invariant

We introduce a stable diffeomorphism invariant for a connected closed oriented smooth four-manifold M whose universal covering possesses a Spin-structure together with an isomorphism α : π_{1}(M) ⟶ Γ_{0} ∗ Γ_{1}. We will surpress base points in the context of fundamental groups since all the group theoretic conditions we will give are invariant under inner automorphisms. Let K(Γ_{0}, 1) ∪ [0, 1] ∪ K(Γ_{1}, 1) be obtained by the disjoint union of the Eilenberg-MacLane spaces and [0, 1] by identifying {i} with the base point of K(Γ_{i}, 1) for i = 0, 1. Choose a map

\overline{α} : M ⟶ K(Γ_{0}, 1) ∪ [0, 1] ∪ K(Γ_{1}, 1)

which is transversal to 1/2 ∈ [0, 1] and up to homotopy determined by the property that it induces on the fundamental groups the isomorphism α up to inner automorphisms if we identify the fundamental group of K(Γ_{0}, 1) ∪ [0, 1] ∪ K(Γ_{1}, 1) for the base point 1/2 with Γ_{0}∗Γ_{1} in the obvious way. We orient [0, 1] by the direction from 0 to 1. We get a trivialization of the normal bundle of 1/2 in [0, 1]. This induces by transversality a trivialization of the normal bundle ν(S, M) of S in M where S is the preimage of 1/2. In particular S splits M into the pieces M_{0} and M_{1}, i.e. M = M_{0} ∪_{S} M_{1} where M_{0} respectively M_{1} is mapped by \overline{α} to K(Γ_{0}, 1) ∪ [0, 1/2] respectively [1/2, 1] ∪ K(Γ_{1}, 1). The inclusion j : S ⟶ M induces the trivial map on the fundamental groups and lifts to a map j̃: S ⟶ M̃. The unique Spin-structure on M̃ restricts to a Spin-structure on j̃^{∗}T M̃ = T S ⊕ ν(S, M). Since we have already fixed a trivialization of ν(S, M), this induces a Spin-structure on S. Denote by R(S) ∈ ℤ/16 the Rohlin invariant of the closed three-dimensional Spin-manifold S which is the signature modulo 16 of any smooth Spin-nullbordism of S. Our invariant is defined by

Definition 2.1 σ(M, α) = (sign(M_{0}) − R(S), sign(M_{1}) + R(S)) ∈ ℤ/16 × ℤ/16

Next we show that this invariant is well-defined and examine its dependency on α. Recall that a finitely generated group Γ is called indecomposable if Γ is non-trivial and Γ ≅ Γ′ ∗ Γ′′ implies that Γ′ or Γ′′ is trivial. Finite non-trivial groups are obviously indecomposable. We want to show

Lemma 2.2 Let M and M′ be connected closed oriented smooth four-manifolds, whose universal coverings possess Spin-structures, together with isomorphisms α : π_{1}(M) ⟶ Γ_{0} ∗ Γ_{1} and α′ : π_{1}(M′) ⟶ Γ′_{0} ∗ Γ′_{1}. Suppose Γ_{0}, Γ_{1}, Γ′_{0} and Γ′_{1} are indecomposable and not infinite cyclic. Assume that there is an oriented diffeomorphism

f : M ♯ k(S^{2} × S^{2}) ⟶ M′ ♯ k′(S^{2} × S^{2}).

Then we get

σ(M, α) = σ(M′, α′).

where we may have to interchange the order of the summands ℤ/16×ℤ/16 in the case where Γ_{0} and Γ_{1} are isomorphic.

\underline{\underline{Proof}} : In the first step we show the existence of isomorphisms β_{i} : Γ_{i} ⟶ Γ′_{i} such that the composition α′ ∘ f_{∗} ∘ α^{−1} is (β_{0} ∗ β_{1}) up to inner automorphisms after possibly renumbering Γ_{0} and Γ_{1}. By Kurosh Subgroup Theorem [12, Theorem 1.10 on page 178] and after possibly renumbering Γ_{0} and Γ_{1} the composition α′ ∘ f_{∗} ∘ α^{−1} sends Γ_{0} respectively Γ_{1} to a conjugate of Γ′_{0} respectively Γ′_{1}. Hence there are isomorphisms β_{i} : Γ_{i} ⟶ Γ′_{i} and an automorphism ϵ of Γ_{0} ∗ Γ_{1} sending γ_{0} ∈ Γ_{0} to γ_{0} and γ_{1} ∈ Γ_{1} to δγ_{1}δ^{−1} for some δ ∈ Γ_{0} ∗ Γ_{1} such that α′ ∘ f_{∗} ∘ α^{−1} is mboxϵ∘(β_{0}∗β_{1}) up to inner automorphisms. Without destroying this property one can change δ, β_{0} and β_{1} such that δ is trivial or δ begins with a non-trivial letter in Γ_{1} and ends with a non-trivial letter in Γ_{0}. In the second case no element of Γ_{1} can lie in the image of ϵ and hence the surjectivity of ϵ forces ϵ to be the identity and the claim follows.

