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Dual decompositions of 4-manifolds. (English) Zbl 0993.57017
A dual decomposition of a closed connected smooth 4-manifold \(N\) is a representation of \(N\) as the union of two handlebodies along their boundary, each one with handles of index \(\leq 2\). The two pieces are “dual” in the sense that each is the complement of the other. In dimensions \(\geq 5\) results of Smale (simply connected case) and Wall (general case) describe analogous decompositions, up to diffeomorphism, in terms of homotopy type of skeleta or chain complexes. In dimension 4, decompositions are described up to 2-deformation of their spines. Given two CW-complexes \(K\) and \(L\), a 2-deformation \(K\to L\) is a sequence of moves starting with \(K\) and ending with \(L\). Each move is either an elementary expansion or collapse of dimension \(\leq 2\), or change of attaching map of a 2-cell by homotopy. To state the main results of this fundamental paper, recall that chain complexes are finitely generated free based complexes over the group ring \(\mathbb{Z} [\pi_1(N)]\).
If \(K \to N\) is a CW-complex, then \(C_*(K)\) denotes the cellular chains of the cover of \(K\) induced from the universal cover of \(N\). The first theorem (chain realization) says that if \(D_*\) is a chain complex with a chain map \(D_*\to C_* (N)\), then \(D_*\) is simply chain equivalent to the cellular chains of one side of a dual decomposition of \(N\) if and only if \(D_*\) is homologically 2-dimensional and \(H_0(D_*)\to H_0(C_*(N))\cong\mathbb{Z}\) is an isomorphism. The second theorem (spine realization) says that if \(N=M\cup W\) is a dual decomposition of \(N\) and \(K\to N\) is a 2-complex, then there is an ambient deformation of \(M\) to a decomposition \(M'\cup W'\) with a 2-deformation of \(K\) to the spine of \(W'\) if and only if there is a simple chain equivalence \(C_*(K)\to C_*(W)\) that chain-homotopy commutes with the inclusions. A lot of consequences and corollaries of the main theorems are discussed. We recall some of them.
(1) (Spine characterization): A finite CW-2-complex \(K\to N\) 2-deforms to the spine of half a dual decomposition if and only if \(K\) is connected and \(\pi_1(K) \to\pi_1(N)\) is onto;
(2) (Characterization of dual spines): Two CW-2-complexes \(K\) and \(L\) 2-deform to spines of a dual decomposition iff there is a chain nullhomotopy giving simple duality on the chain level.
(3) (Dual spines in \(\mathbb{S}^4)\): Two 2-complexes \(K\) and \(L\) occur, up to 2-deformation, as spines of a dual decomposition of \(\mathbb{S}^4\) if and only if they are connected and there are isomorphisms \(H_1(K)\cong H^2(L)\) and \(H_2(K)\cong H^1(L)\);
(4) If \((N, \partial N)\) is simply connected, then there is a “pseudo” handle decomposition without 1-handles, in the sense that there is a pseudo-collar \((M, \partial N)\) (i.e., a relative 2-handlebody with spine that 2-deforms to \(\partial N)\) and \(N\) is obtained from it by attaching handles of index \(\geq 2\).

MSC:
57R65 Surgery and handlebodies
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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