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**Stratified mappings - structure and triangulability.**
*(English)*
Zbl 0543.57002

Lecture Notes in Mathematics. 1102. Subseries: Mathematisches Institut der Universität und Max-Planck-Institut für Mathematik, Bonn, Vol. 4. Berlin etc.: Springer-Verlag. IX, 160 p. DM 26.50 (1984).

One of the basic problems in (algebraic) topology is to show that certain spaces and mappings are triangulable. During the last fifty years, and by the efforts of many mathematicians (Cairns, Whitehead, Lefschetz, Lojasiewicz, Hironaka and Hardt, to mention only a few) larger and larger categories of spaces and maps were considered: smooth manifolds, algebraic sets, subanalytic sets. But it turned out, in the last years, that the most suitable category for discussing triangulability is the category of stratified sets introduced by Thom (alias the abstract stratifications of Mather). This category includes all the types of spaces mentioned before and also the orbit spaces of smooth actions of compact Lie groups. The triangulability of the stratified sets is due to R. M. Goresky [Proc. Am. Math. Soc. 72, 193-200 (1978; Zbl 0392.57001)].

The purpose of this book is to give a proof of the triangulability of these stratified sets and of some classes of mappings (including the proper, topologically stable smooth mappings between two manifolds). This last part is the main new contribution and can be regarded as an answer to R. Thom’s implicite conjecture: ”almost all” smooth mappings are triangulable [Enseign. Math., II. Sér. 8, 24-33 (1962; Zbl 0109.400)].

The first three chapters are devoted to a quick and rather complete introduction to stratified spaces (stratifications, controlled vector fields, Thom mappings). In the next two chapters the theory is slightly extended, allowing the strata to be manifolds with corners (called here manifolds with faces). The sixth chapter deals with the structure of Thom mappings, a \(C^{\infty}\)-analogue of the resolution of singularities. The final two chapters contain the proofs of the triangulability results mentioned above. Some facts from PL-topology are collected in an appendix. This very clearly written book can be of great help to a patient reader determined to tackle this highly technical and important subject.

The purpose of this book is to give a proof of the triangulability of these stratified sets and of some classes of mappings (including the proper, topologically stable smooth mappings between two manifolds). This last part is the main new contribution and can be regarded as an answer to R. Thom’s implicite conjecture: ”almost all” smooth mappings are triangulable [Enseign. Math., II. Sér. 8, 24-33 (1962; Zbl 0109.400)].

The first three chapters are devoted to a quick and rather complete introduction to stratified spaces (stratifications, controlled vector fields, Thom mappings). In the next two chapters the theory is slightly extended, allowing the strata to be manifolds with corners (called here manifolds with faces). The sixth chapter deals with the structure of Thom mappings, a \(C^{\infty}\)-analogue of the resolution of singularities. The final two chapters contain the proofs of the triangulability results mentioned above. Some facts from PL-topology are collected in an appendix. This very clearly written book can be of great help to a patient reader determined to tackle this highly technical and important subject.

Reviewer: A.Dimca

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

57N80 | Stratifications in topological manifolds |

58A35 | Stratified sets |

57R05 | Triangulating |

57Q99 | PL-topology |