Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 97–110

SIMPLY CONNECTED QUASIREGULARLY ELLIPTIC 4-MANIFOLDS

Seppo Rickman

Department of Mathematics and Statistics, P.O. Box 68 FI-00014 University of Helsinki, Finland; seppo.rickman@helsinki.fi

Abstract. We construct a quasiregular map of R^{4} onto the connected sum of S^{2} × S^{2} with itself. The proof is based on a symmetric splitting of T^{4} to two pieces together with some technique from Piergallini’s article [9].

1. Introduction

An oriented and connected Riemannian n-manifold N is (K)-quasiregularly elliptic if there exists a nonconstant (K)-quasiregular map of R^{n} into N. This terminology was introduced by Bonk and Heinonen in [1] and was suggested from the discussion by Gromov in [4, pp. 63–67]. In this paper we prove the following result.

Theorem 1. The connected sum S^{2} × S^{2}#S^{2} × S^{2} is quasiregularly elliptic.

Theorem 1 gives the first example of a nontrivial simply connected closed quasiregularly elliptic 4-manifold and it solves one question posed by Gromov [3, p. 200], [4, 2.41] and the author [12, p. 183].

In the other direction, a break-through was recently obtained by Bonk and Heinonen [1], namely, if N is a connected, closed, and K -quasiregularly elliptic n-manifold, then the dimension of the de Rham cohomology ring H^{∗}(N) of N has an upper bound depending only on n and K. Earlier results on the closed case were essentially based on the behavior of the first homotopy group and are contained in [5], [8], and [18]. For example, from [18, pp. 146–147] it follows, as observed in [1, Corollary 1.6] that dimH^{1}(N) ≤ n. This gives for n = 2 and n = 3 the bound 2^{n} for dimH^{∗}(N). This is optimal as is seen from dimH^{∗}(T^{n}) = 2^{n} for the n-torus T^{n}, which equality is true for every n ≥ 2. It is an interesting question whether one has a bound for dimH^{∗}(N) independent of K also for n ≥ 4. Furthermore, if this is the case, is the bound again 2^{n}? We refer to [1] for a more detailed discussion on the question of quasiregular ellipticity. For the theory of quasiregular mappings, see [13].

The proof of Theorem 1 is reduced to showing the existence of a branched covering T^{4} → S^{2} × S^{2}#S^{2} × S^{2} and composing it with the projection R^{4} → T^{4} .

2000 Mathematics Subject Classification: Primary 30C65; Secondary 57M12.

An outline of the various steps is as follows. We use a 2-fold branched covering T^{2} → S^{2} to perform the product map F: T^{2} × T^{2} → S^{2} × S^{2} . With the help of a handle decomposition of T^{4} we split T^{4} in a symmetric way into two pieces V and W such that F takes V onto S^{2} × S^{2} with a 4-ball deleted and such that boundary goes to boundary. By a preliminary shifting of coordinates in T^{4} we get similarly a map G which takes W to another copy of S^{2} × S^{2} with a 4-ball deleted. On the common boundary M of V and W we then get two 4-fold branched coverings onto 3-spheres. The task is reduced to the construction of a branched covering M × [0, 1] → S^{3} × [0, 1] which coincides with given maps on M × 0 and M × 1 respectively. Here we make use of the symmetry property of F and G and apply Section 2 from Piergallini’s paper [9]. Piergallini later used [9] in [10] to show that each closed oriented PL 4-manifold is a 4-fold covering of S^{4} .

Acknowledgements. I thank Juha Heinonen for inspiring discussions connected to the subject and to the proof of Theorem 1.

