On surgery inside a manifold.

*(English)*Zbl 1190.57015
Kubarski, Jan (ed.) et al., Special issue: Proceedings of the 8th conference on geometry and topology of manifolds (Lie algebroids, dynamical systems and applications), Luxembourg-Poland-Ukraine conference, Przemyśl, Poland, L’viv, Ukraine, April 30–May 6, 2007. Luxembourg: University of Luxembourg, Faculty of Science, Technology and Communication (ISBN 978-2-87971-025-9/pbk). Travaux Mathématiques 18, 45-64 (2008).

The authors’ introduction: “In surgery theory, one sometimes looks at submanifolds to get additional information about surgery obstruction groups and natural maps (see A. Bak and Yu.V. Muranov [Sb. Math. 197, No. 6, 791–811 (2006); translation from Mat. Sb. 197, No. 6, 3–24 (2006; Zbl 1148.57042)], W. Browder and G. R. Livesay [Tohoku Math. J., II. Ser. 25, 69–87 (1973; Zbl 0276.57013)], Sylvain E. Cappell and Julius L. Shaneson [Algebraic topology, Proc. Symp. Aarhus 1978, Lect. Notes Math. 763, 395–447 (1979; Zbl 0416.57020)], Alberto Cavicchioli, Yurij V. Muranov and Fulvia Spaggiari [Glasg. Math. J. 48, No. 1, 125–143 (2006; Zbl 1110.57019)], Ian Hambleton [Algebraic K-theory, Proc. Conf., Oberwolfach 1980, Part II, Lect. Notes Math. 967, 101–131 (1982; Zbl 0503.57018)], Ian Hambleton, Laurence Taylor and Bruce Williams [Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 49–127 (1984; Zbl 0556.57026)], I. Hambleton, R. J. Milgram, L. Taylor and B. Williams [Proc. Lond. Math. Soc., III. Ser. 56, No. 2, 349-379 (1988; Zbl 0665.57026)], I. Hambleton and A. F. Kharshiladze [Russ. Acad. Sci., Sb., Math. 77, No. 1, 1–9 (1994); translation from Mat. Sb. 183, No. 9, 3–14 (1992; Zbl 0791.57022)], Friedrich Hegenbarth and Yuri V. Muranov [Int. Math. Forum 3, No. 5–8, 209–228 (2008; Zbl 1164.57013)], A. F. Kharshiladze [Russ. Math. Surv. 42, No. 4, 65–103 (1987); translation from Usp. Mat. Nauk 42, No. 4(256), 55–85 (1987; Zbl 0671.57020)], S. López de Medrano [Ergebnisse der Mathematik und ihrer Grenzgebiete. 59. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0214.22501)], Andrew Ranicki [Mathematical Notes, 26. Princeton, New Jersey: Princeton University Press; University of Tokyo Press (1981; Zbl 0471.57012)] and C. T. C. Wall [Mathematical Surveys and Monographs. 69. Providence, RI: American Mathematical Society (1999; Zbl 0935.57003)]).

To study surgery on manifold pairs, C. T. C. Wall [op. cit.] introduced the concept of splitting of a simple homotopy equivalence along a submanifold in the case of piecewise linear and smooth manifolds and applied this approach to various geometric problems. For topological manifolds this approach was developed by A. A. Ranicki [Algebraic topology, Pro. Symp., Aarhus 1978, Lect. Notes Math. 763, 275–316 (1979; Zbl 0428.57012); Cambridge Tracts in Mathematics 102. Cambridge: Cambridge University Press. ix, 358 p. (2008; Zbl 1143.57001) and Mathematical Notes, 26. op. cit.]. The advantage of topological category is exhibited by the possibility of realizing various obstruction groups, structure sets, and natural maps on the spectra level (see A. Bak and Yu.V. Muranov [J. Math. Sci., New York 123, No. 4, 4169–4184 (2004; Zbl 1078.57030)], Yu. V. Muranov, D. Repovš and R. Jimenez [Trans. Mosc. Math. Soc. 2006, 261–288 (2006); translation from Tr. Mosk. Mat. O.-va 67, 294–325 (2006; Zbl 1163.57022)], A. A. Ranicki [op. cit.], C. T. C. Wall [op. cit.] and Shmuel Weinberger [Chicago, IL: University of Chicago Press (1994; Zbl 0826.57001)]).