In the next step we show that the choice of \overline{α} does not matter. Suppose we have two choices of maps \overline{α} and \overline{α}′ : M ⟶ K(Γ_{0}, 1) ∪ [0, 1] ∪ K(Γ_{1}, 1) which are transversal to 1/2. Let M = M′_{0} ∪_{S′} M′_{1} be the splitting induced by \overline{α}′. Since \overline{α} and \overline{α}′ induce up to inner automorphisms the same homomorphism on the fundamental groups, they are homotopic. Hence there is a map h : M × [0, 1] ⟶ K(Γ_{0}, 1) ∪ [0, 1] ∪ K(Γ_{1}, 1) which is transveral to 1/2 and h_{0} = \overline{α} and h_{1} = \overline{α}′. As explained above h induces a splitting M × [0, 1] = W_{0} ∪_{Z} W_{1} and Z inherits a Spin structure (and in particular an orientation) from h. The orientation of M induces orientations on M×[0, 1], W_{0} and W_{1}. We use the convention for an oriented manifold V with boundary ∂V that ∂V inherits the orientation determined by the decomposition T V |_{∂V} = T ∂V ⊕ ν(∂V, V) and the orientation on the normal bundle ν(∂V, V) given by the outward normal field. Notice that the orientations on S and Z coming from the Spinstructures as described above agree with the ones coming from S = ∂M_{0} and Z = ∂W_{0} and are the opposites of the orientations coming from S = ∂M_{1} and Z = ∂W_{1}. We laim that the orientation of S ⊂ ∂Z agrees with the one coming from S = ∂M_{0}. Namely, the decompositions T W_{0}|_{Z} = T Z ⊕ ν(Z, W_{0}) and T Z|_{S} = T S ⊕ ν(S, Z) induce a decomposition

T W_{0}|_{S} = T S ⊕ ν(S, Z) ⊕ ν(S, M_{0}).

The orientation of S given by S ⊂ ∂Z is compatible with this decomposition if one uses the outward normal fields on ν(S, Z) and ν(S, M_{0}). The decompositions T M_{0}|_{S} = T S ⊕ ν(S, M_{0}) and T W_{0}|_{M_{0}} = T M_{0} ⊕ ν(M_{0}, W_{0}) yield

T W_{0}|_{S} = T S ⊕ ν(S, M_{0}) ⊕ ν(S, Z).

The orientation of S given by S = ∂M_{0} is compatible with this decomposition if one uses the outward respectively inward normal field on ν(S, M_{0}) respectively ν(S, Z). One treats the other component S′ similiarly and gets ∂Z = S ∐(S′)^{−}. This implies

sign(Z) = R(S) − R(S′) ∈ ℤ/16 × ℤ/16.

The boundary of W_{0} is M_{0}^{−} ∪ Z ∪ M′_{0} and of W_{1} is M_{1}^{−} ∪ Z^{−} ∪ M′_{1}. This shows

−sign(M_{0}) + sign(Z) + sign(M′_{0}) = 0 and − sign(M_{1}) − sign(Z) + sign(M′_{1}) = 0.

and hence

(sign(M_{0})−R(S), sign(M_{1})+R(S)) = (sign(M′_{0})−R(S′), sign(M′_{1})+R(S′)) ∈ ℤ/16×ℤ/16.

In the final step we can assume that f is an oriented diffeomorphism from M to M′. Choose base point preserving maps \overline{β_{i}} : K(Γ_{i}, 1) ⟶ K(Γ′_{i}, 1) inducing β_{i} on the groups for i = 0, 1. By our first step \overline{α}′ is homotopic to the composition

M′ ⟶^{f^{−1}} M ⟶^{\overline{α}} K(Γ_{0}, 1) ∪ [0, 1] ∪ K(Γ_{1}, 1) ⟶^{\overline{β_{0}}∪id∪\overline{β_{1}}} K(Γ′_{0}, 1) ∪ [0, 1] ∪ K(Γ′_{1}, 1)

Obviously the invariant for the splitting of M′ with respect to this composition is the same as the one for the splitting of M with respect to \overline{α} and the claim follows.