2. Maps F and G

We let T^{n} be [−1, 1]^{n} with opposite sides identified. We define a handle decomposition of T^{4} consisting of one 0-handle H^{0} , four 1-handles H_{i}^{1}, i = 1, …, 4, six 2-handles H_{ij}^{2} , 1 ≤ i < j ≤ 4, four 3-handles H_{i}^{3}, 1, …, 4, and one 4handle H^{4} . These are fixed as follows:

H^{0} = {x ∈ T^{4} : |x_{k}| ≤ 1/3, k = 1, …, 4},
H_{i}^{1} = {x ∈ T^{4} : 1/3 ≤ |x_{i}| and |x_{k}| ≤ 1/3 for k ≠ i},
H_{ij}^{2} = {x ∈ T^{4} : 1/3 ≤ |x_{i}|, |x_{j}| and |x_{k}| ≤ 1/3 for k ≠ i, j},
H_{i}^{3} = {x ∈ T^{4} : |x_{i}| ≤ 1/3 and 1/3 ≤ |x_{k}| for k ≠ i},
H^{4} = {x ∈ T^{4} : 1/3 ≤ |x_{k}|, k = 1, …, 4}.

We work throughout with PL maps. Therefore our constructed map in the end will be quasiregular. First we define a 2-to-1 branched cover f: T^{2} → S^{2} such that the square [0, 1]^{2} is mapped onto the upper half sphere S2_{+} of S^{2} and such that the extension to the rest of T^{2} satisfies f(x_{1}, x_{2}) = f(−x_{1}, −x_{2}) = r ∘ f(−x_{1}, x_{2}), where r is the reflection in the equator of S^{2} . The map f is shown in Figure 1, where points a_{i} and b_{i} = f(a_{i}), i = 1, …, 4, are indicated. Then set

(2.1)  F = f × f: T^{2} × T^{2} → S^{2} × S^{2}.

Let λ: T^{4} → T^{4} be the map (x_{1}, x_{2}, x_{3}, x_{4}) ↦ (x_{1}, x_{3}, x_{4}, x_{2}) and define

(2.2)  G: T^{4} → S^{2} × S^{2}

Figure 1.

to be F ∘ λ followed by the identity to another copy of S^{2} × S^{2} . The degree of F and G is clearly four.

Next let

\begin{aligned}X & = H^{0} ∪ H_{1}^{1} ∪ ⋯ ∪ H_{4}^{1} ∪ H_{12}^{2} ∪ H_{34}^{2}, \\ Y & = H^{4} ∪ H_{1}^{3} ∪ ⋯ ∪ H_{4}^{3} ∪ H_{13}^{2} ∪ H_{24}^{2}, \\ V & = X ∪ H_{23}^{2}, \\ W & = Y ∪ H_{14}^{2}. \end{aligned}

We first study F | X and F | V . For this we write

E = {x ∈ T^{2} : |x_{1}|, |x_{2}| ≤ 1/3}

and observe that fE is a disk with f∂E = ∂fE. We have

\begin{aligned}X & = (T^{2} × E) ∪ (E × T^{2}), \\ T^{4} \ X & = (T^{2} \ E) × (T^{2} \ E), \\ F X & = S^{2} × S^{2} \ U, \end{aligned}

where U is the open 4-ball (S^{2} \ fE) × (S^{2} \ fE) = F (T^{4} \ X). The boundary of X is the 3-manifold

∂X = ((T^{2} \ int E) × ∂E) ∪ (∂E × (T^{2} \ int E))

and

(2.3)  F ∂X = ((S^{2} \ int fE) × ∂fE) ∪ (∂fE × (S^{2} \ int fE)) = ∂U.

Hence F ∂X = ∂F X, which means that F | X is a closed map. Moreover, F | ∂X: ∂X → ∂U is a branched cover.

In (2.3) we see the 3-sphere ∂U splitted into two solid tori. The image of the branch set of F | ∂X (notice the difference of terminology in literature) is the union of the following six circles:

s_{i} = b_{i} × ∂fE, S_{i} = ∂fE × b_{i}, i = 1, 2, 4.
We write H_{23}^{2} = A_{2} × A_{1} , where

\begin{aligned}A_{1} & = \{x ∈ T^{2} : |x_{2}| ≤ 1/3 ≤ |x_{1}|\}, \\ A_{2} & = \{x ∈ T^{2} : |x_{1}| ≤ 1/3 ≤ |x_{2}|\}. \end{aligned}

Clearly ∂F H_{23}^{2} = F ∂H_{23}^{2} , so F | H_{23}^{2} is closed. Moreover, F H_{23}^{2} is a 4-ball contained in \overline{U} . The common boundary of X and H_{23}^{2} is

