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**Intrinsic transversality structures.**
*(English)*
Zbl 0661.57005

This paper tackles the fundamental problem of the existence and classification of topological manifold structures on Poincaré duality spaces \(X^ n\), with Spivak normal fibration \(\nu\), through various notions of transversality. Classically, this is a two step problem. The first step is to check if \(\nu\) has a TOP reduction. This can be accomplished by viewing \(\nu\) as a high, but finite dimensional spherical fibration and requiring that all maps f: \(M^ n\to T(\nu)\) can be made Poincaré-transverse in a mutually consistent way, or, in the terminology of the authors, \(X^ n\) has an extrinsic transversality structure. It is shown by N. Levitt and J. Morgan [Bull. Am. Math. Soc. 78, 1064-1068 (1972; Zbl 0267.55021)] that \(X^ n\) has a TOP- reducible Spivak normal fibration iff it has an extrinsic transversality structure. The second step is to see whether such a reduction exists such that the resulting surgery obstruction in \(L_ n(\pi_ 1X)\) vanishes. The authors show that this can be accomplished by considering the notion of an intrinsic transversality structure. Very roughly speaking, \(X^ n\) has an intrinsic transversality structure if, for any PL-stratified space Q, all maps f: \(X^ n\to Q\) can be made Poincaré-transverse to the stratification of the range Q. After making this notion precise the authors prove: Theorem. Intrinsic transversality structures on \(X^ n\) are in one-to-one correspondence with topological manifold structures on \(X^ n\).

Reviewer: R.Stern

### MSC:

57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |

57P10 | Poincaré duality spaces |

57N75 | General position and transversality |

57Q65 | General position and transversality |

57N80 | Stratifications in topological manifolds |