In the present paper, we compare the abstract surgery with the surgery inside an ambient manifold. We consider only topological manifolds and topological normal maps. All manifold pairs \(X\subset Y\) will be topological manifold pairs in the sense of A. A. Ranicki [op. cit.]; in particular, \(X\) will be a locally flat submanifold. We consider the case of higher dimensions, that is the dimension of all closed manifolds will be \(\geq 5\), and the dimension of all manifolds with boundary will be \(\geq 6\). In section 2 we recall necessary definitions of surgery theory for manifold pairs. Afterwards, we describe various natural maps between exact sequences containing surgery obstruction groups and structure sets. Then we give an example of exact computations. In section 3 we consider manifolds with filtration. At first, we describe the surgery and splitting problem in this case and give a short summary of results on this subject (see A. Bak and Yu.V. Muranov [Sb. Math. 199, No. 6, 787–809 (2008); translation from Mat. Sb. 199, No. 6, 3–26 (2008; Zbl 1166.57019)], William Browder and Frank Quinn [Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 27–36 (1975; Zbl 0343.57017)], Alberto Cavicchioli, Yurij V. Muranov and Fulvia Spaggiari [“On the elements of the second type in surgery groups”, Preprint-Max-Plank Institut für Mathematik, 111 (2006)], I. Hambleton and A. F. Kharshiladze [op. cit.], Friedrich Hegenbarth and Yuri V. Muranov [op. cit.], Yu. V. Muranov, D. Repovš and R. Jimenez [op. cit.] and Shmuel Weinberger [op. cit.]).

Then we obtain new relations between various obstruction groups and structure sets for filtered manifolds. We describe also applications of the obtained results to the problem of realizing elements of various surgery and splitting obstruction groups by normal maps of closed manifolds.”

For the entire collection see [Zbl 1168.57001].

To study surgery on manifold pairs, C. T. C. Wall [op. cit.] introduced the concept of splitting of a simple homotopy equivalence along a submanifold in the case of piecewise linear and smooth manifolds and applied this approach to various geometric problems. For topological manifolds this approach was developed by A. A. Ranicki [Algebraic topology, Pro. Symp., Aarhus 1978, Lect. Notes Math. 763, 275–316 (1979; Zbl 0428.57012); Cambridge Tracts in Mathematics 102. Cambridge: Cambridge University Press. ix, 358 p. (2008; Zbl 1143.57001) and Mathematical Notes, 26. op. cit.]. The advantage of topological category is exhibited by the possibility of realizing various obstruction groups, structure sets, and natural maps on the spectra level (see A. Bak and Yu.V. Muranov [J. Math. Sci., New York 123, No. 4, 4169–4184 (2004; Zbl 1078.57030)], Yu. V. Muranov, D. Repovš and R. Jimenez [Trans. Mosc. Math. Soc. 2006, 261–288 (2006); translation from Tr. Mosk. Mat. O.-va 67, 294–325 (2006; Zbl 1163.57022)], A. A. Ranicki [op. cit.], C. T. C. Wall [op. cit.] and Shmuel Weinberger [Chicago, IL: University of Chicago Press (1994; Zbl 0826.57001)]).

In the present paper, we compare the abstract surgery with the surgery inside an ambient manifold. We consider only topological manifolds and topological normal maps. All manifold pairs \(X\subset Y\) will be topological manifold pairs in the sense of A. A. Ranicki [op. cit.]; in particular, \(X\) will be a locally flat submanifold. We consider the case of higher dimensions, that is the dimension of all closed manifolds will be \(\geq 5\), and the dimension of all manifolds with boundary will be \(\geq 6\). In section 2 we recall necessary definitions of surgery theory for manifold pairs. Afterwards, we describe various natural maps between exact sequences containing surgery obstruction groups and structure sets. Then we give an example of exact computations. In section 3 we consider manifolds with filtration. At first, we describe the surgery and splitting problem in this case and give a short summary of results on this subject (see A. Bak and Yu.V. Muranov [Sb. Math. 199, No. 6, 787–809 (2008); translation from Mat. Sb. 199, No. 6, 3–26 (2008; Zbl 1166.57019)], William Browder and Frank Quinn [Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 27–36 (1975; Zbl 0343.57017)], Alberto Cavicchioli, Yurij V. Muranov and Fulvia Spaggiari [“On the elements of the second type in surgery groups”, Preprint-Max-Plank Institut für Mathematik, 111 (2006)], I. Hambleton and A. F. Kharshiladze [op. cit.], Friedrich Hegenbarth and Yuri V. Muranov [op. cit.], Yu. V. Muranov, D. Repovš and R. Jimenez [op. cit.] and Shmuel Weinberger [op. cit.]).

Then we obtain new relations between various obstruction groups and structure sets for filtered manifolds. We describe also applications of the obtained results to the problem of realizing elements of various surgery and splitting obstruction groups by normal maps of closed manifolds.”

For the entire collection see [Zbl 1168.57001].

Reviewer: Mircea Craioveanu (Timişoara)

##### MSC:

57N99 | Topological manifolds |

57R65 | Surgery and handlebodies |

57Q10 | Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. |