(2.4)  X ∩ H_{23}^{2} = (A_{2} × (E ∩ A_{1})) ∪ ((E ∩ A_{2}) × A_{1}),

which is a solid torus. The map F takes this onto the 3-ball V_{1} ∪ V_{2} ⊂ ∂U where V_{1} and V_{2} are the following closed 3-balls:

\begin{aligned}V_{1} & = fA_{2} × f(E ∩ A_{1}), \\ V_{2} & = f(E ∩ A_{2}) × fA_{1}. \end{aligned}

These have disjoint interior and

V_{1} ∩ V_{2} = f(E ∩ A_{2}) × f(E ∩ A_{1})

is a disk. The circle s_{2} has the arc l_{2}^{1} = b_{2} × f(E ∩ A_{1}) in V_{1} and the circle S_{4} the arc L_{4}^{2} = f(E ∩ A_{2}) × b_{4} in V_{2} .

We have

Y = (C_{2} × C_{2}) ∪ (C_{1} × C_{1}),

where

C_{i} = {x ∈ T^{2} : 1/3 ≤ |x_{i}|}, i = 1, 2.

The common boundary of Y and H_{23}^{2} is the solid torus

(2.5)  Y ∩ H_{23}^{2} = (A_{2} × (C_{2} ∩ A_{1})) ∪ ((C_{1} ∩ A_{2}) × A_{1}).

The map F takes this onto the 3-ball V_{3} ∪ V_{4} ⊂ \overline{U} , where

\begin{aligned}V_{3} & = fA_{2} × f(C_{2} ∩ A_{1}), \\ V_{4} & = f(C_{1} ∩ A_{2}) × fA_{1}. \end{aligned}

Figure 2. Link L.

The intersection is now the disk

V_{3} ∩ V_{4} = f(C_{1} ∩ A_{2}) × f(C_{2} ∩ A_{1}).

The boundary ∂H_{23}^{2} ≃ S^{3} is the union of the solid tori given in (2.4) and (2.5) with a common 2-torus. The map F takes this 2-torus onto the common 2-sphere of V_{1} ∪ V_{2} and V_{3} ∪ V_{4} .

We get F | ∂V by replacing in F | ∂X the part F | X ∩ H_{23}^{2} by F | Y ∩ H_{23}^{2} . In particular, F (X ∩ H_{23}^{2}) = V_{1} ∪ V_{2} will be replaced by F (Y ∩ H_{23}^{2}) = V_{3} ∪ V_{4} . We observe that F | V is a closed map and that F | ∂V is a branched cover onto a 3-sphere.

To get a picture of the link diagram of the image of the branch set of F | ∂V we observe the following. The image F (H_{23}^{2}) gives a cobordism between (V_{1} ∪ V_{2}, l_{2}^{1} ∪ L_{4}^{2}) and (V_{3} ∪ V_{4}, l_{2}^{3} ∪ L_{4}^{4}). This affects the link diagram for F | ∂X by changing the crossing between s_{2} and S_{4} so that the resulting link L for F | ∂V has the form given in Figure 2. There we have denoted the modified s_{2} and S_{4} by s̃_{2} and S̃_{4} respectively.

To study G | Y and its extension G | W we write

E′ = {x ∈ T^{2} : |x_{1}|, |x_{2}| ≥ 1/3}.

We have

GY = S^{2} × S^{2} \ U′,

where U′ is the open 4-ball (S^{2} \ fE′) × (S^{2} \ fE′) = G(T^{4} \ Y). Instead of (2.3) we have in an analogous way

G∂Y = ((S^{2} \ int fE′) × ∂fE′) ∪ (∂fE′ × (S^{2} \ int fE′)) = ∂U′.

The image of the branch set of G | ∂Y now consists of the circles

s′_{i} = b_{i} × ∂fE′, S′_{i} = ∂fE′ × b_{i}, i = 2, 3, 4.

