Surgery on a pair of transversal manifolds.

*(English)*Zbl 1257.57034The main focus of classical surgery theory is the classification, up to homeomorphism, of closed manifolds homotopy equivalent to a closed manifold. There are variants of the classical theory for manifolds with boundary, for pairs of manifolds or, even more general, for manifold \(n\)-ads. In the paper under review, the authors study a certain case of stratified spaces. More precisely, the authors study the case when the space is the union of two manifolds over a common locally flat submanifold. It is assumed that the two manifolds are of the same dimension and all three manifolds that appear are connected. The authors point out that their methods, being functorial enough, work without these assumptions. The approach uses the classical approach. First, the correct definition of normal maps (called \(t\)-triangulation) in this setting is given. Essentially, these are maps between the ambient spaces such that the pre-images of the pieces are the corresponding pieces and the restriction to the two manifold pairs that appear are \(t\)-triangulations of pairs [M. Cencelj, Yu. V. Muranov and D. Repovš, Homology Homotopy Appl. 11, No. 2, 195–222 (2009; Zbl 1185.57030); 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 equivalence relation between such maps is concordance. It is shown that the right spectrum whose homotopy groups, in the right dimension, are the equivalence classes of the \(t\)-structures, is the pullback of the \(\mathbf{L}_{\bullet}\)-homology spectra of the two manifold pieces over the \(\mathbf{L}_{\bullet}\)-homology spectrum of the intersection with a dimension shift, depending on the codimension of the intersection in each space. A similar construction works for the obstruction groups. The spectrum whose homotopy groups are the obstruction groups is the pullback of the spectra of the \(\mathbb{L}P\)-spectra of the corresponding pairs (and their normal data) over the \(\mathbb{L}\)-spectrum of the intersection. Alternatively, the authors characterize the obstruction spectrum as the pullback of the \(\mathbb{L}P\)-spectrum of one of the piece, the \(\mathbb{L}\)-spectrum of the other piece with a dimension shift, over the \(\mathbb{L}\)-spectrum of the inclusion of the complement of the intersection in the last piece to the whole submanifold, again with the same dimension shift. The dimension shift is positive and to the order of the codimension of the intersection. All the above constructions ensure that the objects defined for the stratified space fit into braids of exact sequences that connect them with the classical objects. Furthermore, the homotopy groups of the spectra that are defined also fit into the corresponding exact sequence that generalizes the classical one.

As an application the authors give the example of the space \(\mathcal{X} = (\mathbb{R}P^n, \mathbb{R}P^n; \mathbb{R}P^{n-1})\), \(n \geq 6\). In this case, the components for building up the objects of \(\mathcal{X}\) are known. Using the fact that \(\mathbb{R}P^n \setminus \mathbb{R}P^{n-1} \cong D^n\), direct calculation for the structure sets shows that: \[ \mathcal{S}^{\text{TOP}}(\mathcal{X}) \cong \mathcal{S}^{\text{TOP}}(\mathbb{R}P^n, \mathbb{R}P^{n-1}) \cong \mathcal{S}^{\text{TOP}}(\mathbb{R}P^{n-1}). \]

As an application the authors give the example of the space \(\mathcal{X} = (\mathbb{R}P^n, \mathbb{R}P^n; \mathbb{R}P^{n-1})\), \(n \geq 6\). In this case, the components for building up the objects of \(\mathcal{X}\) are known. Using the fact that \(\mathbb{R}P^n \setminus \mathbb{R}P^{n-1} \cong D^n\), direct calculation for the structure sets shows that: \[ \mathcal{S}^{\text{TOP}}(\mathcal{X}) \cong \mathcal{S}^{\text{TOP}}(\mathbb{R}P^n, \mathbb{R}P^{n-1}) \cong \mathcal{S}^{\text{TOP}}(\mathbb{R}P^{n-1}). \]

Reviewer: Stratos Prassidis (Karlovassi)

##### MSC:

57R67 | Surgery obstructions, Wall groups |

19J25 | Surgery obstructions (\(K\)-theoretic aspects) |

57N99 | Topological manifolds |

##### Keywords:

surgery obstruction groups; splitting obstruction groups; stratified manifolds; surgery exact sequence
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\textit{A. Bak} and \textit{Y. V. Muranov}, J. Homotopy Relat. Struct. 7, No. 2, 255--279 (2012; Zbl 1257.57034)

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##### References:

[1] | Adams, J.F.: Stable homotopy and generalised homology. In: Chicago Lectures in Mathematics. The University of Chicago Press, Chicago (1974) · Zbl 0309.55016 |

[2] | Bak, A.; Muranov, Y. V., Splitting along submanifolds and L-spectra, J. Math. Sci. (N.Y), 123, 4169-4184, (2004) · Zbl 1078.57030 |

[3] | Bak, A., Muranov, Y.V.: Splitting a simple homotopy equivalence along a submanifold with filtration Matem. Sbornik 199(6), 3-26 (2008). English transl. in Sbornik: Math. 199, 787-809 (2008) |

[4] | Bak, A., Muranov, Y.V.: Normal invariants of manifold pairs and assembly maps Matem. Sbornik 197(6), 3-24 (2006). English transl. in Sbornik: Math. 197, 791-811 (2006) |

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

[6] | Cencelj, M.; Muranov, Y. V.; Repovš, D., On structure sets of manifold pairs, Homol. Homotopy Appl., 11, 195-222, (2009) · Zbl 1185.57030 |

[7] | Lopezde Medrano S.: Involutions on Manifolds. Springer, Berlin (1971) |

[8] | Milnor, J.: Lectures on the \(h\)-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton (1965) · Zbl 0161.20302 |

[9] | Muranov, Y.V.: Splitting obstruction groups and quadratic extensions of anti-structures Izvestiya: Mathematics 59(6), 1207-1232 (1995). English transl. from Izvestiya RAN: Ser. Mat. 59(6), 107-132 · Zbl 0996.57518 |

[10] | Ranicki, A.A.: The total surgery obstruction. In: Lecture Notes in Math., vol. 763, pp. 275-316. Springer, Berlin (1979) · Zbl 0428.57012 |

[11] | Ranicki, A.A.: Exact sequences in the algebraic theory of surgery. In: Math. Notes 26. Princeton Univ. Press, Princeton (1981) · Zbl 0471.57012 |

[12] | Wall, C.T.C.: Ranicki, A.A. (ed.) Surgery on Compact Manifolds. Academic Press, London (1970); 2nd edn. Amer. Math. Soc., Providence 1999 · Zbl 0219.57024 |

[13] | Weinberger S.: The Topological Classification of Stratified Spaces. The university of Chicago Press, Chicago (1994) · Zbl 0826.57001 |

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