by Andrei Verona

Abstract

For several reasons, most of them stemming from algebraic topology, it is important to know whether a topological space, or more generally a continuous map, is triangulable or not. Cairns [1] proved the triangulability of smooth manifolds; another proof, also providing a uniqueness result, is due to J.H.C. Whitehead [18]. First attempts to prove the triangulability of algebraic sets are due van der Waerden [16], Lefschetz [9], Koopman and Brown [8], and Lefschetz and Whitehead [10]. Lojasiewicz [10] and Giesecke [2] gave rigorous proofs in the more general case of semianalytic sets. Later, Hironaka [5] and Hardt [4] proved the triangulability of subanalytic sets. The most general spaces known to be triangulable are the stratified sets introduced by Thom [15] and the abstract stratifications introduced by Mather [12]. Mather's notion is slightly different from Thom's one, but it is more or less clear that the two classes of spaces coincide, at least in the compact case. They include all the spaces mentioned above and also the orbit spaces of smooth actions of compact Lie groups. Their triangulability was proved by several authors: Goresky [3], Johnson [6], Kato [7] and Verona [17], to mention only the published proofs. A more detailed discussion of these proofs and of others can be found in the introduction of Johnson's paper or at the end of Section 7 of the present work. Much fewer authors considered the more difficult problem of the triangulability of mappings: Putz [13] proved the triangulability of smooth submersions, Hardt [4] proved the triangulability of some, very special, subanalytic maps and we proved in [17] the triangulability of certain stratified maps. In [14] Thom considered the problem of the triangulability of smooth maps and (implicitly) conjectured that "almost all" smooth mappings are triangulable. It is the aim of this paper to prove this conjecture. To be precise, we shall prove:

Theorem. Let M and N be smooth manifolds without boundary. Then any proper, topologically stable smooth map from M to N is triangulable.

Since the set of proper and topologically stable smooth maps from M to N is dense in the set of all proper smooth maps from (Thom-Mather theorem) we obtain a positive answer to the above-mentioned conjecture. As a matter of fact, we prove a more general result concerning the triangulability of certain stratified mappings (Theorem 8.9). It implies the theorem stated above and also the following result (first proved by Hardt [4]): any proper light subanalytic map is triangulable.Our Theorem 8.9 is not as general as one would expect it. It applies only to proper and nice abstract Thom mappings (nice means that the mapping is finite to one when restricted to a certain subspace). It is natural to conjecture that the theorem is true for any proper abstract Thom mapping. A positive answer to this would solve another conjecture of Thom [14]: any proper Thom mapping (in Thom's terminology "application sans eclatement") is triangulable. The main difficulty in proving this more general version of Theorem 8.9 is explained in Section 8.15.3.

Since we are dealing with stratified spaces as introduced by Mather in [l2] and since these lecture notes have never been published, we thought that it would be useful to collect in a first part of the present work (Chapters 1, 2, and 3) the main results of the theory. Some of the proofs presented here are new and simpler than the original ones. For technical reasons, we are forced to work with certain manifolds with corners, called here manifolds with faces. The necessary facts concerning them are presented in Chapter 4. In Chapter 5, we extend the theory of abstract stratifications and abstract Thom mappings to the case when the strata are allowed to be manifolds with faces; most of the proofs are copies of the proofs presented in the first three chapters and so they are omitted. In Chapter 6, we prove some theorems concerning the structure of abstract stratifications and of abstract Thom mappings. In some sense, they can be viewed as a kind of "resolution of the singularities" in the differentiable case. For example, Theorem 6.5 can be interpreted as saying that any abstract stratification of finite depth can be obtained from a manifold with faces by making certain identifications on the faces. Chapter 8 contains the main results of the paper (they were mentioned above). In an appendix, we have collected some facts from PL-topology, which are needed in Chapters 7 and 8.

Acknowledgement. For their support and/or hospitality during the preparation of this work, I wish to express my gratitude to the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, the Alexander von Humboldt Foundation, the Sonderforschungsbereich Theoretische Mathematik, of the University of Bonn, the National Science Foundation (Grant HCS-8L08814 A01), the Institute for Advanced Study in Princeton, and, last but not least, the Max-Planck-Institut fur Mathematik in Bonn. Especially, I wish to thank Professor P. Hirzebruch, whose confidence and understanding during the preparation of this paper were of great help.

References:
 

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