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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.
Reviewer: Yang Su (Beijing)
MSC:
 57R65 Surgery and handlebodies 57R67 Surgery obstructions, Wall groups
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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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