The \(\pi\)-\(\pi\)-theorem for manifold pairs with boundaries.

*(English. Russian original)*Zbl 1141.57012
Math. Notes 81, No. 3, 356-364 (2007); translation from Mat. Zametki 81, No. 3, 405-416 (2007).

The \(\pi\)-\(\pi\) theorem of [C. T. C. Wall, Surgery on Compact Manifolds, second edition. Mathematical Surveys and Monographs. 69. (Providence), RI: American Mathematical Society (AMS). (1999; Zbl 0935.57003)] states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair \((X,\partial X)\) is normally bordant to a simple homotopy equivalence if the map \(\pi_1(\partial X) \to \pi_1(X)\) is an isomorphism. Wall also defined the surgery obstruction groups \(LP_*\) for normal maps to manifold pairs and splitting obstruction groups \(LS_*\). From this point of view, the authors generalized the \(\pi\)-\(\pi\) theorem in the following sense: Let \((f, \partial f): (M, \partial M) \to (X, \partial X)\) be a topological normal map to a pair of manifolds with boundary \((Y, \partial Y) \subset (X, \partial X)\) with topological normal block fibration \(\xi_Y=\xi_{Y\subset X}\) and \(\dim Y \geq 6\). If the inclusion \(\partial X \to X\) induces an isomorphism between the push-out squares of fundamental groups of \((X; Y, X-Y; S(\xi_Y))\) and \((\partial X; \partial Y, \partial X -\partial Y; S(\xi_{\partial Y}))\), then

(1) \((f,\partial f)\) is normally bordant to an \(s\)-triangulation of the pair of manifolds.

(2) if \((f, \partial f)\) is a simple homotopy equivalence of the pairs, then it is concordant to an \(s\)-triangulation of the pair of manifolds.

(3) if the restricted map \((f,\partial f) :(f^{-1}(Y), f^{-1}(\partial Y)) \to (Y, \partial Y)\) is a simple homotopy equivalence of pairs, then \((f, \partial f)\) is normally bordant to an \(s\)-triangulation of the pair of manifolds.

Where an \(s\)-triangulation of a pair of manifolds is in the sense of [A. A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery. Mathematical Notes, 26. Princeton, New Jersey: Princeton University Press; University of Tokyo Press. (1981; Zbl 0471.57012)]. The results can be carried over to \(Diff\)- and \(PL\)-categories. The authors applied the obtained results to surgery on filtered manifolds.

(1) \((f,\partial f)\) is normally bordant to an \(s\)-triangulation of the pair of manifolds.

(2) if \((f, \partial f)\) is a simple homotopy equivalence of the pairs, then it is concordant to an \(s\)-triangulation of the pair of manifolds.

(3) if the restricted map \((f,\partial f) :(f^{-1}(Y), f^{-1}(\partial Y)) \to (Y, \partial Y)\) is a simple homotopy equivalence of pairs, then \((f, \partial f)\) is normally bordant to an \(s\)-triangulation of the pair of manifolds.

Where an \(s\)-triangulation of a pair of manifolds is in the sense of [A. A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery. Mathematical Notes, 26. Princeton, New Jersey: Princeton University Press; University of Tokyo Press. (1981; Zbl 0471.57012)]. The results can be carried over to \(Diff\)- and \(PL\)-categories. The authors applied the obtained results to surgery on filtered manifolds.

Reviewer: Yang Su (Beijing)

##### Keywords:

surgery obstruction groups; surgery on manifold pairs; normal maps; homotopy triangulation; splitting obstruction groups; \(\pi\)-\(\pi\) theorem; surgery on filtered manifolds
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\textit{Yu. V. Muranov} et al., Math. Notes 81, No. 3, 356--364 (2007; Zbl 1141.57012); translation from Mat. Zametki 81, No. 3, 405--416 (2007)

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##### References:

[1] | C. T. C. Wall, Surgery on Compact Manifolds, Ed. by A. A. Ranicki, 2nd ed. (Amer. Math. Soc., Providence, RI, 1999). · Zbl 0935.57003 |

[2] | A. A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery (Mathem. Notes, Princeton, 1981). · Zbl 0471.57012 |

[3] | W. Browder and F. Quinn, ”A surgery theory for G-manifolds and stratified spaces,” in Manifolds (Univ. Tokyo Press, 1975), pp. 27–36. · Zbl 0343.57017 |

[4] | Yu. V. Muranov, D. Repovš, and R. Jimenez, ”Surgery spectral sequence and manifolds with filtration,” Trudy Moskov. Mat. Obshch. 67, 294–325 (2006) [Trans. Moscow Math. Soc.]. |

[5] | S. Weinberger, The Topological Classification of Stratified Spaces (Univ. Chicago Press, Chicago-London, 1994). · Zbl 0826.57001 |

[6] | S. Lopez de Medrano, Involutions on Manifolds (Springer-Verlag, Berlin-Heidelberg-New York, 1971). · Zbl 0199.58603 |

[7] | M. M. Cohen, A Course in Simple-Homotopy Theory (Springer-Verlag, New York, 1973). · Zbl 0261.57009 |

[8] | Yu. V. Muranov, D. Repovš, and F. Spaggiari, ”Surgery on triples of manifolds,” Mat. Sb. 8, 139–160 (2003) [Russian Acad. Sci. Sb. Math. 194 (8), 1251–1271 (2003)]. · Zbl 1067.57032 · doi:10.4213/sm764 |

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