3. Examples Splitting Topologically but not Smoothly

In this section we give an example which splits topologically but not smoothly. Let us recall from [7] that every minimal elliptic surface V_{n}(p_{1}, p_{2}, … p_{k}) (whose elliptic fibration with base ℂP^{1} has 6n cusp fibers and k multiple fibers with multiplicities p_{i}) can be decomposed as a union N_{n}(p_{1}, p_{2}, … p_{k}) ∪_{Σ} Φ_{n} along the Seifert fibered homology three-sphere Σ(2, 3, 6n−1) which as the link of a singularity bounds the Milnor fiber Φ_{n} of Σ_{n}. The piece N_{n}(p_{1}, p_{2}, …, p_{k}) is called nucleus of V_{n}(p_{1}, p_{2}, … p_{k}).

Theorem 3.1 Let N = N_{1}(2, 2) be the nucleus of the Enriques surface V = V_{1}(2, 2). Put M = N ∪_{∂N} N^{−}.

1. M is homeomorphic to M_{0}♯M_{1} for two closed topological four-manifolds M_{0} and M_{1} with π_{1}(M_{i}) = ℤ/2 for i = 0, 1.

2. M ♯ k(S^{2}×S^{2}) is diffeomorphic to M_{0}♯M_{1} for non-simply connected closed smooth fourmanifolds M_{0} and M_{1} if and only f k ≥ 8. In fact, M ♯ 8(S^{2} × S^{2}) is diffeomorphic to V ♯V^{−}.

\underline{\underline{Proof}} : First recall from [7] that the nucleus of a minimal elliptic surface is constructed by taking a regular neighborhood of one cusp fiber and a section of the elliptic fibration. This gives a four-manifold in the homotopy type of S^{2} ∨ S^{2}. Then one performs all t he logarithmic transformations inside this neighborhood. The following properties of N are easy consequences from this construction, for details see [7, Section 3]. We also remind the reader that the Enriques surface V has even intersection form E_{8} ⊕ H and its universal covering is the Kummer surface which is Spin.

1. sign(N) = 0 and χ(N) = 3.

2. The inclusion N ⟶ V induces an isomorphism on the fundamental groups and π_{1}(N) = ℤ/2. Since the Milnor fiber Φ_{n} is simply connected this implies that Σ is π_{1}-null in N, i.e. the inclusion of Σ = Σ(2, 3, 5) = ∂N into N induces the trivial map on the fundamental groups.

3. The intersection form of N is even and Ñ is Spin.

The first assertion of our theorem follows directly from the following lemma by setting X = X′ = N.

Lemma 3.2 Let X and X′ be two topological four-manifolds with the same boundary Σ, a homology three-sphere. Assume that Σ is π_{1}-null in X and that X has a good fundamental group. Let C be a contractible four-manifold with boundary Σ. Then there exists a homeomorphism

X ∪_{Σ} X′ ≈ (X ∪_{Σ} C) ♯ (X′ ∪_{Σ} C).

\underline{\underline{Proof}} : Recall that a good fundamental group is one for which the topological s-cobordism theorem holds. By [6] examples of good groups are poly-finite or -cyclic groups. Also, by [5, Theorem 1.4’ on page 367] a manifold C as in the lemma exists.

By taking the connected sum inside the contractible parts C, we obtain a homeomorphism

(X ∪_{Σ} C) ♯ (X′ ∪_{Σ} C) ≈ (X ∪_{Σ} (C♯C)) ∪_{Σ} X′.

Hence it suffices to show that X and (X ∪_{Σ} (C♯C)) are homeomorphic relative boundary. By assumption, the topological s-cobordism theorem holds for this fundamental group and thus it remains to construct an s-cobordism between X and X ∪_{Σ} (C♯C) relative boundary.

Note that C ∪_{Σ} C is a simply connected closed topological four-manifold with the same integral homology as S^{4} and hence by [5, Theorem 1.6 page 371] is homeomorphic to S^{4}. Let C_{0} be a complement of the interior of an embedded disk D^{4} ⊂ int(C). Then we obtain a homeomorphism

(C♯C) ∪_{Σ×{0,1}} Σ × [0, 1] ⟶ C_{0} ∪_{Σ} C_{0} ∪_{S^{3}×{0,1}} S^{3} × [0, 1] ⟶ S^{3} × S^{1}.

