STABLE MAPPING CLASS GROUPS OF 4-MANIFOLDS WITH BOUNDARY

OSAMU SAEKI

Abstract. We give a complete algebraic description of the mapping class groups of compact simply connected 4-manifolds with boundary up to connected sum with copies of S^{2} × S^{2}.

1. Introduction

It has been shown in [15] that two orientation preserving homeomorphisms of a closed simply connected 4-manifold are homotopic if and only if the induced automorphisms on the second homology group coincide with each other (see also [4, §5]). Using this, Quinn [15] has shown that two homeomorphisms inducing the same automorphism on the second homology group are actually topologically isotopic (see also [13]). Quinn has also shown that two diffeomorphisms inducing the same automorphism are smoothly isotopic after the connected sum with the identity diffeomorphism of some copies of S^{2} × S^{2}. In this paper we give a similar smooth stable isotopy criterion for the case where the simply connected 4-manifold has nonempty connected boundary.

Let M be a compact 1-connected 4-manifold with nonempty connected boundary. In this paper we first give an algebraic criterion for two diffeomorphisms of M to be stably isotopic (see §2), where two diffeomorphisms h_{0} and h_{1} of M which are the identity on the boundary are stably isotopic relative to boundary if h_{0}♯k(id) and h_{1}♯k(id) are smoothly isotopic relative to boundary as diffeomorphisms of M♯k(S^{2}× S^{2}) for some k ≥ 0. We will see that certain homomorphisms H_{2}(M, ∂M; Z) → H_{2}(M; Z), called variation maps, associated with such diffeomorphisms play an essential role.

Then in §3, we introduce the notion of the stable mapping class group of M and give its complete algebraic description using the result of §2. The algabraic objects that play important role here are homomorphisms, called variational homomorphisms, H_{2}(M, ∂M; Z) → H_{2}(M; Z) which are defined to be abstract homomorphisms satisfying certain algebraic conditions. We will see that the set of variational homomorphisms form a group with respect to a certain multiplication. The main content of §3 is a construction of a diffeomorphism which realizes a given variational homomorphism.

In §4, we study the relationship between the group of variational homomorphisms and the group of isometries of H_{2}(M) endowed with the intersection form, which will be used in §5.

In §5, we consider the stable mapping class group of boundary free diffeomorphims of 4-manifolds whose boundaries are certain spherical 3-manifolds.

The author has been supported in part by Grant-in-Aid for Scientific Research (No. 11440022), Ministry of Education, Science and Culture, Japan.

2000 Mathematics Subject Classification. Primary 57N13; Secondary 57R52, 57N37.

Key words: isotopy, 4-manifold with boundary, stable mapping class group, variation map, isometry.

Throughout the paper, we work in the smooth category unless otherwise specified. All the homology and cohomology groups are with integer coefficients unless otherwise indicated. We use the symbol “≅” to denote a diffeomorphism between smooth manifolds or an appropriate isomorphism between algebraic objects.

The author would like to thank the referee for drawing his attention to Kreck’s paper [10]. He also would like to thank the people at IRMA, Strasbourg, for their hospitality during the preparation of the manuscript.

2. Isotopy of 1-connected 4-manifolds with boundary

Let M be a compact 1-connected 4-manifold with nonempty connected boundary. We first define the variation map associated with a diffeomorphism of M, which will play an important role throughout the paper.

Definition 2.1. Let h : M → M be a diffeomorphism which is the identity on the boundary. We define the variation map Δ_{h} : H_{2}(M, ∂M) → H_{2}(M) as follows. For a homology class γ ∈ H_{2}(M, ∂M), take a 2-cycle (D, ∂D) in (M, ∂M) representing γ. Then D ∪ (−h(D)) is a 2-cycle in M and we define Δ_{h}(γ) to be the class represented by D ∪ (−h(D)). Note that this does not depend on the choice of (D, ∂D) and that Δ_{h} is a homomorphism. See also [8].

Note that if two diffeomorphisms of M which are the identity on the boundary are isotopic relative to boundary, then their variation maps coincide with each other. The main result of this section is that the converse is also true “stably” as follows.

Theorem 2.2. Let M be a compact 1-connected 4-manifold with nonempty connected boundary. Suppose that h_{0} and h_{1} : M → M are two diffeomorphisms with h_{0}|_{∂M} = h_{1}|_{∂M} being the identity map. Then h_{0} and h_{1} are stably isotopic relative to boundary if and only if Δ_{h_{0}} = Δ_{h_{1}} : H_{2}(M, ∂M) → H_{2}(M).

Proof. In the following, for an integer k ≥ 0, M_{k} will denote the 4-manifold M♯k(S^{2} × S^{2}).

If h_{0} and h_{1} are stably isotopic relative to boundary, then it is easy to see that their variation maps coincide with each other, since on the direct summand of H_{2}(M_{k}, ∂M_{k}) corresponding to H_{2}(♯^{k}S^{2} × S^{2}), the variation maps of h_{j}♯k(id), j = 0, 1, are the zero homomorphisms.

In order to prove the converse, let us consider the following construction, which is called an open book construction (see [8]). For a diffeomorphism h : M → M which is the identity on the boundary, let L_{h} be the 5-dimensional manifold with boundary obtained from M × [0, 1] by identifying M × {1} with M × {0} using h : M × {1} → M × {0}. Note that ∂L_{h} is canonically diffeomorphic to ∂M × S^{1}. Then, let N_{h} be the closed 5-dimensional manifold obtained by attaching ∂M × D^{2} to L_{h} along the boundary. Note that N_{h} is 1-connected and that M is naturally identified with M × {0} ⊂ N_{h}.

Then, by an argument similar to that in [9, §2], we easily obtain the following.

Lemma 2.3. A diffeomorphism h : M → M which is the identity on the boundary is pseudo-isotopic to the identity relative to boundary if and only if N_{h} bounds a compact 1-connected 6-dimensional manifold W such that π_{j}(W, M) = 0 for all j.

We also have the following.