A treatment similar to the one for F shows that the image of the branch set of G | ∂W is the link L′ given in Figure 3. There s̃′_{4} and S̃ ′_{4} are obtained from s′_{4} and S′_{4} when G(Y ∩H_{14}^{2}) is replaced by G(X ∩H_{14}^{2}). We denote the set ∂V = ∂W by M. The sets V and W induce opposite orientations on M, and F | V and G | W induce orientations on F M and GM.

Figure 3. Link L′.

3. Monodromies and change to simple maps

In order to relate F | M and G | M as explained in the introduction we will in this section present preparatory material in order to apply Piergallini’s paper [9].

To present the monodromy of F | M we choose q ∈ F M ≃ S^{3} for which F^{−1}(q) = {p_{1}, p_{2}, p_{3}, p_{4}}, where

\begin{aligned}p_{1} & = (−1/3, 0, 0, −1/3), \\ p_{2} & = (1/3, 0, 0, −1/3), \\ p_{3} & = (−1/3, 0, 0, 1/3), \\ p_{4} & = (1/3, 0, 0, 1/3). \end{aligned}

Let α_{i}, β_{i}, i = 1, 2, 4, be paths in F M with base point q shown in Figure 4. We observe that these paths stay in F (M ∩ ∂X) so that we see the various lifts by looking at F | ∂X. When we identify p_{j} with j in the presentation of elements of the symmetric group of degree four, the monodromy map takes these paths as follows:

α_{i} ↦ (12)(34), β_{i} ↦ (13)(24), i = 1, 2, 4.

Paths α_{i}, β_{i} represent a part of the Wirtinger generators of π_{1}(F M \ L) (see [14, p. 56]). By studying the behavior at the bridges of L we find that the other Wirtinger generators give permutations so that permutations stay constant

Figure 4. Link L.

for each circle of L. In Figure 4 we have indicated this by placing the corresponding permutations at each subarc and by using abbreviations x = (12)(34), y = (13)(24).

The monodromy of G | M is similar. This time we take q′ ∈ GM ≃ S^{3} for which G^{−1}(q′) = {p′_{1}, p′_{2}, p′_{3}, p′_{4}}, where

\begin{aligned}p′_{1} & = (−1/3, 1, 1, −1/3), \\ p′_{2} & = (1/3, 1, 1, −1/3), \\ p′_{3} & = (−1/3, 1, 1, 1/3), \\ p′_{4} & = (1/3, 1, 1, 1/3), \end{aligned}

and paths α′_{i}, β′_{i}, i = 2, 3, 4, in GM with base point q′ shown in Figure 5. Then we can repeat word by word for the link L′ what was said for the link L above. Without loss of generality we can therefore assume that F | V and G | W are maps onto two copies of S^{2} × S^{2} \ B^{4} with a common boundary S^{3} where L and L′ are identified together with their monodromies. Because of this latter property the boundary maps F | M and G | M are conjugated by a homeomorphism of M. We cannot use directly this fact because such a homeomorphism is not necessarily isotopic to identity. The solution to this problem is given in the next section by the technique presented in [9, Section 2]. The rest of this section is devoted to perform moves for the maps that result into so called normalized forms. Note that the induced orientations on S^{3} are opposite, hence F | M: M → S^{3} and G | M: M → S^{3} are orientation preserving because M is equipped with opposite orientations for F | M and G | M.

By isotopy on S^{3} we then move L together with monodromy labelling to the link presented in Figure 6. There we have decomposed S^{3} into two 3-balls B_{1}

Figure 5. Link L′.

and B_{2} and a ring R = S^{2} × [0, 1]. The monodromies stay in our case fixed on circles in the link during the isotopy. The part in R of the link is called a braid. We denote F | M and G | M followed by the isotopies described above by φ and ψ respectively.

By slightly perturbing maps φ and ψ in small tubes arround the preimages of the circles of the link in Figure 6 we change φ and ψ to simple maps, i.e., the inverse of a point consists of at least three points. The monodromy is then represented by transpositions. Each arc of the link in B_{1} and B_{2} is then replaced by two arcs. In Figure 7 we see the part in B_{1} of the new link together with the corresponding transpositions. There are simple rules for the behavior of the transpositions at each bridge of the braid. To see this we look at a bridge of the braid with transpositions u, v, w as in Figure 8. Then u and v determine w by the following rules. If u and v have no common index or if u = v, then w = v. If u = (ij) and v = (kj) with k ≠ i, then w = (ik).

