Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid developยญ ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of -or the connected components of-a constructible set (Le. , a locally finite union of differยญ ences of real analytic sets) need not be constructible (e. g. , R - {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 -=I- O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of anaยญ lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic
CONTENT
I. Preliminaries -- ยง1.1. Whitney stratifications -- ยง1.2. Subanalytic sets and semialgebraic sets -- ยง1.3. PL topology and C? triangulations -- II. X-Sets -- ยง11.1. X-sets -- ยง11.2. Triangulations of X-sets -- ยง11.3. Triangulations of X functions -- ยง11.4. Triangulations of semialgebraic and X0 sets and functions -- ยง11.5. Cr X-manifolds -- ยง11.6. X-triviality of X-maps -- ยง11.7. X-singularity theory -- III. Hauptvermutung For Polyhedra -- ยงIII.1. Certain conditions for two polyhedra to be PL homeomorphic -- ยงIII.2. Proofs of Theorems III.1.1 and III.1.2 -- IV. Triangulations of X-Maps -- ยงIV.l. Conditions for X-maps to be triangulable -- ยงIV.2. Proofs of Theorems IV.1.1, IV.1.2, IV.1.2? and IV.1.2? -- ยงIV.3. Local and global X-triangulations and uniqueness -- ยงIV.4. Proofs of Theorems IV.1.10, IV.1.13 and IV.1.13? -- V. D-Sets -- ยงV.1. Case where any D-set is locally semilinear -- ยงV.2. Case where there exists a D-set which is not locally semilinear -- List of Notation