Lemma 2.4. Let h : M → M be a diffeomorphism which is the identity on the boundary. Then we have the following exact sequence.

0 → H_{3}(N_{h}) → H_{2}(M, ∂M) ⟶^{Δ_{h}} H_{2}(M) → H_{2}(N_{h}) → 0.

Proof. Consider the exact sequence for the pair (N_{h}, M):

0 → H_{3}(N_{h}) → H_{3}(N_{h}, M) ⟶^{∂} H_{2}(M) → H_{2}(N_{h}) → H_{2}(N_{h}, M). (2.1)

Let M̃ ≅ M × [0, 1] be a normal 1-disk bundle neighborhood of M in N_{h}. Note that the closure of N_{h} ∖ M̃ is again diffeomorphic to M × [0, 1]. Hence, by excision together with Poincaré-Lefschetz duality, we have

\begin{aligned}H_{i}(N_{h}, M) & ≅ H_{i}(N_{h}, M̃) ≅ H_{i}(M × [0, 1], ∂(M × [0, 1])) \\ & ≅ H^{5−i}(M × [0, 1]) ≅ H^{5−i}(M) ≅ H_{i−1}(M, ∂M). \end{aligned}

In particular, we have H_{2}(N_{h}, M) ≅ H_{1}(M, ∂M) = 0, since M is simply connected and ∂M is connected. Furthermore, under the isomorphism H_{3}(N_{h}, M) ≅ H_{2}(M, ∂M) given above, the boundary homomorphism ∂ : H_{3}(N_{h}, M) → H_{2}(M) appearing in (2.1) is identified with the variation map Δ_{h} : H_{2}(M, ∂M) → H_{2}(M). This completes the proof of Lemma 2.4.

Let us go back to the proof of Theorem 2.2. Let h_{0} and h_{1} : M → M be diffeomorphisms which are the identity on the boundary such that Δ_{h_{0}} = Δ_{h_{1}}. Set h = h_{0} ∘ (h_{1})^{−1}. Then it is easy to show that Δ_{h} is the zero homomorphism (for example, see [16, §9]). Then by following the same argument as in [9, §2] and by replacing the Wang exact sequence for a bundle over S^{1} with that of Lemma 2.4, we can construct a 6-dimensional manifold W as in Lemma 2.3. Therefore, h is pseudo-isotopic to the identity relative to boundary (see also the last paragraph of [10]).

Thus h_{0} and h_{1} are pseudo-isotopic relative to boundary. Then by Quinn [15], they are stably isotopic relative to boundary. This completes the proof of Theorem 2.2.

Compare Theorem 2.2 with [16, Conjecture 9.6].

Remark 2.5. In the above situation, we can prove that h_{0}♯k(id) and h_{1}♯k(id) are pseudo-isotopic relative to boundary for some k ≥ 0 also by using an argument based on the theory of open books on 5-dimensional manifolds [16, 17].

3. Stable mapping class group and variational homomorphisms

In this section, we define the stable mapping class group of a compact 4-manifold with boundary and give a result which describes it algebraically.

Definition 3.1. Let M be a compact 4-manifold with boundary. A stable diffeomorphism of M is a diffeomorphism of M_{k} = M♯k(S^{2} × S^{2}) for some k. Two stable diffeomorphisms h_{0} : M_{k_{0}} → M_{k_{0}} and h_{1} : M_{k_{1}} → M_{k_{1}} which are the identity on the boundary are said to be stably isotopic relative to boundary if h_{0}♯k′_{0}(id) : M_{k_{0}+k′_{0}} → M_{k_{0}+k′_{0}} and h_{1}♯k′_{1}(id) : M_{k_{1}+k′_{1}} → M_{k_{1}+k′_{1}} are isotopic relative to boundary for some k′_{0} and k′_{1} with k_{0} + k′_{0} = k_{1} + k′_{1}. Finally, the set of all stable isotopy classes relative to boundary of stable diffeomorphisms of M which are the identity on the boundary is called the stable mapping class group of (M, ∂M) and is denoted by SM(M, ∂M). Note that if we denote by M(M, ∂M) the usual mapping class group relative to boundary, then we can naturally identify SM(M, ∂M) with the inductive limit

lim_{k→∞} M(M_{k}, ∂M_{k}).

This obviously forms a group with respect to the composition of maps.

Let M be as above and h : M_{k} → M_{k} a stable diffeomorphism of M which is the identity on the boundary. Let us denote the variation map of h by Δ_{h} : H_{2}(M_{k}, ∂M_{k}) → H_{2}(M_{k}). We also denote the induced map h_{∗} : H_{2}(M_{k}) → H_{2}(M_{k}) by Λ_{h}. Since the intersection form

H_{2}(M_{k}) × H_{2}(M_{k}, ∂M_{k}) → Z (3.1)

of M_{k} is unimodular, we may regard H_{2}(M_{k}) and H_{2}(M_{k}, ∂M_{k}) dual to each other. For a homomorphism ρ involving these spaces, ρ^{∗} will denote its dual homomorphism. Let j : M_{k} → (M_{k}, ∂M_{k}) denote the inclusion map.

In order to give an algebraic description of the stable mapping class group, let us prepare some lemmas.

Lemma 3.2. We have the following.

(1) (j_{∗})^{∗} = j_{∗} : H_{2}(M_{k}) → H_{2}(M_{k}, ∂M_{k}).

(2) Λ_{h} = id − Δ_{h} ∘ j_{∗} : H_{2}(M_{k}) → H_{2}(M_{k}).

(3) j_{∗} = Λ_{h}^{∗} ∘ j_{∗} ∘ Λ_{h} : H_{2}(M_{k}) → H_{2}(M_{k}).

(4) Δ_{h} + Δ_{h}^{∗} = Δ_{h} ∘ j_{∗} ∘ Δ_{h}^{∗} : H_{2}(M_{k}, ∂M_{k}) → H_{2}(M_{k}).

