Dual decompositions of 4-manifolds.

*(English)*Zbl 0993.57017A 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 \(\geq 2\).

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 \(\geq 2\).

Reviewer: Alberto Cavicchioli (Modena)

##### 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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\textit{F. Quinn}, Trans. Am. Math. Soc. 354, No. 4, 1373--1392 (2002; Zbl 0993.57017)

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##### References:

[1] | S. Akbulut and R. Matveyev, A convex decomposition theorem for 4-manifolds, Internat. Math. Res. Notices 7 (1998), 371 – 381. · Zbl 0911.57025 · doi:10.1155/S1073792898000245 · doi.org |

[2] | J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192 – 195. · Zbl 0131.38301 |

[3] | S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. · Zbl 0820.57002 |

[4] | Maria Rita Casali, The average edge order of 3-manifold coloured triangulations, Canad. Math. Bull. 37 (1994), no. 2, 154 – 161. · Zbl 0819.57003 · doi:10.4153/CMB-1994-022-x · doi.org |

[5] | M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton University Press, 1990. · Zbl 0705.57001 |

[6] | Robert E. Gompf and András I. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. · Zbl 0933.57020 |

[7] | Günther Huck, Embeddings of acyclic 2-complexes in \?\(^{4}\) with contractible complement, Topology and combinatorial group theory (Hanover, NH, 1986/1987; Enfield, NH, 1988) Lecture Notes in Math., vol. 1440, Springer, Berlin, 1990, pp. 122 – 129. · Zbl 0765.57004 · doi:10.1007/BFb0084457 · doi.org |

[8] | Vyacheslav S. Krushkal, Embedding obstructions and 4-dimensional thickenings of 2-complexes, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3683 – 3691. · Zbl 0957.57016 |

[9] | Vyacheslav S. Krushkal and Peter Teichner, Alexander duality, gropes and link homotopy, Geom. Topol. 1 (1997), 51 – 69. · Zbl 0885.55001 · doi:10.2140/gt.1997.1.51 · doi.org |

[10] | Frank Quinn, Handlebodies and 2-complexes, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 245 – 259. · doi:10.1007/BFb0075228 · doi.org |

[11] | C. T. C. Wall, Geometrical connectivity. I, J. London Math. Soc. (2) 3 (1971), 597 – 604. , https://doi.org/10.1112/jlms/s2-3.4.597 C. T. C. Wall, Geometrical connectivity. II. Dimension 3, J. London Math. Soc. (2) 3 (1971), 605 – 608. · doi:10.1112/jlms/s2-3.4.605 · doi.org |

[12] | C. T. C. Wall, Formal deformations, Proc. London Math. Soc. (3) 16 (1966), 342 – 352. · Zbl 0151.31302 · doi:10.1112/plms/s3-16.1.342 · doi.org |

[13] | C. T. C. Wall, Finiteness conditions for \?\?-complexes, Ann. of Math. (2) 81 (1965), 56 – 69. · Zbl 0152.21902 · doi:10.2307/1970382 · doi.org |

[14] | C. T. C. Wall, Finiteness conditions for \?\? complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129 – 139. |

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