Next we perform isotopies on S^{3} in order to rearrange the braid into a so called normalized form. First we move each pair of arcs in B_{1} and B_{2} coming from a single circle in Figure 6 as shown in Figure 9 for the first pair from left in B_{1} . Then, by successive use of the isotopies presented in [9, Figure 8, p. 910] in a way that only three indices are taken at a time, we obtain a new braid diagram where the parts B_{1} and B_{2} are shown in Figure 10. This we call a normalized form. We call the new φ and ψ by ξ and η respectively.

Figure 6.

Figure 7.

4. Stable equivalence and its realization

An easy argument shows that the preimages

\begin{aligned}H_{1} & = ξ^{−1}B_{1}, H_{2} = ξ^{−1}B_{2} \\ H′_{1} & = η^{−1}B_{1}, H′_{2} = η^{−1}B_{2} \end{aligned}

are handlebodies. Furthermore, P = ξ^{−1}R and P′ = η^{−1}R have product presentations

\begin{aligned}P & = ∂H_{1} × [0, 1], \\ P′ & = ∂H′_{1} × [0, 1], \end{aligned}

such that ξ^{−1}(S^{2} × t) and η^{−1}(S^{2} × t) are identified with ∂H_{1} × t and ∂H′_{1} × t respectively. Maps ξ and η define Heegard splittings (H_{1}, P ∪ H_{2}) and (H′_{1}, P′∪ H′_{2}) of M with same genus g.

Figure 8.

Next we fix a handlebody T_{g} ⊂ R^{3} of genus g with boundary surface F_{g} and a standard map s with degree four of T_{g} onto B_{1} (with B_{1} deformed slightly) as shown in Figure 11. Without loss of generality we may assume that there are homeomorphisms θ_{i}: H_{i} → T_{g} and θ′_{i}: H′_{i} → T_{g}, i = 1, 2, such that ξ | H_{1} = s∘θ_{1} , η | H′_{1} = s ∘ θ′_{1} , ξ | H_{2} = ϱ ∘ s ∘ θ_{2} , η | H′_{2} = ϱ ∘ s ∘ θ′_{2} , where ϱ: B_{1} → B_{2} is the obvious homeomorphism. In [9, Figure 4, p. 906] a map similar to s is shown for degree three.

Figure 9.

The braid in Figure 10 can be realized by an isotopy on ∂B_{1} . Such an isotopy gives a homeomorphism h: F_{g} → F_{g} through lifting by s.

Let W be a closed oriented 3-manifold and (W_{1}, W_{2}) a Heegard splitting of W. Let U ⊂ W_{2} be a disk such that ∂U consists of two arcs Γ_{1} ⊂ ∂W_{1} and Γ_{2} ⊂ W_{2} . We form a new Heegard splitting of W by adding to W_{1} a closed tubular neighborhood V of Γ_{2} so that (W_{1} ∪ V, \overline{W_{2} \ V}) is a Heegard splitting of one genus higher. A Heegard splitting (Z_{1}, Z_{2}) is obtained from (W_{1}, W_{2}) by stabilization if it is obtained by a finite number of steps as above.

Given two Heegard splittings (W_{1}, W_{2}) and (W′_{1}, W′_{2}) of W, then they are stably equivalent, that is, there exist stabilizations of these to splittings (Z_{1}, Z_{2}) and (Z′_{1}, Z′_{2}) of same genus and a homeomorphism ζ: W → W isotopic to identity such that ζZ_{i} = Z′_{i}, i = 1, 2. This result is known as the stabilization theorem by Reidemeister [11] and Singer [16]. See the paper [19] by Waldhausen. Later proofs are given by Craggs [2] and by Lei [6].

Figure 10. Normalized form.