(5) For two stable diffeomorphisms h and h′ : M_{k} → M_{k} which are the identity on the boundary, we have

Δ_{h∘h′} = Δ_{h} + Λ_{h} ∘ Δ_{h′} : H_{2}(M_{k}, ∂M_{k}) → H_{2}(M_{k}).

Proof. (1) This follows from the fact that α · j_{∗}β = β · j_{∗}α for all α, β ∈ H_{2}(M_{k}), where “·” denotes the intersection form (3.1).

(2) For each α ∈ H_{2}(M_{k}), we have

(id − Δ_{h} ∘ j_{∗})(α) = α − (α − h_{∗}α) = h_{∗}α = Λ_{h}α.

Thus the result follows.

(3) This follows from the fact that h_{∗} : H_{2}(M_{k}) → H_{2}(M_{k}) is an isometry of (H_{2}(M_{k}), ·), where “·” denotes the intersection form of M_{k} on H_{2}(M_{k}).

(4) We have

Δ_{h} + Δ_{h}^{∗} = Δ_{h} ∘ j_{∗} ∘ Δ_{h}^{∗} ⟺ (id − Δ_{h} ∘ j_{∗}) ∘ Δ_{h}^{∗} = −Δ_{h}
(3.2)  ⟺ Λ_{h} ∘ Δ_{h}^{∗} = −Δ_{h}
(3.3)  ⟺ Δ_{h}^{∗} = −Λ_{h}^{−1} ∘ Δ_{h},

where (3.2) follows from (2). This is equivalent to that

Δ_{h}β · α = −(Λ_{h}^{−1} ∘ Δ_{h}α) · β (3.4)

for all α, β ∈ H_{2}(M_{k}, ∂M_{k}). Let a and b be 2-cycles in (M_{k}, ∂M_{k}) representing α and β respectively. Then the right hand side of the above equation is equal to

−h^{−1}(a − ha) · b = (a − h^{−1}a) · b
= a · b − h^{−1}a · b
= b · a − hb · a
= (b − hb) · a.

Thus we have (3.4) and have proved the required equality.

(5) For every 2-cycle a of (M_{k}, ∂M_{k}), we have

a − (h ∘ h′)a = a − ha + ha − h(h′a) = a − ha + h(a − h′a).

Thus we have

Δ_{h∘h′} = Δ_{h} + Λ_{h} ∘ Δ_{h′}.

This completes the proof.

Definition 3.3. Let M be a compact 4-manifold with boundary. A homomorphism Δ : H_{2}(M, ∂M) → H_{2}(M) is variational if Δ + Δ^{∗} = Δ ∘ j_{∗} ∘ Δ^{∗}. Let V(M, ∂M) denote the set of all variational homomorphisms.

Lemma 3.4. Let Δ : H_{2}(M, ∂M) → H_{2}(M) be a variational homomorphism. Then Λ = id − Δ ∘ j_{∗} is an isometry of the inner product space (H_{2}(M), ·), where “ · ” denotes the intersection form of M on H_{2}(M).

Proof. Since Δ+Δ^{∗} = Δ∘j_{∗} ∘Δ^{∗} holds by our assumption, we have j_{∗} ∘(Δ+Δ^{∗}) = j_{∗} ∘ Δ ∘ j_{∗} ∘ Δ^{∗}. This is equivalent to the equality

(id − j_{∗} ∘ Δ) ∘ (id − j_{∗} ∘ Δ^{∗}) = id : H_{2}(M) → H_{2}(M).

Since H_{2}(M) is free, we see that id − j_{∗} ∘ Δ and Λ^{∗} = id − j_{∗} ∘ Δ^{∗} are isomorphisms which are inverses to each other. Thus Λ is an isomorphism and

(id − j_{∗} ∘ Δ^{∗}) ∘ (id − j_{∗} ∘ Δ) = id

also holds. In particular, we have

(id − j_{∗} ∘ Δ^{∗}) ∘ (id − j_{∗} ∘ Δ) ∘ j_{∗} = j_{∗},

which is equivalent to Λ^{∗} ∘ j_{∗} ∘ Λ = j_{∗}. Thus Λ is an isometry. This completes the proof.

Lemma 3.5. The set V(M, ∂M) of all variational homomorphisms forms a group under the multiplication given by

Δ_{1} ∗ Δ_{2} = Δ_{1} + (id − Δ_{1} ∘ j_{∗}) ∘ Δ_{2},

for Δ_{1}, Δ_{2} ∈ V(M, ∂M).

Proof. Put V = V(M, ∂M) for simplicity. Let us first show that Δ_{1} ∗ Δ_{2} ∈ V. Putting Λ_{i} = id − Δ_{i} ∘ j_{∗}, i = 1, 2, we have

(Δ_{1} ∗ Δ_{2}) + (Δ_{1} ∗ Δ_{2})^{∗} = (Δ_{1} + Λ_{1} ∘ Δ_{2}) + (Δ_{1} + Λ_{1} ∘ Δ_{2})^{∗}
(3.5)  = Δ_{1} + Δ_{1}^{∗} + Λ_{1} ∘ Δ_{2} + (Λ_{1} ∘ Δ_{2})^{∗}.

On the other hand, we have (Δ_{1} ∗ Δ_{2}) ∘ j_{∗} ∘ (Δ_{1} ∗ Δ_{2})^{∗}

= (Δ_{1} + Λ_{1} ∘ Δ_{2}) ∘ j_{∗} ∘ (Δ_{1} + Λ_{1} ∘ Δ_{2})^{∗}

= Δ_{1} ∘ j_{∗} ∘ Δ_{1}^{∗} + Δ_{1} ∘ j_{∗} ∘ Δ_{2}^{∗} ∘ Λ_{1}^{∗} + Λ_{1} ∘ Δ_{2} ∘ j_{∗} ∘ Δ_{1}^{∗} + Λ_{1} ∘ Δ_{2} ∘ j_{∗} ∘ Δ_{2}^{∗} ∘ Λ_{1}^{∗}

