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Nash manifolds. (English) Zbl 0629.58002
Lecture Notes in Mathematics, 1269. Berlin etc.: Springer-Verlag. VI, 223 p.; DM 35.00 (1987).
The book is devoted to the study of \(C^ r\) Nash maps and \(C^ r\) Nash manifolds, \(r=0,1,...,\omega\). Let U, V be open semialgebraic subsets of \({\mathbb{R}}^ n\) and \({\mathbb{R}}^ m\) respectively. A \(C^ r\) map \(U\to V\) is a \(C^ r\) Nash map if its graph is semialgebraic in \({\mathbb{R}}^ n\times {\mathbb{R}}^ m\). A \(C^ r\) Nash manifold is a \(C^ r\) manifold with a finite system of coordinate neighborhoods \(\{\psi_ i: U_ i\to {\mathbb{R}}^ m\}\) such that the corresponding transition maps \(\psi_ j\circ \psi_ i^{-1}: \psi_ i(U_ i\cap U_ j)\to \psi_ j(U_ i\cap U_ j)\) are \(C^ r\) Nash diffeomorphisms. Starting from these definitions, one can develop a natural analogue of many basic facts of differential manifold theory. An important new concept is that of affine manifold. A \(C^ r\) Nash manifold is affine if there exists a \(C^ r\) Nash imbedding of it into some \({\mathbb{R}}^ n\). The main results of the book are grouped around this notion and the regularity properties of noncompact Nash manifolds. They are the following.
(1) Let \(0<r<\infty\). Then a \(C^ r\) Nash manifold is affine and admits a unique affine \(C^{\omega}\) Nash manifold structure (Chapter III).
(2) Every compact \(C^{\omega}\) differential manifold admits many pairwise nonisomorphic non-affine \(C^{\omega}\) Nash manifold structures. Moreover there exists a continuum number of such structures. Similar results hold for interiors of compact \(C^{\omega}\) manifolds with boundary (Chapter IV).
(3) Every \(C^ 0\) Nash manifold has a natural compactification to a \(C^ 0\) Nash manifold with boundary. A compact \(C^ 0\) Nash manifold possibly with boundary admits a unique PL manifold structure. Moreover, there is a 1-1 correspondence between isomorphism classes of compact PL manifolds possibly with boundary and isomorphism classes of \(C^ 0\) Nash manifolds without boundary (given by \(M\mapsto int M)\) (Chapter V).
(4) Similarly, one can uniquely compactify a noncompact affine \(C^{\omega}\) Nash manifold by attaching boundary. This \(C^{\omega}\) Nash compactification can be realized by nonsingular algebraic varieties in a natural way (Chapter VI).
There are similar results on \(C^ r\) Nash vector and fibre bundles. The results on affine Nash manifolds are based on an approximation theorem of \(C^ r\) Nash maps by \(C^{\omega}\) Nash maps in an appropriate \(C^ r\) topology. Such a theorem is the main result of Chapter II. Chapter I is devoted to some preliminaries.
Reviewer: N.Ivanov

58A07 Real-analytic and Nash manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
14Pxx Real algebraic and real-analytic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry