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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

##### MSC:
 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