= (Δ_{1} + Δ_{1}^{∗}) + Δ_{1} ∘ j_{∗} ∘ Δ_{2}^{∗} ∘ Λ_{1}^{∗} + Λ_{1} ∘ Δ_{2} ∘ j_{∗} ∘ Δ_{1}^{∗} + Λ_{1} ∘ (Δ_{2} + Δ_{2}^{∗}) ∘ Λ_{1}^{∗}

= (Δ_{1} + Δ_{1}^{∗}) + (Δ_{1} ∘ j_{∗} + Λ_{1}) ∘ Δ_{2}^{∗} ∘ Λ_{1}^{∗} + Λ_{1} ∘ Δ_{2} ∘ (j_{∗} ∘ Δ_{1}^{∗} + Λ_{1}^{∗})

= (Δ_{1} + Δ_{1}^{∗}) + Δ_{2}^{∗} ∘ Λ_{1}^{∗} + Λ_{1} ∘ Δ_{2}.

Thus by (3.5), we see that Δ_{1} ∗ Δ_{2} ∈ V.

The associativity holds, since we have

id − (Δ_{1} ∗ Δ_{2}) ∘ j_{∗} = id − (Δ_{1} + (id − Δ_{1} ∘ j_{∗}) ∘ Δ_{2}) ∘ j_{∗}
(3.6)  = (id − Δ_{1} ∘ j_{∗}) ∘ (id − Δ_{2} ∘ j_{∗}).

and hence

(Δ_{1} ∗ Δ_{2}) ∗ Δ_{3} = (Δ_{1} + Λ_{1} ∘ Δ_{2}) + (Λ_{1} ∘ Λ_{2}) ∘ Δ_{3}
= Δ_{1} + Λ_{1} ∘ (Δ_{2} + Λ_{2} ∘ Δ_{3})
= Δ_{1} ∗ (Δ_{2} ∗ Δ_{3}).

It is obvious that the zero homomorphism is variational and is the identity element.

Finally, for Δ ∈ V, its inverse Δ^{−1} is given by

Δ^{−1} = −Λ^{−1} ∘ Δ,

where Λ = id − Δ ∘ j_{∗} is an isometry of (H_{2}(M), ·) by Lemma 3.4. This is seen as follows. We have

(−Λ^{−1} ∘ Δ) ∘ j_{∗} ∘ (−Λ^{−1} ∘ Δ)^{∗} = Λ^{−1} ∘ Δ ∘ j_{∗} ∘ Δ^{∗} ∘ (Λ^{−1})^{∗}
= Λ^{−1} ∘ (Δ + Δ^{∗}) ∘ (Λ^{−1})^{∗}
= Λ^{−1} ∘ Δ ∘ (Λ^{−1})^{∗} + Λ^{−1} ∘ Δ^{∗} ∘ (Λ^{−1})^{∗}
= −Δ^{∗} ∘ (Λ^{−1})^{∗} − Λ^{−1} ∘ Δ
= (−Λ^{−1} ∘ Δ) + (−Λ^{−1} ∘ Δ)^{∗}

by an argument similar to (3.3). Thus we have Δ^{−1} = −Λ^{−1} ∘Δ ∈ V. Furthermore, since

id − (−Λ^{−1} ∘ Δ) ∘ j_{∗} = Λ^{−1} ∘ (Λ + Δ ∘ j_{∗}) = Λ^{−1},

we have

Δ ∗ (−Λ^{−1} ∘ Δ) = Δ + Λ ∘ (−Λ^{−1} ∘ Δ) = 0

and

(−Λ^{−1} ∘ Δ) ∗ Δ = −Λ^{−1} ∘ Δ + Λ^{−1} ∘ Δ = 0.

Thus V forms a group. This completes the proof.

Definition 3.6. A stable variational homomorphism of M is a variational homomorphism Δ : H_{2}(M_{k}, ∂M_{k}) → H_{2}(M_{k}) for some k ≥ 0. Two stable variational homomorphisms Δ_{0} : H_{2}(M_{k_{0}}, ∂M_{k_{0}}) → H_{2}(M_{k_{0}}) and Δ_{1} : H_{2}(M_{k_{1}}, ∂M_{k_{1}}) → H_{2}(M_{k_{1}}) of M are said to be stably equivalent if

\begin{aligned}Δ_{0} ⊕ 0_{k′_{0}} : H_{2}(M_{k_{0}+k′_{0}}, ∂M_{k_{0}+k′_{0}}) & ≅ H_{2}(M_{k_{0}}, ∂M_{k_{0}}) ⊕ H_{2}(♯^{k′_{0}}(S^{2} × S^{2})) \\ → H_{2}(M_{k_{0}+k′_{0}}) & ≅ H_{2}(M_{k_{0}}) ⊕ H_{2}(♯^{k′_{0}}(S^{2} × S^{2})) \end{aligned}

coincides with

\begin{aligned}Δ_{1} ⊕ 0_{k′_{1}} : H_{2}(M_{k_{1}+k′_{1}}, ∂M_{k_{1}+k′_{1}}) & ≅ H_{2}(M_{k_{1}}, ∂M_{k_{1}}) ⊕ H_{2}(♯^{k′_{1}}(S^{2} × S^{2})) \\ → H_{2}(M_{k_{1}+k′_{1}}) & ≅ H_{2}(M_{k_{1}}) ⊕ H_{2}(♯^{k′_{1}}(S^{2} × S^{2})) \end{aligned}

for some k′_{0} and k′_{1} with k_{0} + k′_{0} = k_{1} + k′_{1}, where

0_{k′} : H_{2}(♯^{k′}(S^{2} × S^{2})) → H_{2}(♯^{k′}(S^{2} × S^{2}))

stands for the zero map. The set of all equivalence classes of stable variational homomorphisms H_{2}(M_{k}, ∂M_{k}) → H_{2}(M_{k}), k ≥ 0, of M is called the stable variational group and is denoted by SV(M, ∂M). It is not difficult to see that SV(M, ∂M) naturally forms a group, which is nothing but the inductive limit

lim_{k→∞} V(M_{k}, ∂M_{k}),

where we may naturally regard V(M_{k}, ∂M_{k}) ⊂ V(M_{k+1}, ∂M_{k+1}) for each k ≥ 0.