This gives an embedding j : Σ × [0, 1] ⟶ ∂(S^{3} × D^{2}). Let f : Σ × [0, 1] ⟶ X × [0, 1] be the inclusion. Define W by S^{3} × D^{2} ∪_{f} X × [0, 1], i.e. by the push out

Σ × [0, 1] \underrightarrow{j} S^{3} × D^{2}
f ↓  ↓
X × [0, 1] \underrightarrow{\overline{\overline{j}}}  W

We want to show that \overline{\overline{j}} : X × [0, 1] ⟶ W is a simple homotopy equivalence. Since the simple homotopy type of W relative X × [0, 1] depends only on the homotopy class of f [3, II.5.5] and Σ is π_{1}-null in X, we can assume that j factorizes as

f : Σ × [0, 1] ⟶^{f_{1}} Z ⟶^{f_{2}} X × [0, 1]

where Z is obtained from Σ × [0, 1] by collapsing the 1-skeleton to a point. Define Y by the push out Y = S^{3} × D^{2} ∪_{f_{1}} Z. Then W is also the push out

Z  \underrightarrow{\overline{j}} Y
f_{2} ↓  ↓
X × [0, 1] \underrightarrow{\overline{\overline{j}}} W

The map \overline{j} : Z ⟶ Y is a homology equivalence as j : Σ × [0, 1] ⟶ S^{3} × D^{2} is. Since Z and S^{3} × D^{2} are simply connected, \overline{j} and hence \overline{\overline{j}} are simple homotopy equivalences [3, II.8.5. and I.5.9]. This shows that the inclusion of X into W is a simple homotopy equivalence.

Similarly, one verifies that the inclusion of the other part X ∪_{Σ} (C♯C) of the boundary of W into W is a homotopy equivalence. Hence W is a s-cobordism. This finishes the proof the lemma and thus also of the first assertion.

2.) Suppose that f : M ♯ k(S^{2} × S^{2}) ⟶ M_{0}♯M_{1} is a diffeomorphism for connected smooth four-manifolds M_{0} and M_{1} which are non-simply connected. By Kurosh Subgroup Theorem cite[Theorem 1.10 on page 178]Lyndon-Schupp (1977) π_{1}(M_{i}) = ℤ/2 for i = 0, 1. There is an obvious choice of isomorphisms α : π_{1}(M) ⟶ ℤ/2 ∗ ℤ/2 and α′ : π_{1}(M_{0}♯M_{1}) ⟶ ℤ/2 ∗ ℤ/2 such that

σ(M, α) = (sign(N) − R(Σ), sign(N^{−}) + R(Σ)) = (8, 8) ∈ ℤ/16 × ℤ/16

and

σ(M_{0}♯M_{1}, α) = (sign(M_{0}), sign(M_{1})) ∈ ℤ/16 × ℤ/16.

From Lemma 2.2 we get σ(M, α) = σ(M_{0}♯M_{1}, α′). This shows for i = 0, 1

|sign(M_{i})| ≥ 8.

The intersection form of N is even and hence its signature is divisible by eight and its rank is even cite[Corollary 1 on page 53]Serre (1970). Suppose that b_{2}(M_{i}) ≤ 9. Then the rank of the intersection orm must be eight and its signature must be ±8. Hence we can find an orientation such that the intersection form on the smooth oriented closed four-manifold M_{i} is he definite form E_{8}. This is impossible by Donaldson’s result [4, Theorem 1 on page 397] hat a definite intersection form of a smooth closed oriented 4-manifold is equivalent up to sign to the standard Euclidean form. Therefore b_{2}(M_{i}) ≥ 10 for i = 0, 1. Since π_{1}(M_{i}) is finite, we conclude

χ(M_{i}) = 2 + b_{2}(M_{i}) ≥ 12.

Since χ(N) = 3, we have χ(M) = 6. Now we get

6 + 2k = χ(M ♯ k(S^{2} × S^{2})) = χ(M_{0}♯M_{1}) = χ(M_{0}) + χ(M_{1}) − 2 ≥ 12 + 12 − 2 ≥ 22

and hence

k ≥ 8.