We apply the stabilization theorem to Heegard splittings (H_{1}, P ∪ H_{2}) and (H′_{1}, P′ ∪ H′_{2}). We may assume that in the first step of the stabilization process the new map corresponding to s takes the form obtained by adding one (12)-part to the braid in Figure 10 as shown in Figure 12. For this, see [9, Figure 9] and [15, 3.1]. We continue similarly. We retain the original notation for the stabilized case, like H_{i}, H′_{i}, g, s etc. The map h: F_{g} → F_{g} is now a lift with respect to s | F_{g}: F_{g} → ∂B_{1} of some isotopy realizing the braid diagram D for the stabilized case corresponding to the braid diagram in Figure 10. So we have an isotopy ζ^{t}: M → M, t ∈ [0, 1], with ζ^{0} = id, ζ^{1}H_{1} = H′_{1} , ζ^{1}(P ∪ H_{2}) = P′ ∪ H′_{2} . We may also assume ζ^{1}H_{2} = H′_{2} and ζ^{1}P = P′.

Next we transfer ζ^{1} to T_{g} via our maps θ_{i}: H_{i} → T_{g}, θ′_{i}: H′_{i} → T_{g}, and set ζ_{i} = θ′_{i} ∘ ζ^{1} ∘ θ_{i}^{−1} , i = 1, 2. Then we have the following diagram:

This corresponds to the diagram in [9, p. 910] where instead of one map h one has two different maps. We emphasize here that our map h in both places in the

Figure 11.

above diagram is the result of lifting the same braid diagram D by s.

Our final task is to find moves that correct the difference presented by homeomorphisms ζ_{1} and ζ_{2} . The idea is to replace η by another map η̃ through adding braids D_{1} and D_{2} to top and bottom of D that lift by s and ϱ ∘ s to homeomorphisms ζ_{1} and ζ_{2} . Suppose we have done this. Let λ_{1}^{t} and λ_{2}^{t}, t ∈ [0, 1], be isotopies on B_{1} and B_{2} realizing the braids D_{1} and D_{2} on ∂B_{1} and ∂B_{2} . We use the isotopies ζ^{t}, λ_{1}^{t}, and λ_{2}^{t} to define a level preserving branched covering σ: M × [0, 1] → S^{3} × [0, 1] of degree four such that σ^{0} = ξ, σ^{1} = η̃ with the notation σ^{t}(x) = pr_{1}(σ(x, t)). In particular,

\begin{aligned}σ^{t} | ζ^{t}H_{1} & = λ_{1}^{t} ∘ s ∘ θ_{1} ∘ (ζ^{t})^{−1} | ζ^{t}H_{1}, \\ σ^{t} | ζ^{t}H_{2} & = λ_{2}^{t} ∘ ϱ ∘ s ∘ θ_{2} ∘ (ζ^{t})^{−1} | ζ^{t}H_{2}, \end{aligned}

t ∈ [0, 1]. To obtain the braids D_{1} and D_{2} we apply [9, Section 2] in a slightly modified form.

Following the notation in [9] let M^{∗}(g) be the subgroup of the mapping class group M(g) of F_{g} whose elements extend to T_{g}. The isotopy classes of ζ_{1} and ζ_{2} belong to M^{∗}(g). The task is to find moves that result in adding braids that generate, through lifting by s, homeomorphisms of F_{g} which give generators for M^{∗}(g). Generators for M^{∗}(g) are given in [17] and listed in [9, p. 912]. The

Figure 12.

Figure 13. Move I.

solution for the case of degree three is given in [9, pp. 912–916]. By looking at the steps given there, we see that we can in our case be limited to the restriction of s to the right of the dashed lines in Figure 11. The reason is that the braid generator involving the transposition (14) is not needed and that the parts to the left of the dashed lines remain unchanged under braids that we need. We also observe that the action on the 4th sheet is trivial for our needed braids: They lead to compositions of disk twists in that sheet. According to [9] our conclusion therefore is that we need to use only isotopy on S^{3} with transposition labelling (usually called colored isotopy in the case of degree three) and move I (Figure 13) according to the terminology in [9], see third paragraph on p. 912 in [9]. Details how move I is sitting in our map w is presented in [7]. Note that move I is called move C^{±} in [7] and [10]. Observe that move I is local and is restricted to a part where the degree is three. This finishes the proof of Theorem 1.

110 Seppo Rickman

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Received 19 November 2004