We have a natural map

Θ : SM(M, ∂M) → SV(M, ∂M),

which maps each class of a stable diffeomorphism to the class of its variation map. This is a homomorphism of groups by Lemma 3.2 (5).

Theorem 3.7. If M is a smooth compact 1-connected 4-manifold with nonempty connected boundary, then the above correspondence Θ gives an isomorphism of groups.

Proof. The well-definedness and the injectivity follow from Theorem 2.2.

Let Δ : H_{2}(M_{k}, ∂M_{k}) → H_{2}(M_{k}) be a stable variational homomorphism of M. Note that

Λ = id − Δ ∘ j_{∗} : H_{2}(M_{k}) → H_{2}(M_{k})

is an isometry of (H_{2}(M_{k}), ·) by Lemma 3.4.

The following lemma has been proved implicitly in the proof of Lemma 3.2 (see (3.4)).

Lemma 3.8. Δβ · α = −(Λ^{−1} ∘ Δα) · β for all α, β ∈ H_{2}(M_{k}, ∂M_{k}).

Let V = M^{0} ∪_{id} (−M^{1}) with M^{0} = M^{1} = M_{k} be the double of M_{k}, which is a smooth closed 1-connected 4-manifold. Furthermore, let δ : H_{2}(M_{k}, ∂M_{k}) → H_{2}(V) be the doubling homomorphism, which is given by

δ([a]) = [i_{0}a − i_{1}a],

where a is an arbitrary 2-cycle of (M_{k}, ∂M_{k}) and i_{0} : M^{0} → V and i_{1} : M^{1} → V are the inclusion maps. Then put

J′ = im(δ + i_{1∗} ∘ Δ : H_{2}(M_{k}, ∂M_{k}) → H_{2}(V)).

Furthermore, let G(−Λ) denote the subgroup of H_{2}(M^{0})⊕H_{2}(M^{1}) consisting of the elements of the form (α, −Λα), α ∈ H_{2}(M^{0}), where we regard Λ as an isomorphism from H_{2}(M^{0}) to H_{2}(M^{1}). Let ι : H_{2}(M^{0}) ⊕ H_{2}(M^{1}) → H_{2}(V) be the natural homomorphism induced by the inclusions. Then we have the following.

Lemma 3.9. ιG(−Λ) ⊂ J′.

Proof. For every α ∈ H_{2}(M_{k}), we have

(δ + i_{1∗} ∘ Δ)(j_{∗}α) = (i_{0∗}α − i_{1∗}α) + i_{1∗} ∘ Δ ∘ j_{∗}α
= (i_{0∗}α − i_{1∗}α) + i_{1∗}(α − Λα)
= i_{0∗}α − i_{1∗} ∘ Λα.

This shows our required inclusion.

Lemma 3.10. The subgroup J′ of H_{2}(V) is isotropic.

Proof. For every α, β ∈ H_{2}(M_{k}, ∂M_{k}), we have

(δ + i_{1∗} ∘ Δ)(α) · (δ + i_{1∗} ∘ Δ)(β)
= δα · δβ + δα · (i_{1∗} ∘ Δβ) + (i_{1∗} ∘ Δα) · δβ + (i_{1∗} ∘ Δα) · (i_{1∗} ∘ Δβ)
= Δβ · α + Δα · β − Δα · (j_{∗} ∘ Δβ)
= (Δβ · α + Δ^{∗}β · α) − (Δ^{∗} ∘ j_{∗} ∘ Δβ) · α = 0,

since δα · δβ = 0 and Δ is variational. This completes the proof.

Consider the Mayer-Vietoris exact sequence for the pair (M^{0}, M^{1}):

0 ⟶ H_{2}(∂M_{k}) ⟶ H_{2}(M^{0}) ⊕ H_{2}(M^{1}) ⟶^{ι} H_{2}(V) ⟶^{∂′} H_{1}(∂M_{k}) ⟶ 0.

Lemma 3.11. We have ∂′J′ = H_{1}(∂M_{k}).

Proof. For every α ∈ H_{2}(M_{k}, ∂M_{k}), we have

∂′ ∘ (δ + i_{1∗} ∘ Δ)α = ∂′ ∘ δα = ∂α,

where ∂ : H_{2}(M_{k}, ∂M_{k}) → H_{1}(∂M_{k}) is the boundary homomorphism in the exact sequence of the pair (M_{k}, ∂M_{k}):

0 ⟶ H_{2}(∂M_{k}) ⟶ H_{2}(M_{k}) ⟶ H_{2}(M_{k}, ∂M_{k}) ⟶^{∂} H_{1}(∂M_{k}) ⟶ 0.

Since ∂ is surjective, we have the conclusion.

Let J be a maximal isotropic subgroup of H_{2}(V) containing J′. Then by [3, (4.2)] (see also [18]), there exists a smooth h-cobordism W relative to boundary between M^{0} and M^{1} such that

ker(κ_{∗} : H_{2}(V) → H_{2}(W)) = J,

where κ : V → W is the inclusion map. Let η : M^{0} → M^{1} be the homotopy equivalence relative to boundary induced by the h-cobordism W. We can define the variation map Δ_{η} : H_{2}(M^{0}, ∂M^{0}) → H_{2}(M^{1}) of η as before.

Lemma 3.12. The variation map Δ_{η} coincides with Δ.

Proof. Since J′ ⊂ ker κ_{∗}, we have

−κ_{∗} ∘ δ = κ_{∗} ∘ i_{1∗} ∘ Δ : H_{2}(M^{0}, ∂M^{0}) → H_{2}(W).