It remains to prove that M ♯ 8(S^{2} × S^{2}) s diffeomorphic to V ♯V^{−}. Since M = N ∪_{Σ} N^{−}, V = N ∪_{Σ} Φ and he connected sum of V and V^{−} may be taken inside the Milnor fibers Φ, it suffices to show that N ♯ 8(S^{2} × S^{2}) is diffeomorphic (relative boundary) to N ∪_{Σ} (Φ♯Φ^{−}). Here we take the connected sum always in the interior of the manifolds. It was shown in [7, Fig. 27] (we are in the case n = 1) that Φ has a handle decomposition with one 0-handle and eight 2-handles. Therefore, inside Φ♯Φ^{−} we find eight disjointly embedded 2-spheres with trivial normal bundle. These are given by gluing together in pairs the cores of corresponding 2-handles. It is easy to check that after doing 2-surgeries on these eight 2-spheres, i.e. cutting out S^{2} × D^{2} and eplacing it by D^{3} × S^{1}, one gets the product ∂Φ × [0, 1] = Σ × [0, 1]. Reversing this procedure, we see that one can do eight 1-surgeries on (the collar of) = N ∪_{Σ} (Σ × [0, 1]) to obtain N ∪_{Σ} (Φ♯Φ^{−}). We point out that by [7, Fig. 27] all framings for the 2-handles in Φ are even and thus N ∪_{Σ} (Φ♯Φ^{−}) has an even intersection form. Changing slightly our point of view, we see that since all the surgered circles are nullhomotopic in N, each of these 1-surgeries has the effect of taking a connected sum with an oriented S^{2}-bundle over S^{2}. But the nontrivial bundle cannot occure because the resulting manifold must have an even intersection form. Hence we do end up with N ♯ 8(S^{2} × S^{2}) which finishes the proof of our last claim in Theorem 3.1.

References

[1] Cappell, S.E.: “On connected sums of manifolds”, Topology 13, 395 - 400 (1974)

[2] Cappell, S.E.: “A spitting theorem for manifolds”, Inventiones Math. 33, 69 - 170 (1976)

[3] Cohen, M.M.: “A course in simple homotopy theory”, Graduate Texts in Mathematics 10, Springer (1973)

[4] Donaldson, S.K: “The orientation of Yang-Mills moduli spaces and 4-manifold topology”, J. of Differential Geometry 26, 397 - 428 (1987)

[5] Freedman, M.H.: “The topology of four-dimensional manifolds”, J. of Differential Geometry 17, 357 - 453 (1982)

[6] Freedman, M.H.: “The disk theorem for four-dimensional manifolds”, Proc. Int. Cong. Math. Warsaw 1983 volume I, 647 - 663 (1983)

[7] Gompf, R. E.: “Nuclei of elliptic surfaces”, Topology 30, No. 3 , 479-511 (1991)

[8] Hillman, J.A.: “Free products and 4-dimensional connected sums”, preprint (1993)

[9] Hog, C., Lustig, M. and Metzler, W.: “Presentation classes, 3-manifolds and free products”, Proceedings of the conference “Geometry and Topology” 1983 - 1984, University of Maryland, editors: Alexander, J and Harer, J., Lecture Notes in Mathematics 1167, 154 - 167 (1985)

[10] Kirby, R.C. and Siebenmann, L.: “Foundational essays on topological manifolds, smoothings and triangulations”, Annals of Mathematics Studies 88, Princeton University Press (1977)

[11] Kreck, M., Lück, W. and Teichner, P.: “Stable Prime Decompositions of Four-Manifolds”, preprint, Mainz (1994)

[12] Lyndon, R.C. and Schupp, P.E.: “Combinatorial Group Theory”, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer (1977)

[13] Serre, J.P.: “A course in arithmetic”, Graduate Texts in Mathematics 7, Springer (1977)

Matthias Kreck, Fachbereich Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Bundesrepublik Deutschland & Mathematisches Forschungsinstitut Oberwolfach, 77709 Oberwolfach-Walke, Bundesrepublik Deutschland

Wolfgang Lück, Fachbereich Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Bundesrepublik Deutschland

Peter Teichner, University of California, San Diego, Department of Mathematics, 9500 Gilman Drive, LaJolla, CA 92093-0112, U.S.A.

email: kreck/lueck@topologie.mathematik.uni-mainz.de, teichner@euclid.ucsd.edu

FAX: 06131 393867

Version of November 8, 1996