Let a be an arbitrary 2-cycle of (M^{0}, ∂M^{0}). Then we have

κ_{∗} ∘ i_{1∗} ∘ Δ_{η}[a] = κ_{∗} ∘ i_{1∗}([a − ηa])
= [κ ∘ i_{1}a − κ ∘ i_{0}a]
= κ_{∗}(−δ[a])
= κ_{∗} ∘ i_{1∗} ∘ Δ[a].

Since κ_{∗} ∘ i_{1∗} : H_{2}(M^{1}) → H_{2}(V) is an isomorphism, we have Δ_{h} = Δ. This completes the proof.

Now taking the connected sum along cobordisms of the smooth h-cobordism W with the trivial cobordism (♯^{k′}S^{2} ×S^{2})×[0, 1] for some k′, we see that Δ is realized stably by a stable diffeomorphism of M [11], [14]. Hence the correspondence Θ is surjective. This completes the proof of Theorem 3.7.

Remark 3.13. Using an argument similar to the above together with the topological h-cobordism theorem of Freedman [5], we can also obtain some results about the topological mapping class group of topological 4-manifolds with boundary without stabilization. More precisely, let M be a compact 1-connected topological 4-manifold with nonempty connected boundary. Let M^{TOP}(M, ∂M) denote the group of topological isotopy classes relative to boundary of the homeomorphisms of M which are the identity on the boundary. Then we have the natural map

Θ^{TOP} : M^{TOP}(M, ∂M) → V(M, ∂M)

which maps each isotopy class of a homeomorphism to its variation map. Then the argument in the proof of Theorem 3.7 shows that Θ^{TOP} is surjective. We do not know if it is injective or not.

4. Relation to stable isometry group

In this section, we study the relationship between the stable variational group and the stable isometry group of a 4-manifold with boundary.

Definition 4.1. Let M be a compact 4-manifold with boundary. We denote by I(M) the isometry group of the inner product space (H_{2}(M), ·), where “ · ” denotes the intersection form. A stable isometry of M is an isometry of (H_{2}(M_{k}), ·) for some k. Two stable isometries Λ_{0} : H_{2}(M_{k_{0}}) → H_{2}(M_{k_{0}}) and Λ_{1} : H_{2}(M_{k_{1}}) → H_{2}(M_{k_{1}}) of M are said to be stably equivalent if the isometry Λ_{0} ⊕ id of (H_{2}(M_{k_{0}+k′_{0}}), ·) = (H_{2}(M_{k_{0}}) ⊕ H_{2}(♯^{k′_{0}}S^{2} × S^{2}), ·) and the isometry Λ_{1} ⊕ id of (H_{2}(M_{k_{1}+k′_{1}}), ·) = (H_{2}(M_{k_{1}}) ⊕ H_{2}(♯^{k′_{1}}S^{2} × S^{2}), ·) coincide with each other for some k′_{0} and k′_{1} with k_{0}+k′_{0} = k_{1}+k′_{1}. The set of all equivalence classes of stable isometries of M is called the stable isometry group of M and is denoted by SI(M), which can naturally be identified with the inductive limit

lim_{k→∞} I(M_{k}).

This obviously forms a group with respect to the composition of isometries.

Let M be a 1-connected 4-manifold with nonempty connected boundary ∂M = K. For each variational homomorphism Δ ∈ V(M, ∂M), the endomorphism Λ = id − Δ ∘ j_{∗} is an isometry of (H_{2}(M), ·) by Lemma 3.4. Let us define

Ξ : V(M, ∂M) → I(M)

by ΞΔ = Λ. By (3.6), Ξ is a homomorphism of groups.

Let us denote by ∧^{2}H_{1}(K)^{∗} the set of all skew-symmetric bilinear forms H_{1}(K)× H_{1}(K) → Z. Note that this naturally forms an additive group, which is finitely generated and free abelian. Note that an element κ ∈ ∧^{2}H_{1}(K)^{∗} can also be regarded as a homomorphism H_{1}(K) → Hom(H_{1}(K), Z) ≅ H^{1}(K) ≅ H_{2}(K). For such an element κ, define κ̃ : H_{2}(M, ∂M) → H_{2}(M) by the composition

H_{2}(M, ∂M)⟶^{∂}H_{1}(K)⟶^{κ}H_{2}(K)⟶^{ι_{∗}}H_{2}(M),

where ∂ : H_{2}(M, ∂M) → H_{1}(K) is the boundary homomorphism and ι : K → M is the inclusion. Then we can easily check that κ̃ is a variational homomorphism and that the map S : ∧^{2}H_{1}(K)^{∗} → V(M, ∂M) defined by Sκ = κ̃ is an injective homomorphism.

The main result of this section is the following.

Proposition 4.2. For a compact 1-connected 4-manifold M with nonempty connected boundary ∂M = K, the sequence

0⟶ ∧^{2} H_{1}(K)^{∗}⟶^{S}V(M, ∂M)⟶^{Ξ}I(M)

is exact.

Proof. We have already observed that S is injective.

It is easy to check that im S ⊂ ker Ξ. Suppose that Δ is an element of ker Ξ. Thus ΞΔ = id − Δ ∘ j_{∗} is the identity of H_{2}(M) and hence Δ = 0 on im j_{∗} = ker ∂. Thus we may regard Δ as a homomorphism

H_{2}(M, ∂M)/ker ∂ ≅ H_{1}(K) → H_{2}(M).

On the other hand, since Δ is variational, we have Δ + Δ^{∗} = Δ ∘ j_{∗} ∘ Δ^{∗} = 0. Hence, we have 0 = Δ ∘ j_{∗} + Δ^{∗} ∘ j_{∗} = Δ^{∗} ∘ j_{∗}, which implies that j_{∗} ∘ Δ = 0. Thus im Δ ⊂ ker j_{∗} = H_{2}(K). Hence Δ can be regarded as a homomorphism κ : H_{1}(K) → H_{2}(K). Now it is easy to check that κ ∈ ∧^{2}H_{1}(K)^{∗} and Δ = Sκ. This completes the proof.

We can characterize the image of Ξ as follows. Let us first take a basis

{α_{1}, α_{2}, …, α_{u}, α_{u+1}, …, α_{u+v}}

of H_{2}(M) over the integers, where u = rank H_{2}(K), u + v = rank H_{2}(M), and

{α_{1}, α_{2}, …, α_{u}}

is a basis of ι_{∗}H_{2}(K). This is possible, since ι_{∗}H_{2}(K) is a direct summand of H_{2}(M). For H_{2}(M, ∂M), we take the dual basis

{α_{1}^{∗}, α_{2}^{∗}, …, α_{u}^{∗}, α_{u+1}^{∗}, …, α_{u+v}^{∗}}

such that the intersection number in M satisfies α_{k} · α_{l}^{∗} = δ_{kl}, where

δ_{kl} = \{ \begin{matrix}1, & k = l, \\ 0, & k ≠ l. \end{matrix}

Let

( \begin{matrix}A & B \\ C & D \end{matrix} )

be the matrix representation of a homomorphism Δ : H_{2}(M, ∂M) → H_{2}(M) with respect to the above basis, where A, B, C and D are u × u, u × v, v × u and v × v integral matrices respectively. Furthermore, we see easily that the matrix representation of j_{∗} with respect to the above basis must be of the form

Q̃ = ( \begin{matrix}0 & 0 \\ 0 & Q \end{matrix} )

for some nonsingular v × v integral matrix Q. Then it is easy to see that Δ is variational if and only if

D +^{t}D = D Q^{t}D, A +^{t}A = B Q^{t}B, B +^{t}C = B Q^{t}D.

Thus, an isometry Λ ∈ I(M) is in the image of Ξ if and only if, with respect to the above basis, Λ is represented by a matrix of the form

( \begin{matrix}E_{u} & −BQ \\ 0 & E_{v} − DQ \end{matrix} )

such that

D +^{t}D = D Q^{t}D, A +^{t}A = B Q^{t}B, B +^{t}C = B Q^{t}D

for some integral matrices A, B, C and D, where E_{w} denotes the w ×w unit matrix.

If the diagonal entries of Q are even, or if H_{2}(K) = 0, then we can show that this condition is equivalent to that the following diagram is commutative:

\begin{aligned}0 & → & H_{2}(K) & ⟶^{ι_{∗}} & H_{2}(M) & ⟶^{j_{∗}} & H_{2}(M, ∂M) & ⟶^{∂} & H_{1}(K) & → & 0 \\ & & ↓^{id} & & ↓^{Λ} & & ↑_{Λ^{∗}} & & ↓^{id} \\ 0 & → & H_{2}(K) & ⟶^{ι_{∗}} & H_{2}(M) & ⟶^{j_{∗}} & H_{2}(M, ∂M) & ⟶^{∂} & H_{1}(K) & → & 0. \end{aligned} (4.1)

In this case, we have the exact sequence

0⟶ ∧^{2} H_{1}(K)^{∗}⟶^{S}V(M, ∂M)⟶^{Ξ}I(M)⟶^{∂}AQ̃(K)⟶0,

where ∂ and AQ̃(K) are a homomorphism and a group as defined in [3, §1].

If some diagonal entries of Q are not even and H_{2}(K) ≠ 0, then the commutativity of the diagram (4.1) is still necessary, but it is not sufficient in general.

In fact, using Boyer’s obstruction class θ [3], we can show the following.

Proposition 4.3. An isometry Λ ∈ I(M) is in the image of Ξ : V(M, ∂M) → I(M) if and only if ∂Λ = id ∈ AQ̃(K) and Boyer’s obstruction θ(id, Λ) ∈ I^{1}(K) vanishes, where I^{1}(K) = im(H^{1}(K) → H^{1}(K; Z_{2})).

Proof. Suppose ∂Λ = id and θ(id, Λ) = 0. Then by Boyer [3], there exists a homeomorphism h : M → M such that h|_{∂M} = id and h_{∗} = Λ. Let Δ = Δ_{h} ∈ V(M, ∂M) be the variation map of h. Then it is easy to see that ΞΔ = Λ.

Conversely, if Λ = ΞΔ for some Δ, then we have already seen that ∂Λ = id. Furthermore, by Remark 3.13, there exists a homeomorphism h : M → M such that h|_{∂M} = id and Δ_{h} = Δ. Hence by Boyer [3], the obstruction θ(id, Λ_{h}) vanishes, where Λ_{h} = h_{∗}. Since Λ_{h} = ΞΔ_{h} = Λ by Lemma 3.2 (2), we have θ(id, Λ) = 0. This completes the proof.

Note that when the diagonal entries of Q are even or H_{2}(K) = 0, the obstruction θ(id, Λ) always vanishes (see [3, (0.8) Proposition]).

The above proposition can also be proved purely algebraically by using the abovementioned characterization of the image of Ξ in terms of matrices, together with the definition of the obstruction θ.

All the above results hold stably as well. More precisely, we can naturally define the homomorphisms

SS : ∧^{2}H_{1}(K)^{∗} → SV(M, ∂M), SΞ : SV(M, ∂M) → SI(M),

and the sequence

0⟶ ∧^{2} H_{1}(K)^{∗}⟶^{SS}SV(M, ∂M)⟶^{SΞ}SI(M)

is exact. The characterization of the image of Ξ holds similarly for SΞ as well.

5. Boundary free diffeomorphisms

In this section, we study the stable mapping class group of boundary free diffeomorphisms of compact 4-manifolds whose boundaries are certain spherical 3manifolds.

Definition 5.1. Let M be a compact 4-manifold with boundary. Two stable diffeomorphisms h_{0} : M_{k_{0}} → M_{k_{0}} and h_{1} : M_{k_{1}} → M_{k_{1}} of M, which may not necessarily be the identity on the boundary, are said to be stably isotopic if the diffeomorphisms h_{0}♯k′_{0}(id) : M_{k_{0}+k′_{0}} → M_{k_{0}+k′_{0}} and h_{1}♯k′_{1}(id) : M_{k_{1}+k′_{1}} → M_{k_{1}+k′_{1}} are smoothly isotopic for some k′_{0} and k′_{1} with k_{0} + k′_{0} = k_{1} + k′_{1}. Note that the isotopy should leave the boundary invariant as a set, but can move points on the boundary. The set of all stable isotopy classes of stable diffeomorphisms of M is called the stable mapping class group of boundary free diffeomorphisms of M and is denoted by SM(M). This obviously forms a group with respect to the composition of maps.

Let Λ : H_{2}(M) → H_{2}(M) be an isometry of a compact 4-manifold M with boundary K. Then we have the commutative diagram

\begin{aligned}0 & → & H_{2}(K) & ⟶^{ι_{∗}} & H_{2}(M) & ⟶^{j_{∗}} & H_{2}(M, ∂M) & ⟶^{∂} & H_{1}(K) & → & 0 \\ & & ↓^{φ_{2}} & & ↓^{Λ} & & ↑_{Λ^{∗}} & & ↓^{φ_{1}} \\ 0 & → & H_{2}(K) & ⟶^{ι_{∗}} & H_{2}(M) & ⟶^{j_{∗}} & H_{2}(M, ∂M) & ⟶^{∂} & H_{1}(K) & → & 0 \end{aligned} (5.1)

for some isomorphisms φ_{1} and φ_{2}, where ι : ∂M → M and j : M → (M, ∂M) denote the inclusions. Let A(K) denote the group of automorphisms of H_{∗}(K), where an automorphism is required to preserve the torsion linking pairing T_{1}(K) × T_{1}(K) → Q/Z on the torsion subgroup T_{1}(K) of H_{1}(K) together with the intersection pairing H_{1}(K) × H_{2}(K) → Z (for details, see [3, §1]). Note that the isomorphism of H_{∗}(K) determined by (φ_{1}, φ_{2}) in (5.1) is an automorphism and that the map

∂ : I(M) → A(K)

defined by ∂Λ = (φ_{1}, φ_{2}) is a homomorphism.

Furthermore, let H_{+}(K) denote the subgroup of A(K) consisting of those automorphisms which are realized by an orientation preserving homeomorphism of K.

Set I^{0}(M) = ∂^{−1}(H_{+}(K)), which is a subgroup of I(M). It is easy to observe that we naturally have

I^{0}(M) ⊂ I^{0}(M_{1}) ⊂ ⋯ ⊂ I^{0}(M_{k}) ⊂ I^{0}(M_{k+1}) ⊂ · · · .

Thus we can define the inductive limit

SI^{0}(M) = lim_{k→∞} I^{0}(M_{k}),

which is also a group.

Note that if H_{1}(K) = 0, then we have SI^{0}(M) = SI(M). Note also that if h : M_{k} → M_{k} is an orientation preserving stable diffeomorphism of M, then h_{∗} : H_{2}(M_{k}) → H_{2}(M_{k}) lies in SI^{0}(M).

The main result of this section is the following.

Theorem 5.2. Let M be a smooth compact 1-connected 4-manifold with boundary ∂M homeomorphic to a lens space L(p, q) (p ≥ 2), the 3-sphere S^{3}, or the Poincaré homology 3-sphere Σ(2, 3, 5). Then the correspondence

Θ^{fr} : SM(M) → SI^{0}(M)

which maps each class of a stable diffeomorphism to the class of its induced isometry of the second homology group gives an isomorphism of groups.

Proof. Suppose that h : M_{k} → M_{k} is a stable diffeomorphism of M such that Θ^{fr}h is the identity. Then the automorphism (h|_{∂M})_{∗} of H_{∗}(∂M) must be the identity by a commutative diagram similar to (5.1). Then by results of [1], [2], [6], and [7] together with our assumption on the boundary, we see that h|_{∂M} is smoothly isotopic to the identity. Thus, by extending the isotopy to the whole of M using the collar neighborhood, we may assume that h|_{∂M} is the identity. On the other hand, since H_{1}(∂M) is finite, the homomorphism Ξ : V(M_{k}, ∂M_{k}) → I(M_{k}) is injective by Proposition 4.2. Hence the variation map of h must be the zero homomorphism. Thus by Theorem 2.2, h is stably isotopic to the identity diffeomorphism (relative to boundary). Thus the homomorphism Θ^{fr} is injective.

Let Λ : H_{2}(M_{k}) → H_{2}(M_{k}) be a stable isometry of M such that Λ ∈ I^{0}(M_{k}). Then by the definition of I^{0}(M_{k}), there exists an orientation preserving homeomorphism f : ∂M_{k} → ∂M_{k} such that ∂Λ = f_{∗}. We may assume that f is a diffeomorphism (see [12]). Then by [3, (0.8) Proposition] together with the fact that H_{∗}(∂M; Q) = 0, we see that Boyer’s obstruction θ(f, Λ) vanishes, which implies that there exists a homeomorphism f̃ : M_{k} → M_{k} such that f̃_{∗} = Λ and f̃|_{∂M} = f. Then by using the stable h-cobordism theorem [11], [14] together with an argument as in [3, §4] for constructing a smooth h-cobordism, we can show that there exists a diffeomorphism f̃′ of M_{k+k′} for some k′ ≥ 0 such that f̃′|_{∂M} = f̃|_{∂M} and f̃′_{∗} and f̃_{∗} are stably equivalent. Hence Θ^{fr} is surjective. This completes the proof.

Remark 5.3. For the lens space L(p, q), p ≥ 2, the structure of H_{+}(L(p, q)) has been determined. In fact, according to [2], H_{+}(L(p, q)) coincides with

(1) {id},  if p = 2,
(2) {±id, ±q}, if q^{2} ≡ 1 (mod p) and q ≢ ±1 (mod p),
(3) {±id},  otherwise,

where ±q stands for the multiplication by ±q. Thus, if ∂M is diffeomorphic to L(p, q) which satisfies (1) or (3), then SI^{0}(M) coincides with SI(M).

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Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan

E-mail address: saeki@math.sci.hiroshima-u.ac.jp